# Operational Quantum Reference Frame Transformations

Titouan Carette  caretteatlixdotpolytechniquedotfr Basic Research Community
for Physics, Leipzig, Germany  LIX, CNRS, École polytechnique, Institut
Polytechnique de Paris, 91120 Palaiseau, France  Jan Głowacki
<EMAIL_ADDRESS>Basic Research Community for Physics, Leipzig,
Germany  Center for Theoretical Physics, Polish Academy of Sciences, Al.
Lotników 32/46, 02-668 Warsaw, Poland  International Center for Theory of
Quantum Technologies, University of Gdańsk, Jana Bażyńskiego 1A, 80-309
Gdańsk, Poland  Leon Loveridge<EMAIL_ADDRESS>Basic Research
Community for Physics, Leipzig, Germany  Department of Science and Industry
Systems, University of South-Eastern Norway, 3616 Kongsberg, Norway  Okinawa
Institute of Science and Technology Graduate University, 1919-1 Tancha, Onna-
son, Okinawa 904-0495, Japan

###### Abstract

Quantum reference frames are needed in quantum theory for much the same
reasons as reference frames are in classical relativity theories: to manifest
invariance in line with fundamental relativity principles. Though around since
the 1960s, and used in a wide range of applications, only recently has the
means for transforming descriptions between different frames been tackled in
detail. Such transformations are needed for an internally consistent theory of
quantum reference frames. In this work, we provide a general, operationally
motivated framework for quantum reference frames and their transformations,
holding for locally compact groups. The work is built around the notion of
operational equivalence, in which theoretical objects that cannot be
physically distinguished are identified. For example, we describe the
collection of observables relative to a given frame as a subspace of the
algebra of invariants on the composite of system and frame, and from here the
set of relative states can be constructed as a convex subset of the predual.
Besides being invariant, the relative observables are also framed, meaning
that they can be realized with the chosen frame observable. The frame
transformations are then maps between equivalence classes of relative states
that can be distinguished by both initial and final frames. We give an
explicit realisation in the setting that the initial frame admits a highly
localized state with respect to the frame observable. The transformations are
invertible exactly when the final frame also has such a localizability
property. The procedure we present is in operational agreement with other
recent inequivalent constructions on the domain of common applicability, but
extends them in a number of ways which we describe.

##  1 Introduction

A means for describing one quantum system relative to another is needed to
accommodate the imposition of a symmetry principle. For example, in the
absence of absolute space, and in order to respect Galilean or special
relativistic invariance, the position of a quantum system has meaning only in
relation to other quantum systems, which then function as _quantum reference
frames_. A full account of such relative descriptions, at the level of states,
observables and the concomitant probability measures, and how these
descriptions transform when different frames are chosen, is the subject of
this paper. Our contribution is inspired by other recent efforts with the same
aim (e.g. [1, 2, 3]), with the further ambitions of (i) providing a
mathematically precise framework for locally compact groups, and (ii) making
the framework as far as possible _operational_ [4], in the sense of being
grounded ultimately in Born rule probability measures that could in principle
be observed. In so doing, we provide a foundation for the study of quantum
reference frames rooted in the full probabilistic structure of quantum theory
(observables as positive operator valued measures, states as positive trace
class operators with unit trace) with a symmetry principle explicitly
incorporated from the outset. Since there are a number of differing
contemporary frameworks and perspectives on the founding assumptions and
mathematical implementations of quantum reference frames and their
transformations (e.g. [5, 6, 7, 8], with differing takes on fundamental
questions such as whether frame changes are even unitary [9, 10], the need for
a rigorous formulation is clear.

The need to understand quantum reference frames and their transformations
arises in diverse areas, including quantum gravity, in which diffeomorphism
invariance suggests working with relational observables, quantum information
theory (e.g. [5, 6]) where communication tasks require transmission of frame-
invariant information, and in the foundations of quantum mechanics where an
understanding of the basic meaning of the formalism is still being
sought.111In this vein, one can include _relational quantum mechanics_
originally due to Rovelli [11, 12], and the perspectival approach of Bene and
Dieks [13]. These are not based on quantum reference frames explicitly, but
the motivation is largely aligned. This assortment of applications explains
the variety of distinct formulations—perhaps four strands are now visible,
each described by one of [5, 1, 8, 7]—with differing founding assumptions. In
quantum information theory the focus is often on e.g. the possibility of
transmission of quantum information between agents who do not share a frame
[14], enforcing an average (of the states) over all (classical) frame
orientations. By contrast, the perspective neutral approach [8, 3], inspired
by Dirac quantization of constrained systems, limits the (pure) states to the
unit vectors in the given space which are invariant under the action of the
group generated by the quantized constraints, which yields different physics
than that given in the quantum information approach. Another major recent
development is given in [1], in which ‘states relative to a frame’ are taken
as primitive, and explores the ensuing change of state description under frame
changes, resulting in e.g. the generation of entanglement. This is extended in
[2], in which the group theoretic aspects are emphasized. In each approach,
akin to special relativity and gauge theories, there is a group which dictates
both what is observable/physical in the theory and also mediates
transformations between frames, and the distinction between the approaches can
in part be understood through the role played by the symmetry group. In this
paper, we take the perspective that the physical _observables_ are invariant
and explore the consequences, culminating in a frame change procedure based on
ideas arising in e.g. [15, 16, 7]. We comment as we go on relative merits,
drawbacks and distinctions between the various approaches as we go.

###  1.1 Survey

The paper is organised as follows. Sec. 2 begins with the provision of some
general background and nomenclature assumed later, focussing on states,
observables, channels and probabilities, and providing some general
topological prerequisites. The state spaces are more general than those
usually encountered in quantum mechanics, and are understood as convex subsets
of real vector spaces. Next, we formulate the notion of _operational
equivalence_ , based on identifying (typically) states which cannot be
probabilistically distinguished, either for principal or practical reasons.
Concretely, the identification (on the trace class) is given as
$T\sim_{\mathcal{O}}T^{\prime}$ if $\tr[TA]=\tr[T^{\prime}A]$ for all
$A\in\mathcal{O}\subset B(\mathcal{H})$; the ensuing quotient on density
matrices is then the generic model for the state spaces we will encounter
throughout. Some general theorems about such spaces are given, most notably we
show that the standard statistical duality between the states and effects can
be given in general operational terms under the above quotient.

After these generalities, we consider the example of operational equivalence
that arises under a symmetry described by a unitary representation of a
locally compact group $G$, and the set $\mathcal{O}$ above is the fixed point
algebra $B(\mathcal{H})^{G}$ (of invariant bounded operators). The state space
comprises the equivalence classes of states that cannot be distinguished by
invariants. This setting is discussed in some detail, since it serves as an
illustration of the conceptual and formal differences between the approach to
implementing invariance employed in this work and in other approaches to
quantum reference frames. We choose the invariant operators rather than
invariant Hilbert space vectors, as in the perspective-neutral approach [3],
on the grounds that the latter set is typically empty in the Hilbert space
setting, and requires distributional techniques which are not known to be
rigorously possible in general. Further operational equivalences are
considered later; the newly introduced notions of relative states and
observables, needed for the frame transformations, are based on this concept.

Next comes the topic of the _localizability_ (norm-$1$) property of positive
operator valued measures (POVMs), which assures the existence of probability
measures arising via the Born rule being concentrated around an arbitrary
point to arbitrary precision. These POVMs, in conjunction with a _covariance_
property described here, play an important role throughout, as discussed in
the next section. Various examples are provided, including those defined by
systems of coherent states, familiar in the perspective-neutral approach (e.g.
[8, 3]).

Sec. 3 provides a full definition of a quantum reference frame $\mathcal{R}$
as a _system of covariance_
$\mathcal{R}=(U_{\mathcal{R}},\mathsf{E}_{\mathcal{R}},\mathcal{H}_{\mathcal{R}})$
based on $\Sigma_{\mathcal{R}}$, i.e, a unitary representation of $G$ in
$\mathcal{H}_{\mathcal{R}}$ and a covariant POVM $\mathsf{E}_{\mathcal{R}}$ on
$\Sigma_{\mathcal{R}}$ with values in $B(\mathcal{H}_{\mathcal{R}})$. Here,
the compound object of system $\mathcal{S}$ and frame $\mathcal{R}$ is
considered for the first time. Frames are understood as (representing)
physical systems, and fixing a frame is viewed as setting a concrete
experimental arrangement, akin to fixing a measuring apparatus and choosing a
particular pointer observable in the quantum theory of measurement (e.g.,
[17]). Thus, the choice of frame reflects _which_ system will execute that
function, and _how_ it will do it. From here we introduce _framing_ , which
yields an operational equivalence relation reflecting the fixing of the frame
and therefore respecting the given operational/experimental scenario under
consideration.

In this work, we are primarily concerned with the setting of _principal_
frames, in which the value space $\Sigma_{\mathcal{R}}$ is a principle
homogeneous $G$-space. In this context, the frame observables are understood
as providing a means of describing, probabilistically, the _orientation_ of
the quantum system, contingent on the state preparation of the frame; using
localizing sequences of states of localizable principal frames we make
rigorous—in terms of a limiting procedure at the level of
probabilities/expectation values—objects such as $\ket{g}$ ($g\in G$) which
are used freely in physics but require explanation mathematically. We note
that the orientation is not per se observable, since we stipulate that truly
observable quantities are invariant, and we define relative orientation
observables which do respect the invariance requirement.

The main reason for which we restrict our attention to principal frames is
that the _relativization maps_ $\yen^{\mathcal{R}}$ introduced in e.g. [18,
7], which we recall next and which are at the heart of the operational
approach to quantum reference frames, are defined in that setting. Moreover,
this already covers a large range of physically relevant examples. The setting
of general homogeneous spaces is considered for finite groups/spaces in [19]
and the locally compact case is work in progress. The $\yen^{\mathcal{R}}$ map
is a quantum channel defined upon fixing the frame observable, and takes
$B(\mathcal{H}_{\mathcal{S}})$ to invariants on the composite system, i.e.,
$\yen^{\mathcal{R}}:B(\mathcal{H}_{\mathcal{S}})\to
B(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})^{G}$. The
(ultraweak closures of the) images of the relativization maps provide what we
call spaces of _relative observables_ , written
$B(\mathcal{H}_{\mathcal{S}})^{\mathcal{R}}:=\yen^{\mathcal{R}}(\mathcal{H}_{\mathcal{S}})^{\rm
cl}$. Such operators are not only framed but also invariant. They are
understood as the ones that can actually be measured, given the imposition of
symmetry (invariance) and the choice of frame.

Following the definition of relative observables, the _relative states_
$\mathcal{S}(\mathcal{H}_{\mathcal{S}})_{\mathcal{R}}$ (and relative trace
class operators $\mathcal{T}(\mathcal{H}_{\mathcal{S}})_{\mathcal{R}}$) are
defined as operational equivalence classes of states (or trace class
operators) on the composite systems that cannot be distinguished by relative
observables. The space of relative observables is in duality with the space of
relative trace class operators with the relative states embedded as
$\mathcal{S}(\mathcal{H}_{\mathcal{S}})_{\mathcal{R}}\subseteq\mathcal{T}(\mathcal{H}_{\mathcal{S}})_{\mathcal{R}}^{\rm
sa}$; this extends the usual duality between the trace class and bounded
operators in Hilbert space to the relational realm considered in this work.
For the comparison with other attempts to formalise the notion of relative
state, we note that our relative state spaces are isomorphic to convex subsets
of the state space of the system, namely to the image of the predual map
$\yen^{\mathcal{R}}_{*}$. This allows for the representation of the relative
states as states of the system alone, which is in line with what is done in
other works (e.g. [1, 2]), except more general, operationally motivated and
mathematically precise.

We then consider descriptions relative to a chosen frame conditioned upon the
state of the frame, which effectively externalises the frame, since the
resulting description refers to the system alone. The main tool here is the
assignment of the restriction maps
$\Gamma_{\omega}:B(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})\to
B(\mathcal{H}_{\mathcal{S}})$ [18, 7] to the states
$\omega\in\mathcal{S}(\mathcal{H}_{\mathcal{R}})$, which are known to be
completely positive normal conditional expectations. This naturally gives rise
to the dual notions of the _conditioned relative observables_ and the _product
relative states_. After spelling out the relevant definitions we analyze the
properties of this particularly tractable class of relative states. Given the
means for conditioning upon a state of the reference, we address the question
of how the conditioned relative descriptions behave upon localizing the
reference. Generalizing previous results (of e.g. [15, 7]), we find that in
the case of a localizable frame any observable of the system can be
represented as a limit of conditioned relative observables, the relativization
map $\yen^{\mathcal{R}}$ is injective, and the set of relative states is dense
in the state space of the system. Thus the ordinary, non-relational framework
of quantum theory is recovered as a limiting case of the frame-relative
description.

Much of the work thus far described, whilst we believe to be of interest in
its own right, is in service of the novel operational take on frame changes,
to which the paper finally turns in Sec. 4. The transformation rule we give is
inspired by [1, 2] but mathematically and physically distinct from it, and all
other quantum frame changes appearing to date. The map we provide is defined
on the relevant convex operational relative state space. Here, three systems
are considered - a system and two frames, and the aim is to transform the
(state) description relative to one frame to that relative to the other. For
example, the state of system and frame two is initially given relative to
frame one, and we seek a ‘corresponding’ description of system and frame one,
relative to frame two. Before giving the frame change, we introduce a final
ingredient called the _lifting maps_ , which serve to ‘attach’ a chosen state
of the second reference to a state relative to the first one, giving rise to a
state on the total system consisting of both frames and the original system.
With all the tools in hand, we may then define the localized frame
transformations as a limiting case of the following procedure. First, given a
state relative to the first frame, use the lifting map to attach a state of
the first frame that is well-localized at the identity of $G$ in analogy to
[2]. This is a way to pass by the ‘global’ description comprising system and
all frames, whilst respecting the symmetry, similar in spirit to what is done
in [3]. Then use the predual of the second relativization map, yielding a
state relative to the second frame. The limit is taken with respect to an
arbitrary localizing sequence centered around the identity of $G$. The
procedure is well-defined and invertible when the second frame is also
localizable, and both framing equivalence relations are taken into account. We
also show that in the setup of three frames, the localized frame
transformations compose in the relevant sense.

We conclude by showing that our map agrees in spirit with [1, 2] in the
following sense. In the setting that the frame changes in the above works are
rigorously defined, those frame changes yield elements of the operational
equivalence classes appearing in our work. The upshot is therefore that if one
accepts the premise of the operational equivalence relations defined here,
there is no measurement that can distinguish our frame change from theirs.
Given that the operational equivalence classes contain both entangled states
and separable states, in our prescription their is no fact of the matter about
whether the post-frame change states are entangled. We also provide a brief
comparison with the perspective-neutral approach, though a full account of it
is a large project due to the difficulties of making the perspective-neutral
framework fully rigorous. However, we again observe no operational
disagreement between our maps and those of [3] in the setting that the initial
frame is ideal. After a brief conclusion, in Appendix A we provide a glossary
of the various spaces we consider for reference. Appendix B contains all the
diagrams in one place for ease of use, and finally Appendix C provides proofs
of some theorems omitted from the main text.

##  2 Preliminaries

We briefly recall in this section the essentials of the Hilbert space
framework of quantum mechanics pertinent to the subject matter which follows.
We then introduce the notion of operational equivalence, which is used to
identify mathematically distinct entities (most commonly for us these are
states) that cannot be distinguished in a given operational scenario. This
notion recurs throughout the paper; the most important use is to define the
relative states in an operational way, given as classes of density matrices
which cannot be probabilistically told apart by relative observables. We close
by recalling the definitions of covariance and localizability for POVMs,
accompanied by a list of examples.

###  2.1 Basics

In the Hilbert space framework of quantum theory, a quantum system has
associated with it a (complex, separable) Hilbert space $\mathcal{H}$, and
(normal) states are identified with positive trace class operators on
$\mathcal{H}$ of trace one; we write $\mathcal{T}(\mathcal{H})$ for the trace
class (of $\mathcal{H}$), which is a Banach space under the trace norm
$||\cdot||_{1}$, $\mathcal{T}(\mathcal{H}_{\mathcal{S}})^{\rm sa}$ for the
self-adjoint part and $\mathcal{S}(\mathcal{H})$ for the convex subset of
states. The pure states are the extremal elements of
$\mathcal{S}(\mathcal{H})$, characterised as the rank-1 projections and
written $\outerproduct{\varphi}{\varphi}$ for some unit vector
$\varphi\in\mathcal{H}$; the collection of pure states is written
$\mathcal{P}(\mathcal{H})$. The space (which is also a von Neumann algebra) of
bounded operators in $\mathcal{H}$ is denoted $B(\mathcal{H})$, which is the
Banach dual of $\mathcal{T}(\mathcal{H})$. A channel
$\Lambda:B(\mathcal{H})\to B(\mathcal{K})$ is a normal (i.e., continuous with
respect to the ultraweak topologies on $B(\mathcal{H})$ and $B(\mathcal{K})$)
completely positive (CP) map which preserves the unit; the normality implies
there is a unique CP trace-preserving _predual_ map
$\Lambda_{*}:\mathcal{T}(\mathcal{H})\to\mathcal{T}(\mathcal{K})$ defined
through $\tr[T\Lambda(A)]=\tr[\Lambda_{*}(T)A]$ for all
$T\in\mathcal{T}(\mathcal{H})$ and $A\in B(\mathcal{H})$. These maps will also
be called channels. More generally we consider abstract _state spaces_ , which
are convex subsets of real vector spaces, and affine maps between them.
Usually, they will be total convex subsets of real Banach spaces, where total
convexity means that the points are separated by the bounded affine
functionals, which is the case for
$\mathcal{S}(\mathcal{H})\subset\mathcal{T}(\mathcal{H})^{sa}$.

Observables are (identified with) positive operator valued measures (POVMs)
$\mathsf{E}:\mathcal{F}\to B(\mathcal{H})$, where $\mathcal{F}$ is a
$\sigma$-algebra of subsets of some set (or _value space_) $\Sigma$, which
represents outcomes that may be obtained in a measurement of the given
observable. In this paper, $\Sigma$ is a topological space and $\mathcal{F}$
will be the Borel $\sigma$-algebra, written $\mathcal{B}(\Sigma)$. If the
system is prepared in a state $\omega\in\mathcal{S}(\mathcal{H})$, upon
measurement of $\mathsf{E}$ the probability that an outcome in a set
$X\in\mathcal{B}(\Sigma)$ is obtained is given through the Born formula:

$p_{\omega}^{\mathsf{E}}(X)=\tr[\omega\mathsf{E}(X)].$ (1)

Operators in the range of POVMs, that is, positive operators in the unit
operator interval, are called _effects_. Thanks to the duality between the
bounded and trace class operators, the effects can be equivalently
characterized as positive continuous affine functionals on
$\mathcal{S}(\mathcal{H})$ bounded by one. The space of effects on
$\mathcal{H}$ will be denoted by $\mathcal{E}(\mathcal{H})$. If each operator
in the range of a POVM $\mathsf{E}$ is a projection, $\mathsf{E}$ is a
projection-valued measure (PVM) and if defined on (Borel subsets of)
$\mathbb{R}$ the standard description of observables as self-adjoint operators
in $\mathcal{H}$ is recovered through the spectral theorem; we occasionally
write $\mathsf{P}^{A}$ for the spectral measure of $A\in B(\mathcal{H})$, and
write $A=\int xd\mathsf{P}^{A}(x)$ (interpreted weakly). POVMs which are PVMs
will be called _sharp_ (observables), all others _unsharp_ (observables). As
has become standard, in the case that an observable is sharp, we will also
refer to the corresponding self-adjoint operator as an observable (though we
keep this usage to a minimum, since it can occasionally lead to confusion).
Note also that ‘observable algebras’ typically have non-self-adjoint elements,
as one finds in the algebraic QM/QFT literature.

We have already mentioned various topological notions; it is worth briefly
formalizing these since they appear repeatedly. We are motivated by
operational ideas, and therefore the preferred topology on $B(\mathcal{H})$ is
the _topology of pointwise convergence of expectation values_ , i.e.,
$A_{n}\to A\in B(\mathcal{H})$ exactly when $\tr[TA_{n}]\to\tr[TA]$ for all
$T\in\mathcal{T}(\mathcal{H})$ (there is no loss of generality to restrict
further from $\mathcal{T}(\mathcal{H})$ to $\mathcal{S}(\mathcal{H})$). This
is the ultraweak (also called $\sigma$-weak, or weak-$*$ as the dual of
$\mathcal{T}(\mathcal{H})$) topology on $B(\mathcal{H})$ (on norm bounded sets
convergence here agrees with convergence in the weak operator topology arising
from the family of seminorms $A\mapsto|\innerproduct{\varphi}{A\phi}|$, which
can be reconstructed from the pure state expectation values by polarization).
Notice that all these topologies are Hausdorff [20]. On the predual
$\mathcal{T}(\mathcal{H})$ (and by restriction to the state space), we again
use the topology of point-wise convergence of the expectation values, thus
$T_{n}\to T$ exactly when $\tr[T_{n}A]\to\tr[TA]$ for all $A\in
B(\mathcal{H})$. Since the set of effects spans $B(\mathcal{H})$, this can be
equivalently given in $\mathcal{E}(\mathcal{H})$. We will refer to this
topology as the _operational topology_ on the set of states, i.e., it is the
weakest topology that makes all the $T\mapsto\tr[TA]$ continuous. The
superscript "cl" will always refer to ultraweak closure of the subsets in
operator algebras, and operational closure of subsets of trace class
operators.

###  2.2 Operational Equivalence

The notion of operational equivalence allows for the identification of
distinct ‘entities’ that cannot be distinguished physically/operationally. For
instance, in the operational approach to quantum theory (e.g. [4]), states are
defined as equivalence classes of preparation procedures that cannot be
distinguished in any measurement. In gauge theories, fields related by a gauge
transformation may be regarded as identical. In this work we specify a set of
observables, and quotient by an operational equivalence relation defined by
equality of expectation values, i.e., we identify states that cannot be
probabilistically distinguished on the given set of observables. Therefore,
the new view of states is as points which are equivalence classes of the old
states. The prototypical example comes from identifying any states which give
the same statistics on all (gauge-)invariant observables. The state spaces
which arise in this way are naturally identified with the predual of the von
Neumann algebra $B(\mathcal{H}_{\mathcal{S}})^{G}$. The operational state
space is a central object in this work, and in some sense plays the role of
the physical Hilbert space in the perspective-neutral approach [8, 3]. We also
consider an important case of (operationally) equivalent collections of
density matrices that cannot be distinguished by operators in the image of a
given channel.

####  2.2.1 Generalities

######  Definition 2.1.

Let $\mathcal{O}$ be a collection of effects and $\mathcal{S}$ a collection of
states. Then:

  * •

Two states $\rho,\rho^{\prime}\in\mathcal{S}$ are called _operationally
equivalent_ with respect to $\mathcal{O}$ if
$\tr[\rho\mathsf{F}]=\tr[\rho^{\prime}\mathsf{F}]$ for all
$\mathsf{F}\in\mathcal{O}$.

  * •

Two effects $\mathsf{F},\mathsf{F}^{\prime}\in\mathcal{O}$ are called
operationally equivalent with respect to $\mathcal{S}$ if
$\tr[\rho\mathsf{F}]=\tr[\rho\mathsf{F}^{\prime}]$ for all
$\rho\in\mathcal{S}$.

This definition may be adapted to POVMs, self-adjoint and trace class
operators as needed. Since the following will be used throughout, we state is
separately:

######  Definition 2.2.

Given a subset $\mathcal{O}\subseteq B(\mathcal{H})$, the
$\mathcal{O}$-_operational equivalence relation_ on $\mathcal{T}(\mathcal{H})$
is defined as

$T\sim_{\mathcal{O}}T^{\prime}\Leftrightarrow\tr[TA]=\tr[T^{\prime}A]\hskip
3.0pt\forall A\in\mathcal{O}.$

The identification of $\mathcal{O}$-equivalent trace class operators amounts
to taking the quotient $\mathcal{T}(\mathcal{H})/\sim_{\mathcal{O}}$. The
continuity of the map $A\mapsto\tr[TA]$ on $B(\mathcal{H})$ for any
$T\in\mathcal{T}(\mathcal{H})$ means that the set of $\mathcal{O}$-equivalent
trace class operators is equal to the set of
$\mathcal{O}^{\rm{cl}}$-equivalent trace class operators (e.g. [21]). Notice
also that the same equivalence is realized with respect to $\mathcal{O}$,
${\rm span}\\{\mathcal{O}\\}$ and ${\rm conv}\\{\mathcal{O}\\}$ (denoting the
convex hull); these facts will be used throughout.

######  Proposition 2.3.

The space $\mathcal{T}(\mathcal{H})/\sim_{\mathcal{O}}$ equipped with the
quotient norm is the unique Banach predual of the ultraweak closure of the
span of $\mathcal{O}$, so that

$\big{[}{\rm span}\\{\mathcal{O}\\}^{\rm
cl}\big{]}_{*}\cong\mathcal{T}(\mathcal{H})/\sim_{\mathcal{O}}.$

###### Proof.

For $A\in\mathcal{O}$, we write $\phi_{A}$ for the (trace norm) continuous
linear functional $\rho\mapsto\tr[A\rho]$ and identify $A$ with $\phi_{A}$
through the isomorphism $\mathcal{T}(\mathcal{H})^{*}\cong B(\mathcal{H})$.
Then

$\rho\sim_{\mathcal{O}}\rho^{\prime}\Leftrightarrow\forall
A\in\mathcal{O}:\hskip 3.0pt\rho-\rho^{\prime}\in\ker(\phi_{A}).$

The joint kernel $\bigcap_{\phi_{A}\in\hat{\mathcal{O}}}\ker(\phi_{A})$ is
sometimes called the pre-annihilator of $\mathcal{O}$, written
${}^{\perp}\mathcal{O}$, and is always closed in $\mathcal{T}(\mathcal{H})$ as
an intersection of closed sets. Furthermore, it is always a subspace and thus
${}^{\perp}\mathcal{O}={}^{\perp}{\rm span}\\{\mathcal{O}\\}$. Thus
$\mathcal{T}(\mathcal{H})/\sim_{\mathcal{O}}=\mathcal{T}(\mathcal{H})/{}^{\perp}\mathcal{O}$
is a Banach space with the quotient norm:

$||\rho+{}^{\perp}\mathcal{O}||:=\inf_{\mu\in{}^{\perp}\mathcal{O}}||\rho+\mu||_{1}.$

We then have
$\left(\mathcal{T}(\mathcal{H})/\sim_{\mathcal{O}}\right)^{*}=\left(\mathcal{T}(\mathcal{H})/{}^{\perp}\mathcal{O}\right)^{*}\simeq{}^{\perp}\mathcal{O}^{\perp}=\left({}^{\perp}{\rm
span}\\{\mathcal{O}\\}\right)^{\perp}={\rm span}\\{\mathcal{O}\\}^{\rm cl}$
(see Thm. 4.9 and 4.7 in [22]).

For the uniqueness, we note the following. Since the set of compact operators
$K(\mathcal{H})$ is convex, the ultraweak closure $K(\mathcal{H})^{\rm cl}$
equals the $\sigma$-strong∗ closure of $K(\mathcal{H})$ (Thm. 2.6 (iv) in
[23]). Due to the $\sigma$-strong∗ continuity of the ∗-operation,
$K(\mathcal{H})^{\rm cl}$ is a $\sigma$-strongly∗ closed $*$-subalgebra of
$B(\mathcal{H})$ and the bicommutant theorem (Thm. 3.9 in [23]) gives
$K(\mathcal{H})^{\rm cl}=K(\mathcal{H})^{\prime\prime}=B(\mathcal{H})$. Then
clearly $K(\mathcal{H})\cap{\rm span}\\{\mathcal{O}\\}^{\rm cl}$ is
ultraweakly dense in ${\rm span}\\{\mathcal{O}\\}^{\rm cl}$. Since we assume
separability of $\mathcal{H}$, the uniqueness then follows from Thm. 2.1 in
[24].

∎

######  Remark 2.4.

The uniqueness of preduals is known in the theory of von Neumann algebras
[25], extended here to more general operator spaces. Indeed, applying the
above reasoning and Thm. 2.1 in [24] gives uniqueness of Banach preduals for
arbitrary ultraweakly closed subspaces of $B(\mathcal{H})$ for separable
$\mathcal{H}$. These predual spaces are unique up to isometry and can be
equally well characterized as the spaces of normal functionals on such
subspaces (see 2.6 (iii) in [23] and [26]). The uniqueness of the predual
spaces allows for the unambiguous switching between dual descriptions (à la
Schrödinger/Heisenberg pictures), which we apply in various operational
settings later.

The set of classes of indistinguishable density operators can be understood as
a state space on its own right. Indeed, the quotient space
$\mathcal{S}(\mathcal{H})/\sim_{\mathcal{O}}$ is a total convex subset of the
real subspace of classes of self-adjoint trace class operators
$\mathcal{T}(\mathcal{H})^{\rm{sa}}/\sim_{\mathcal{O}}$, as is confirmed by
the following.

######  Proposition 2.5.

The set $\mathcal{S}(\mathcal{H})/\sim_{\mathcal{O}}$ is a total convex state
space as a subset of the real Banach space
$\mathcal{T}(\mathcal{H})^{\rm{sa}}/\sim_{\mathcal{O}}$. Moreover, it is
closed in the quotient operational topology.

###### Proof.

Since the real linear structure of
$\mathcal{T}(\mathcal{H})^{\rm{sa}}/\sim_{\mathcal{O}}$ comes from
$\mathcal{T}(\mathcal{H})^{\rm{sa}}$, convexity is preserved under the
quotient. In particular, writing $[\\_]_{\mathcal{O}}$ for the
$\mathcal{O}$-equivalence classes, for any
$\rho,\rho^{\prime}\in\mathcal{S}(\mathcal{H})$ and $0\leq\lambda\leq 1$ we
have

$\lambda[\rho]_{\mathcal{O}}+(1-\lambda)[\rho^{\prime}]_{\mathcal{O}}=[\lambda\rho+(1-\lambda)\rho^{\prime}]_{\mathcal{O}}\in\mathcal{S}(\mathcal{H})/\sim_{\mathcal{O}}.$

The bounded affine functionals $\mathcal{S}(\mathcal{H})\to[0,1]$ are given by
$\rho\mapsto\tr[\rho\mathsf{F}]$ for with
$\mathsf{F}\in\mathcal{E}(\mathcal{H})$. The effects on
$\mathcal{S}(\mathcal{H})/\sim_{\mathcal{O}}$, i.e, bounded affine functionals
$\mathcal{S}(\mathcal{H})/\sim_{\mathcal{O}}\to[0,1]$, are then given by
$\rho\mapsto\tr[\rho\mathsf{F}]$ with $\mathsf{F}\in\mathcal{E}(\mathcal{H})$
being well-defined on the $\mathcal{O}$-equivalence classes, i.e., such that
for all $\rho,\rho^{\prime}\in\mathcal{S}(\mathcal{H})$ we have

$\rho\sim_{\mathcal{O}}\rho^{\prime}\Rightarrow\tr[\rho\mathsf{F}]=\tr[\rho^{\prime}\mathsf{F}].$

Since the operational equivalence classes taken with respect to $\mathcal{O}$
and $\rm{span}\\{\mathcal{O}\\}^{\rm cl}$ are the same, and clearly the
effects outside of $\rm{span}\\{\mathcal{O}\\}^{\rm cl}$ will not be well-
defined on the $\mathcal{O}$-equivalence classes, we can conclude that
$\mathsf{F}\in\rm{span}\\{\mathcal{O}\\}^{\rm cl}$. The effects on
$\mathcal{S}(\mathcal{H})/\sim_{\mathcal{O}}$ are then given by the operators
in $\mathcal{E}(\mathcal{H})\cap\rm{span}\\{\mathcal{O}\\}^{\rm cl}$. Such
effects separate the elements of $\mathcal{S}(\mathcal{H})/\sim_{\mathcal{O}}$
by construction, providing total convexity.

The state space $\mathcal{S}(\mathcal{H})$ is operationally closed in
$\mathcal{T}(\mathcal{H})$, since for any sequence of states
$(\rho_{n})\subset\mathcal{S}(\mathcal{H})$ such that
$\lim_{n\to\infty}\tr[\rho_{n}A]=\tr[TA]$ for all $A\in B(\mathcal{H})$ and
some $T\in\mathcal{T}(\mathcal{H})$, we can conclude that
$T\in\mathcal{S}(\mathcal{H})$. Indeed, the continuity of the trace gives
positivity and normalization of $T$. The operational topology on
$\mathcal{S}(\mathcal{H})/\sim_{\mathcal{O}}$ is the quotient topology of the
one on $\mathcal{T}(\mathcal{H})$ so we have

$\lim_{n\to\infty}[\rho_{n}]_{\mathcal{O}}=[T]_{\mathcal{O}}\in\mathcal{S}(\mathcal{H})/\sim_{\mathcal{O}}.$

∎

A state space of the form above will be called an _operational state space_.
Total convex subsets of real Banach spaces can be embedded in the general
framework of base-norm spaces and the dual order unit spaces (e.g., [27]),
pointing to potential generalizations of the notions presented in this work.

Often in this paper, the set $\mathcal{O}$ is the image of a unital normal
positive map. In this case, the corresponding state space admits an equivalent
useful characterization.

######  Proposition 2.6.

Any normal, positive, unital map $\Lambda:B(\mathcal{K})\to B(\mathcal{H})$
provides a state space isomorphism

$\mathcal{S}(\mathcal{H})/\sim_{\imaginary\Lambda}\cong\Lambda_{*}(\mathcal{S}(\mathcal{H})),$

###### Proof.

Since $\Lambda$ is normal, we can write
${}^{\perp}\imaginary\Lambda=\ker\Lambda_{*}$, and thus
$\mathcal{T}(\mathcal{H})/\sim_{\imaginary\Lambda}=\mathcal{T}(\mathcal{H})/\ker\Lambda_{*}$.
Then $\Lambda_{*}$ restricts to an _invertible_ bounded linear map
$\mathcal{T}(\mathcal{H})/\sim_{\imaginary\Lambda}\to\imaginary\Lambda_{*}$.
Since $\Lambda$ is linear, unital and positive, $\Lambda_{*}$ restricts
further to an affine bijection
$\mathcal{S}(\mathcal{H})/\sim_{\imaginary\Lambda}\to\Lambda_{*}(\mathcal{S}(\mathcal{H}))$,
providing the state space isomorphism.222Note that in general, this
correspondence doesn’t hold at the level of the ambient Banach spaces since
$\imaginary\Lambda_{*}$ might not be closed. ∎

####  2.2.2 Operational equivalence from symmetry

An important example of operational equivalence arises in the presence of
symmetry. Consider a (strongly continuous) unitary representation $U:G\to
B(H)$ of a (locally compact) group $G$, writing $g.A$ to stand for
$U(g)AU(g)^{*}$ with $A\in B(\mathcal{H})$ and $g.T$ for $U(g)^{*}TU(g)$ with
$T\in\mathcal{T}(\mathcal{H})$. Two density operators are identified if they
cannot be distinguished by observables invariant under the representation, and
therefore _invariant algebra_ $B(\mathcal{H})^{G}:=\\{A\in
B(\mathcal{H})~{}|~{}g.A=A\\}$ (which is a von Neumann algebra as the
commutant of a unitary group) plays a key role. Moreover, as we shall see,
observables with effects in the invariant algebra do not depend on an external
frame.

######  Definition 2.7.

The operational equivalence relation on $\mathcal{T}(\mathcal{H})$ taken with
respect to the invariant algebra $B(\mathcal{H}_{\mathcal{S}})^{G}$ will be
denoted by $T\sim_{G}T^{\prime}$. The space of $G$-_equivalent trace class
operators_ is given by

$\mathcal{T}(\mathcal{H})_{G}:=\mathcal{T}(\mathcal{H})/\sim_{G},$

while the operational state space of $G$-_equivalent states_ is defined to
be333The $G$-equivalent states defined here are similar to the ”symmetry-
equivalent” states defined in [28] (Def.18) in the context of finite abelian
groups.

$\mathcal{S}(\mathcal{H})_{G}:=\mathcal{S}(\mathcal{H})/\sim_{G}.$

Prop. 2.5 and 2.3 then ensure the following:

######  Proposition 2.8.

We have the following Banach space isomorphism

$B(\mathcal{H}_{\mathcal{S}})^{G}_{*}\cong\mathcal{T}(\mathcal{H}_{\mathcal{S}})_{G}.$

Moreover, the subset
$\mathcal{S}(\mathcal{H}_{\mathcal{S}})_{G}\subset\mathcal{T}(\mathcal{H}_{\mathcal{S}})^{sa}_{G}$
is a total convex state space.

Note that in general $\mathcal{S}(\mathcal{H})_{G}$ cannot be identified with
the set $\mathcal{S}(\mathcal{H})^{G}$ of invariant states, i.e. those
$\rho\in\mathcal{S}(\mathcal{H})$ for which $g.\rho=\rho$, this last set being
generically empty if $G$ is not compact. By contrast, there is often an
abundance of invariant observables, which is most clearly seen in the setting
of a composite system. From here the state space
$\mathcal{S}(\mathcal{H})_{G}$ is generically non-trivial and justifies on
mathematical grounds that invariance should be stipulated on the observables
rather than states (see [7, 18, 16] on which the present approach is based).
We are also able to avoid the use of distributions/rigged Hilbert spaces which
are needed for constructing the physical Hilbert space in the perspective-
neutral approach [3], which is defined as the space of invariant
(‘kinematical’) Hilbert space vectors/distributions. We note immediately that
by setting $B(\mathcal{H})^{G}$ as the collection of operators on which the
equivalence of states is defined, any state on a $G$-orbit is operationally
indistinguishable/equivalent to any other. This is as one would expect of
gauge transformations: all states related by a gauge transformation are
physically equivalent.

If the group $G$ is _compact_ the $G$-_twirl_ (or _incoherent group average_)
$\mathcal{G}:B(\mathcal{H})\to B(\mathcal{H})$ is given by

$\mathcal{G}(A)=\int_{G}d\mu(g)U(g)AU(g)^{*},$ (2)

where $\mu$ is (normalised) Haar measure; the integral is understood in terms
of Banach-space valued functions $G\to B(\mathcal{H})$ in the sense of
Bochner. $\mathcal{G}$ is a unital normal map with pre-dual
$\mathcal{G}_{*}:\mathcal{T}(\mathcal{H})\to\mathcal{T}(\mathcal{H})$ taking
the form $\mathcal{G}_{*}(\rho)=\int_{G}d\mu(g)U(g)^{*}\rho U(g)$. Both
$\mathcal{G}$ and $\mathcal{G}_{*}$ are idempotent, respectively onto the sets
$B(\mathcal{H})^{G}$ and $\mathcal{T}(\mathcal{H})^{G}$ of invariant bounded,
respectively trace class, operators. The image under $\mathcal{G}_{*}$ of a
state is often interpreted (though not always with clear operational meaning)
as an invariant ‘version’ of the given state. If $G$ is not compact the
integral does not converge in general on states or operators, and we therefore
avoid it in this setting.

Interestingly, in the case of compact $G$, there is no operational difference
between stipulating the invariance requirement on observables or states (or
both):

######  Proposition 2.9.

$\tr[\mathcal{G}^{*}(A)\rho]=\tr[A\mathcal{G}(\rho)]=\tr[\mathcal{G}^{*}(A)\mathcal{G}(\rho)]$.

The proof uses the invariance of $\mu$ and is straightforward. Moreover, by
Prop. 2.6, for $G$ compact there is an isomorphism of state spaces
$S(\mathcal{\mathcal{H}})^{G}=\mathcal{G}_{*}(\mathcal{S}(\mathcal{H}))\cong\mathcal{S}(\mathcal{\mathcal{H}})_{G}$.
Furthermore, in this case, the correspondence lifts to the ambient Banach
spaces.

######  Proposition 2.10.

If $G$ is compact, then the Banach spaces $\mathcal{T}(\mathcal{H})^{G}$ and
$\mathcal{T}(\mathcal{H})_{G}$ are isometrically isomorphic.

###### Proof.

We have:
$\mathcal{T}(\mathcal{H})_{G}=\mathcal{T}(\mathcal{H})/\ker(\mathcal{G}_{*})$
and $\mathcal{T}(\mathcal{H})^{G}=\imaginary\mathcal{G}_{*}$.
$\mathcal{G}_{*}$ factorizes through a bijective map
$\tilde{\mathcal{G}_{*}}:\mathcal{T}(\mathcal{H})/\ker(\mathcal{G}_{*})\to\imaginary\mathcal{G}_{*}$.
We show that $\tilde{\mathcal{G}_{*}}$ is an isometry. First,
$\mathcal{G}_{*}$ is a contraction with respect to the trace norm:

$||\mathcal{G}_{*}(\rho)||_{1}=\left\|\int_{G}g\cdot\rho
d\mu(g)\right\|_{1}\leq\int_{G}||g\cdot\rho||_{1}d\mu(g)\leq\int_{G}||\rho||_{1}d\mu(g)=||\rho||_{1};$

it follows that for all $\sigma\in\ker(\mathcal{G}_{*})$

$||\tilde{\mathcal{G}_{*}}(\rho+\ker(\mathcal{G}_{*}))||_{1}=||\mathcal{G}_{*}(\rho)||_{1}=||\mathcal{G}_{*}(\rho+\sigma)||_{1}\leq||\rho+\sigma||_{1}.$

Hence
$||\tilde{\mathcal{G}_{*}}(\rho+\ker(\mathcal{G}_{*}))||_{1}\leq||\rho+\ker(\mathcal{G}_{*})||_{1}=\inf_{\mu\in\ker(\mathcal{G}_{*})}||\rho+\mu||_{1}$.
Then, since $\mathcal{G}_{*}$ is idempotent we also have

$||\rho+\ker(\mathcal{G}_{*})||_{1}=\inf_{\mu\in\ker(\mathcal{G}_{*})}||\rho+\mu||_{1}\leq||\rho+(\mathcal{G}_{*}(\rho)-\rho)||_{1}=||\mathcal{G}_{*}(\rho)||_{1}=||\tilde{\mathcal{G}_{*}}(\rho+\ker(\mathcal{G}_{*}))||_{1},$

which thus provides an isometry between
$\mathcal{T}(\mathcal{H}_{\mathcal{S}})_{G}$ and
$\mathcal{T}(\mathcal{H}_{\mathcal{S}})^{G}$. ∎

The setup of invariant observables and operational state spaces therefore
aligns with e.g. the quantum information approach to QRFs [5] in the case of
compact groups, but is better suited for generalization.

###  2.3 Localizability

The localizability property of a POVM described below is used to recover the
standard kinematics of quantum mechanics from the relational one presented in
the next section (see also [29, 18, 7]), and in defining frame transformations
later. Through the Born rule probabilities (eq. (1)), for a fixed observable
$\mathsf{E}$, a state gives rise to a probability measure on $\Sigma$, often
to be denoted by $\mu^{\mathsf{E}}_{\omega}$. We will refer to states as being
‘highly localized’ with respect to $\mathsf{E}$ in some (open) set $X$ or
around some point (i.e., in an open neighbourhood $X$) in $\Sigma$, meaning
that the given probability $\mu^{\mathsf{E}}_{\omega}(X)$ is close to unity on
that $X$. If $\mathsf{P}:\mathcal{B}(\Sigma)\to B(\mathcal{H})$ is a PVM, for
any $X\in\mathcal{B}(\Sigma)$ for which $\mathsf{P}(X)\neq 0$, there is a
state $\omega$ for which $\tr[\omega\mathsf{P}(X)]=1$ (set $\omega$ to be a
projection onto any unit vector in the range of $\mathsf{P}(X)$), and this
state, with respect to $\mathsf{P}$, is perfectly localized (with
probabilistic certainty) in $X$. For example, if $\mathsf{P}=\mathsf{P}^{A}$,
then any eigenvector of $A$ with eigenvalue in $X$ can be understood in this
way. However, since $A$ may have no eigenvalues (e.g. the position operator on
a dense subset of $L^{2}(\mathbb{R})$), the above characterisation is more
general. _A fortiori_ , for a POVM, the probability measures described by (1)
are typically not localized in any open set; there exist POVMs for which there
is no state satisfying $p_{\omega}^{\mathsf{E}}(X)=1$ for any $X\neq\Sigma$.
There are, however, POVMs which ‘almost’ have the localizability property
enjoyed by all PVMs:

######  Definition 2.11 (Norm-$1$ property).

A POVM $\mathsf{E}:\mathcal{B}(\Sigma)\to B(\mathcal{H})$ satisfies the _norm-
$1$ property_ (see e.g. [29]) if for all $X\in\mathcal{B}(\Sigma)$ for which
$\mathsf{E}(X)\neq 0$, it holds that $\left\|\mathsf{E}(X)\right\|=1$. Such
POVMs are called _localizable_.

######  Proposition 2.12.

The following are equivalent [29]:

  1. 1.

$\mathsf{E}$ is localizable (i.e.$\mathsf{E}$ satisfies the norm-$1$
property).

  2. 2.

For every $X$ for which $\mathsf{E}(X)\neq 0$, there is a sequence of unit
vectors $(\varphi_{n})\subset\mathcal{H}$ for which
$\lim_{n\to\infty}\innerproduct{\varphi_{n}}{\mathsf{E}(X)\varphi_{n}}=1$.

  3. 3.

For every $\mathsf{E}(X)\neq 0$ and for any $\epsilon>0$ there exists a unit
vector $\varphi_{\epsilon}\in\mathcal{S}(\mathcal{H})$ for which
$\innerproduct{\varphi_{\epsilon}}{\mathsf{E}(X)\varphi_{\epsilon}}>1-\epsilon$
(this is called the $\epsilon$-_decidability property_).

A useful consequence is the following:

######  Proposition 2.13.

If a POVM $\mathsf{E}:\mathcal{B}(\Sigma)\to B(\mathcal{H})$ satisfies the
norm-1 property and $\Sigma$ is metrizable, then for any $x\in\Sigma$ there
exists a sequence of pure states $(\omega_{n})$ such that the sequence of
probability measures $(\mu^{\mathsf{E}}_{\omega_{n}})$ converges weakly to the
Dirac measure $\delta_{x}$.

###### Proof.

Fix $x\in\Sigma$ and denote by $B_{n}$ the open ball centred at $x$ of radius
$\frac{1}{n}$. Since $\mathsf{E}$ satisfies the norm-$1$ property, using the
$\epsilon$-decidability property of Prop. 2.12 we can choose unit vectors
$\varphi_{n}$ such that
$\innerproduct{\varphi_{n}}{\mathsf{E}(B_{n})\varphi_{n}}>1-1/n$. Denoting by
$\omega_{n}$ the associated pure state
$\omega_{n}=\outerproduct{\varphi_{n}}{\varphi_{n}}$, we have
$\tr[\omega_{n}E(B_{n})]=\innerproduct{\varphi_{n}}{\mathsf{E}(B_{n})\varphi_{n}}>1-1/n$,
and thus $\mu^{\mathsf{E}}_{\omega_{n}}(B_{n})>1-1/n$.

We will show weak convergence using the portemanteau theorem [30]. We must
show that for each measurable set $X$ with negligible boundary (i.e., such
that $\delta_{x}(\partial X)=0$ for all $x$) we have:
$\lim\limits_{n\to\infty}\mu^{\mathsf{E}}_{\omega_{n}}(X)=\delta_{x}(X)$. We
compute:

$\mu^{\mathsf{E}}_{\omega_{n}}(X)=\mu^{\mathsf{E}}_{\omega_{n}}(X\setminus
B_{n})+\mu^{\mathsf{E}}_{\omega_{n}}(X\cap B_{n}).$

For the first term, we have $\mu^{\mathsf{E}}_{\omega_{n}}(X\setminus
B_{n})\leq\mu^{\mathsf{E}}_{\omega_{n}}(\Sigma\setminus
B_{n})=1-\mu^{\mathsf{E}}_{\omega_{n}}(B_{n})\leq\frac{1}{n}$, which vanishes
as $n$ goes to infinity.

For the second term we distinguish two cases:

  * •

If $x\in X$, by hypothesis we may assume that $x\notin\partial X$ so
$x\in\mathring{X}$, the interior of $X$. Since $\mathring{X}$ is open, for $n$
large enough we always have $B_{n}\subseteq\mathring{X}\subseteq X$, so $X\cap
B_{n}=B_{n}$, and therefore $\mu^{\mathsf{E}}_{\omega_{n}}(X\cap
B_{n})=\mu^{\mathsf{E}}_{\omega_{n}}(B_{n})>1-1/n$. Thus, the second term goes
to $1$ as $n$ goes to infinity.

  * •

If $x\notin X$, by hypothesis we may assume that $x\notin\partial X$ so
$x\in\Sigma\setminus\overline{X}$, the complement of the adherence of $X$,
which is an open set. So for $n$ large enough we always have
$B_{n}\subseteq\Sigma\setminus\overline{X}\subseteq\Sigma\setminus X$, hence
$X\cap B_{n}=\emptyset$, leading to $\mu^{\mathsf{E}}_{\omega_{n}}(X\cap
B_{n})=0$. Thus, the second term goes to $0$ as $n$ goes to infinity.

Thus we have shown that
$\lim\limits_{n\to\infty}\mu^{\mathsf{E}}_{\omega_{n}}(X)=\delta_{x}(X)$.
Finally, the portemanteau theorem gives that the sequence
$(\mu^{\mathsf{E}}_{\omega_{n}})$ converges weakly to $\delta_{x}$. ∎

A converse to this theorem is given in [31]. Thus if $\Sigma$ is metrizable
and $\mathsf{E}$ is localizable, as will be the case in the sequel, we can
approximate the Dirac delta measure centred at any $x\in\Sigma$ with measures
of the form
$\mu^{\mathsf{E}}_{\omega_{n}}(X)=\tr[\mathsf{E}(X)\omega_{n}(x)]$. We will
call $\omega_{n}(x)$ a _localizing sequence centered at $x$_.

###  2.4 Covariance

Observables are often characterised by their covariance properties (e.g. [4]):

######  Definition 2.14.

Let $\mathsf{E}:\mathcal{B}(\Sigma)\to B(\mathcal{H})$ be a POVM, $G$ a
locally compact second countable topological group,
$\alpha:G\times\Sigma\to\Sigma$ a continuous transitive action (so that
$\Sigma$ is a homogeneous $G$-space) and $U:G\to B(\mathcal{H})$ a strongly
continuous projective unitary representation. Then
$(U,\mathsf{E},\mathcal{H})$ is called a _system of covariance based on_
$\Sigma$ if for all $X\in\mathcal{B}(\Sigma)$ and all $g\in G$,

$\mathsf{E}(\alpha(g,X))=U(g)\mathsf{E}(X)U(g)^{*}.$ (3)

$\mathsf{E}$ is called a _covariant POVM_ , and if $\mathsf{E}$ is projection-
valued, then $(U,\mathsf{E}\equiv\mathsf{P},\mathcal{H})$ is called a _system
of imprimitivity (SOI)_.

We will often write $g.X$ to stand for $\alpha(g,X)$, and will presume that
$U$ as given above is a true unitary representation.

Systems of imprimitivity are characterised by the Imprimitivity Theorem, which
states that for a closed subgroup $H\subset G$ and $\Sigma=G/H$ with left
$G$-action, there is (up to unitary equivalence) a one-to-one correspondence
between systems of imprimitivity $(U,\mathsf{P},\mathcal{H})$ based on
$\Sigma$ and unitary representations $U_{\chi}$ of $H$ (e.g. [32]).
Irreducible systems of imprimitivity (those with no invariant subspaces of
$(U,\mathsf{P})$) correspond to irreducible representations of $H$.444The
exact correspondence is as follows. Given a representation $U_{\chi}$ of $H$,
construct the SOI $(U^{\chi},\mathsf{P}^{\chi},\mathcal{H}^{\chi})$, with
$\mathcal{H}^{\chi}=L^{2}(G/H)\otimes\mathcal{H}_{\chi}$ as the carrier space
for the representation $U^{\chi}$ of $G$ induced by the representation
$U_{\chi}$ of $H$, and $\mathsf{P}^{\chi}$ acts as multiplication by the
characteristic function on $\mathcal{H}^{\chi}$. In the other direction, for
any SOI $(U,\mathsf{P},\mathcal{H})$, there is a unitary representation
$U_{\chi}$ of $H$ for which $(U,\mathsf{P},\mathcal{H})$ is unitarily
equivalent to the one given above. We note that any space $\Sigma$ with a
continuous transitive $G$-action can be written as $G/H$, where $H=H_{x}$ is
the stabiliser of some point $x\in\Sigma$.

For $G$ finite the canonical irreducible system of imprimitivity based on $G$
can be described very explicitly.

######  Example 2.15.

Let $L^{2}(G)$ denote the Hilbert space of the complex-valued functions
$G\to\mathbb{C}$ on a finite group $G$ (sometimes denoted $\mathbb{C}[G]$),
which has as an orthonormal basis the collection of indicator functions
$\delta_{g}$. These define the rank-$1$ projections $P(g)$, and to make
contact with more common notation in the physics literature we write
$\delta_{g}\equiv\ket{g}$ and $P(g)\equiv\outerproduct{g}{g}$. The left
regular representation is given as
$U_{L}(g)\ket{g^{\prime}}=\ket{gg^{\prime}}$. Then with
$P:g\mapsto\outerproduct{g}{g}\in B(L^{2}(G))$, $(U_{L},P,L^{2}(G))$ is a
system of imprimitivity based on $G$, and up to unitary equivalence is the
unique irreducible one.

The simple case above generalises to the following.

######  Example 2.16.

Let $G$ be a locally compact group acting transitively from the left on the
measure space $(\Sigma,\mu)$, with $\mu$ a $\sigma$-finite invariant measure
on $\Sigma$, and take $\mathcal{H}=L^{2}(\Sigma,\mu)$. Then
$(U,P,\mathcal{H})$ as given below is a system of imprimitivity based on
$\Sigma$.

$\displaystyle P(X)f=\chi_{X}f$ (4) $\displaystyle(U(g)f)(x)=f(g^{-1}.x).$

######  Remark 2.17.

Setting $G=\Sigma$ as a topological space (so that $H=\\{e\\}$) yields the
left-regular representation of $G$, equipped with the canonical (irreducible)
system of imprimitivity based on $G$ as a left $G$-space. Further specialising
to $G=\mathbb{R}$, with $\mu$ Lebesgue measure and $\mathbb{R}$ understood as
the configuration space of a single particle, (4) yields the standard
Schrödinger representation of wave mechanics for $P=P^{Q}$ ($Q$ position) and
$U$ is therefore generated by momentum. The uniqueness (all up to unitary
equivalence) of the irreducible representation of $\\{e\\}$ corresponds to the
uniqueness of the canonical commutation relation, and the imprimitivity point
of view therefore constitutes a generalisation of the Stone-von Neumann
theorem. The uniqueness of the shift-covariant spectral measure of position
(which follows from the imprimitivity theorem) is particular to the sharp
case; there exist many shift-covariant unsharp observables which can be
obtained through convolution with a Markov kernel (e.g. [33]).555It is
possible to be more general in the following sense. Let
$(\mathcal{A},G,\alpha)$ be a $W^{*}$ dynamical system, that is, $\mathcal{A}$
is a $W^{*}$ algebra, $G$ a locally compact group and $\alpha:G\to
Aut(\mathcal{A})$ an ultraweakly continuous homomorphism to the group of
automorphisms of $\mathcal{A}$. A covariant representation of $\mathcal{A}$ is
a pair $(\pi,U)$, where $\pi:\mathcal{A}\to B(\mathcal{H})$ is a linear
$*$-homomorphism $\mathcal{A}\to B(\mathcal{H})$ and $U$ is a unitary
representation for which for all $g$ and $A$,
$\pi(\alpha_{g}(A))=U(g)\pi(A)U(g)^{*}.$ (5) If
$\mathcal{A}=L^{\infty}(G,\mu)$, a representation of $\mathcal{A}$ corresponds
exactly to a PVM, and a covariant representation to a system of imprimitivity.
Example 2.16 is recovered by choosing the representation $f\mapsto M_{f}$
defined by $(M_{f})\varphi=f\varphi$. This suggests that the essential
structure of the approach to quantum reference frames given here is displayed
at the algebraic level.

######  Example 2.18 (Systems of coherent states).

Systems of covariance can be constructed from certain families of generalized
coherent states (see e.g. [34, 35] for coherent states, and [3] for an example
of their use as quantum reference frames). For instance, set $U$ to be a
unitary representation of a locally compact group $G$ in $\mathcal{H}$, with a
cyclic vector $\ket{\eta}$ (i.e., the span of $\\{U(g)\eta\\}$ is dense). Then
the orbit $\\{\eta_{g}:=U(g)\eta\\}$ is called a system of (Perelomov-Gilmore)
coherent states, and if they satisfy a certain square integrability condition,
they resolve the identity in the sense that

$\int_{G}\outerproduct{\eta_{g}}{\eta_{g}}d\mu(g)=\lambda\mathbbm{1},$ (6)

where as usual $\mu$ is Haar measure and $\lambda$ is some positive real
number. Then

$\mathsf{E}^{\eta}(X):=\frac{1}{\lambda}\int_{X}\outerproduct{\eta_{g}}{\eta_{g}}d\mu(g)$

defines a covariant POVM, and therefore $(U,\mathsf{E}^{\eta},\mathcal{H})$ is
a system of covariance. $\mathsf{E}^{\eta}$ does not satisfy the norm-$1$
property unless $\lambda=1$, and rigorously speaking the existence of sharp
coherent state systems, understood in the Hilbert space sense, require $G$ to
be discrete.

We note that this definition can be made much more general, and we refer to
[35] for a precise treatment. A minor generalisation is to proceed as above,
but set $\Sigma:=G/H$, with $H$ the stabilisier (up to a phase) of $\eta$, and
define another system of coherent states $\eta_{\sigma(x)}=U(\sigma(x))\eta$,
with $\sigma:\Sigma\to G$ any Borel section. Note that the measure $\mu$ in
(6) must be replaced by a quasi-invariant measure, and if this is actually
invariant, $H$ is compact. This setting is explored in the perspective-neutral
approach to quantum reference frames in [3] and has some interesting physical
consequences, which are beyond the scope of the present work but will be
addressed systematically within the framework presented in this paper in the
future.

######  Example 2.19 (Canonical Phase).

Let $\\{\ket{n}\in\mathcal{H};n\in\mathbb{N}\\}$ be an orthonormal basis of
$\mathcal{H}$, and $N=\sum_{n\geq 0}n\outerproduct{n}{n}$ a (densely defined)
_number observable_. Then $\mathsf{E}:\mathcal{B}((0,2\pi])\to B(\mathcal{H})$
is a _covariant phase observable_ if it satisfies (3) - explicitly, if
$e^{iN\theta}\mathsf{E}(X)e^{-iN\theta}=\mathsf{E}(X+\theta)$, where addition
is mod-$2\pi$. $U(\theta)=e^{iN\theta}$ is a strongly continuous unitary
representation of the circle group, and $(U,\mathsf{E},\mathcal{H})$ forms a
system of covariance. $\mathsf{E}$ is used to model the phase (observable) of
an electromagnetic field, with the states $\\{\ket{n}\\}$ corresponding to
photon number. These observables are completely characterised and take the
form

$\mathsf{E}(X)=\sum_{n,m=1}^{\infty}c_{n,m}\int_{X}e^{i\theta(n-m)}\outerproduct{n}{m}\frac{d\theta}{2\pi},$
(7)

where $(c_{n,m})$ is a positive matrix with $c_{n,n}=1$ for all $n$. The
boundedness from below of $N$ means that $\mathsf{E}$ is never sharp [36], but
the _canonical phase_ characterised by $c_{n,m}=1$ for all $n,m$ does satisfy
the norm-1 property ([37]).

We note that in general there are obstacles to the existence of covariant
POVMs with good localization properties, for example as given in the following
proposition:

######  Proposition 2.20.

Let $U$ be a strongly continuous unitary representation of $G$ in
$\mathcal{H}$, which is compact, abelian, and generated by $N=\sum
n\outerproduct{n}{n}$, and $G$ acts on itself from the left. Moreover let
$\mathcal{H}$ have finite dimension $d$. Then, for any
$X\in\mathcal{B}(\Sigma)$ ($X\neq G$),

$\tr[\rho\mathsf{E}(X)]\leq d.\mu(X),$ (8)

where $\mu(X):=\int_{X}d\mu(g)$.

The proof is straightforward: it always holds that
$\tr[\rho\mathsf{E}(X)]\leq\tr[\mathsf{E}(X)]$, and since
$\innerproduct{n}{\mathsf{E}(X)n}=\innerproduct{n}{\mathsf{E}(g.X)n}$ and
therefore $\innerproduct{n}{\mathsf{E}(X)n}=\mu(X)$ and $\tr[E(X)]=d.\mu(X)$
and the result follows. This means that for small $X$, a large $d$ is needed
to have localization probability close to $1$.

##  3 Relational Quantum Kinematics

In this section, we introduce further ideas on which the operational approach
to quantum reference frames is built. First, quantum reference frames are
introduced, defined as systems of covariance. As non-invariant objects, the
POVMs defining the frame are _not_ observable themselves, but understood as
part of the overall description reflecting the experimental arrangement. This
understanding suggests the notion of _framing_ , introduced next, which
respects the choice of frame observable for the given experiment—the framed
observables are those that can be realized as joint observables on the system-
frame composite. After this we recall the _relativization maps_ , denoted by
$\yen^{\mathcal{R}}$, which are understood as yielding frame-relative
observables, the collection of which is denoted by
$B(\mathcal{H}_{\mathcal{S}})^{\mathcal{R}}$ (with $\mathcal{R}$ the frame),
given as the (ultraweak closure of the) image of the relativization map. These
observables are both framed and invariant. The relative state spaces are then
taken to be the operational state spaces associated with
$B(\mathcal{H}_{\mathcal{S}})^{\mathcal{R}}\subseteq
B(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})^{\mathsf{G}}$,
denoted by
$\mathcal{S}(\mathcal{H}_{\mathcal{S}})_{\mathcal{R}}=\mathcal{S}(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})/\sim_{B(\mathcal{H}_{\mathcal{S}})^{\mathcal{R}}}$.
We also provide an equivalent characterization of this state space as the
image of the predual of $\yen^{\mathcal{R}}$, denoted by
$\mathcal{S}(\mathcal{H}_{\mathcal{S}})_{\mathcal{R}}\cong\mathcal{S}(\mathcal{H}_{\mathcal{S}})^{\mathcal{R}}\subseteq\mathcal{S}(\mathcal{H}_{\mathcal{S}})$;
this perspective is in keeping with the notions of relative states considered
in other works, but has clear operational motivation and is mathematically
precise.

From here we analyze the structure of the spaces of relative states when the
state of the frame is fixed; these are referred to as $\omega$-_product
relative states_ , for $\omega\in\mathcal{S}(\mathcal{H}_{\mathcal{R}})$. The
relationships between the various spaces of relative observables and relative
states are depicted in a dual pair of commuting diagrams, summarising the
framework developed so far. Various properties of product relative states are
then discussed. Finally, we turn our attention to the case of the state of the
reference being highly localized, in order to show that the relational
framework reproduces, at the probabilistic level, and in a limiting sense, the
predictions of orthodox quantum theory: generalizing previous results, we
prove the equivalence between relative and ‘absolute’ kinematical descriptions
in the context of arbitrary localizable principal (i.e. defined on the
principal $G$-space) frames.

###  3.1 Quantum Reference Frames

We are now in a position to give the definition of a quantum reference frame
suited to the purposes of this paper, defined as a covariant POVM on a
homogeneous space. There are strong physical reasons for the requirement of
covariance (e.g. [3]; see also [38] for a construction of covariant frame
observables on the Poincaré group), but we note that, at least for the
kinematics, more general frames are possible [18].

######  Definition 3.1.

A _quantum reference frame_ $\mathcal{R}$ is a system of covariance
$\mathcal{R}=(U_{\mathcal{R}},\mathsf{E}_{\mathcal{R}},\mathcal{H}_{\mathcal{R}})$
on a homogeneous (left) $G$-space $\Sigma_{\mathcal{R}}$.

When referring to a frame $\mathcal{R}$ we assume $G$ is given and often refer
to $\mathsf{E}_{\mathcal{R}}$ as the frame (observable) when $U_{\mathcal{R}}$
and $\mathcal{H}_{\mathcal{R}}$ are understood, and occasionally drop the
$\mathcal{R}$ subscript when it won’t lead to confusion. A quantum reference
frame is therefore a quantum physical system equipped with an observable which
transforms covariantly under the action of $G$. When based on $G$, such an
observable has a direct interpretation as an ‘orientation’ observable, and
since covariant observables on $G/H$ can be pulled back to those on $G$, this
is quite general.

######  Definition 3.2.

Two frames $\mathcal{R}_{1}=(U_{1},\mathsf{E}_{1},\mathcal{H}_{1})$ and
$\mathcal{R}_{2}=(U_{2},\mathsf{E}_{2},\mathcal{H}_{2})$ defined on the same
value space $\Sigma=\Sigma^{\prime}$ are called _(unitarily) equivalent_ if
there is a unitary map $U:\mathcal{H}_{1}\to\mathcal{H}_{2}$ such that for any
$X\in\mathcal{B}(\Sigma)$ we have

$\mathsf{E}_{2}(X)=U\mathsf{E}_{1}(X)U^{*}.$

There are various special classes of frames:

######  Definition 3.3.

  * •

A frame $\mathcal{R}$ is called _principal_ if $\Sigma_{\mathcal{R}}$ is a
principal homogeneous space, _non-principal_ otherwise.

  * •

A frame $\mathcal{R}$ is called _sharp_ if $\mathsf{E}_{\mathcal{R}}$ is
sharp, unsharp otherwise.

  * •

A frame $\mathcal{R}$ is called _ideal_ if it is principal and sharp.

  * •

A frame $\mathcal{R}$ is called _localizable_ if $\mathsf{E}_{\mathcal{R}}$ is
localizable.

  * •

A frame $\mathcal{R}$ is called _complete_ if there is no (non-trivial)
subgroup $H_{0}\subseteq G$ acting trivially on all the effects of
$\mathsf{E}_{\mathcal{R}}$, and _incomplete_ otherwise. Such an $H_{0}$ will
be called an _isotropy subgroup_ for $\mathsf{E}_{\mathcal{R}}$.

  * •

A frame $\mathcal{R}$ is called a _coherent state frame_ if
$\mathsf{E}_{\mathcal{R}}$ is constructed from a coherent states system as in
(2.18).

Some remarks are in order. The definition of completeness is reminiscent of
that appearing in other works (e.g. [5, 3]), and agrees with that of [3] in
the case of coherent state frames. This can be contrasted with the non-
principal setting, which is not considered in other works, but analysed for
finite homogeneous spaces in [19], in which there is a non-trivial isotropy
group $H<G$ for the action of $G$ on $\Sigma$. General connections between
incomplete frames and non-principal frames is the topic of current
investigation.666Notice, however, that for a localizable frame and any $h\in
H_{0}$ in the isotropy subgroup, the value space of $\mathsf{E}_{\mathcal{R}}$
as $G/H$ and $(\omega_{n})$ the sequence localizing at the identity coset
$eH\in G/H$, we have
$\delta_{hH}(X)=\lim_{n\to\infty}\mu_{\omega_{n}(hH)}^{\mathsf{E}_{\mathcal{R}}}(X)=\lim_{n\to\infty}\tr[h^{-1}.\omega_{n}\mathsf{E}_{\mathcal{R}}]=\lim_{n\to\infty}\tr[\omega_{n}h^{-1}.\mathsf{E}_{\mathcal{R}}(X)]=\lim_{n\to\infty}\tr[\omega_{n}\mathsf{E}_{\mathcal{R}}(X)]=\delta_{eH}(X),$
so that $hH=eH$ for any $h\in H_{0}$, and we can conclude that $H_{0}<H$. In
particular, localizable principal frames are complete, and in general, a
localizable frame observable on $\Sigma_{\mathcal{R}}\cong G/H$ factorizes
through $B(\mathcal{H}_{\mathcal{R}})^{H}$, i.e. we can write
$\mathsf{E}_{\mathcal{R}}:\mathcal{B}(G/H)\to
B(\mathcal{H}_{\mathcal{R}})^{H}\hookrightarrow B(\mathcal{H}_{\mathcal{R}}).$
Sharp frames are described by systems of imprimitivity, which by the
imprimitivity theorem, are unitarily equivalent to $L^{2}(G/H)$. Fixing
$H=\\{e\\}$ and demanding that $(U,\mathsf{P},\mathcal{H})$ is irreducible
yields (up to unitary equivalence) the left regular representation. The term
‘ideal’ frames (and also ‘perfect’ frames [5]) is used in the literature to
mean something similar but not identical, the typical provision being that
there is a collection of states in $\mathcal{H}$ indexed by $g\in G$ (for
example coherent state systems as in [3]), and these states must be orthogonal
(or ‘perfectly distinguishable’) for $g\neq g^{\prime}$ for the frame to be
ideal. However, we avoid this usage since the notion of orthogonality is not
compatible with the Hilbert space framework within which we are working - $G$
is generally continuous, yet $\mathcal{H}$ is assumed to be separable. We note
also that elsewhere, coherent states themselves are understood, through their
$G$-dependence, as encoding ‘frame orientations’; for us, this comes
probabilistically through the measure on $G$ defined by an arbitrary state
paired with $\mathsf{E}_{\mathcal{R}}$, through the Born rule (1), which we
view as being more operationally motivated. There is no operational
distinction between sharp frames and localizable frames based on the same
space, and this is manifested clearly in the next subsection in which standard
and relative observables are operationally compared.

There are various other interesting frames one can consider that we do not
touch upon here, for instance _informationally complete_ frames, in which
$\mathsf{E}$ is an informationally complete POVM, which is necessarily
unsharp.

###  3.2 Framing

After fixing a frame observable $\mathsf{E}_{\mathcal{R}}$, and for now not
imposing a symmetry constraint, one can consider _joint observables_ of system
and frame (e.g. [4]). From here, one can compute conditional probabilities,
i.e., the probability of observing the system observable
$\mathsf{E}_{\mathcal{S}}:\mathcal{B}(\Sigma_{\mathcal{S}})\to
B(\mathcal{H}_{\mathcal{S}})$ to have value in some
$Y\subseteq\Sigma_{\mathcal{S}}$ given that the frame is measured in some
$X\subseteq\Sigma_{\mathcal{R}}$. This captures the idea that a frame is part
of an experimental arrangement, and those observables of system-plus-frame
which can be measured in that experiment are limited by the choice of frame.
In the sequel, we will describe how to measure such objects in a relational
way. The simplest joint observables are of the form
$M(X,Y):=\mathsf{E}_{\mathcal{R}}(X)\otimes\mathsf{E}_{\mathcal{S}}(Y)$. In
order to allow for mixing we take the closed convex hull of the above joint
observables, for a fixed frame but arbitrary system observable, whence:

######  Definition 3.4.

Given a frame $\mathcal{R}$ and a system $\mathcal{S}$, the effects in

$\mathcal{E}(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})^{\mathsf{E}_{\mathcal{R}}}:=\rm{conv}\left\\{\mathsf{E}_{\mathcal{R}}(X)\otimes\mathsf{F}_{\mathcal{S}}\hskip
3.0pt|\hskip 3.0pt\hskip
3.0ptX\in\mathcal{B}(\Sigma_{\mathcal{R}}),\mathsf{F}_{\mathcal{S}}\in\mathcal{E}(\mathcal{H}_{\mathcal{S}})\right\\}^{\rm
cl},$

with $\rm{conv}$ denoting the convex hull, will be called
$\mathsf{E}_{\mathcal{R}}$-_framed effects_ , while the elements of the Banach
space

$B(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})^{\mathsf{E}_{\mathcal{R}}}:=\rm{span}\\{\mathcal{E}(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})^{\mathsf{E}_{\mathcal{R}}}\\}^{\rm
cl}$

will be referred to as $\mathsf{E}_{\mathcal{R}}$-_framed operators_. We
denote by $\sim_{\mathsf{E}_{\mathcal{R}}}$ the operational equivalence
relation on
$\mathcal{T}(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})$ taken
with respect to
$\mathcal{E}(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})^{\mathsf{E}_{\mathcal{R}}}$.
The space of $\mathsf{E}_{\mathcal{R}}$-_framed trace class operators_ is
given by

$\mathcal{T}(\mathcal{H}_{\mathcal{S}})_{\mathsf{E}_{\mathcal{R}}}:=\mathcal{T}(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})/\sim_{\mathsf{E}_{\mathcal{R}}},$

while the operational state space of $\mathsf{E}_{\mathcal{R}}$-_framed
states_ is defined to be

$\mathcal{S}(\mathcal{H}_{\mathcal{S}})_{\mathsf{E}_{\mathcal{R}}}:=\mathcal{S}(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})/\sim_{\mathsf{E}_{\mathcal{R}}}.$

The $\mathsf{E}_{\mathcal{R}}$-framed (or just ’framed’ when the frame is
clear from the context) states are then precisely those that can be
distinguished by the observables on the composite system that respect the
choice of the frame observable. Prop. 2.5 and 2.3 give the following.

######  Proposition 3.5.

There is a Banach space isomorphism

$B(\mathcal{H}_{\mathcal{S}})^{\mathsf{E}_{\mathcal{R}}}_{*}\cong\mathcal{T}(\mathcal{H}_{\mathcal{S}})_{\mathsf{E}_{\mathcal{R}}}.$

Moreover, the subset
$\mathcal{S}(\mathcal{H}_{\mathcal{S}})_{\mathsf{E}_{\mathcal{R}}}\subset\mathcal{T}(\mathcal{H}_{\mathcal{S}})^{\rm
sa}_{\mathsf{E}_{\mathcal{R}}}$ is a total convex state space.

###  3.3 Relative Observables and States

Quantum reference frames are introduced, in analogy to classical physics, to
study properties of some system relative to the given frame. Together they are
described as a compound system. The main principle put forward in [7, 39, 18,
16] is that truly observable quantities are invariant under gauge/symmetry
transformations, understood here as a diagonal action on the compound system.
Here, we make a further distinction between the invariant algebra
$B(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})^{G}$, containing
the ‘strong Dirac observables’, and the observables in the image of the
relativization map (defined below), which will be called _relative
observables/operators_ , and which are close to the relational Dirac
observables as given in e.g. [3]. They can be understood as satisfying both
the framing and invariance properties.

####  3.3.1 Relativization and relative observables

Let
$\mathcal{R}=(U_{\mathcal{R}},\mathsf{E}_{\mathcal{R}},\mathcal{H}_{\mathcal{R}})$
be a principal frame and $U_{\mathcal{S}}$ be any (strongly continuous,
unitary) representation of $G$ in $\mathcal{H}_{\mathcal{S}}$. As has been
shown in [7], there is a map
$\yen^{\mathcal{R}}:B(\mathcal{H}_{\mathcal{S}})\to
B(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})^{G}$ defined by

$\yen^{\mathcal{R}}(A_{\mathcal{S}}):=\int_{G}d\mathsf{E}_{\mathcal{R}}(g)\otimes
U_{\mathcal{S}}(g)A_{\mathcal{S}}U_{\mathcal{S}}(g)^{*},$ (9)

called the relativization map, which has the following properties (not all of
which are independent): $\yen^{\mathcal{R}}$ is linear, unital, adjoint-
preserving, preserves effects, bounded (thus continuous), completely positive,
a contraction, normal, injective if $\mathsf{E}_{\mathcal{R}}$ satisfies the
norm-1 property (see 3.23 below), and multiplicative exactly when
$\mathsf{E}_{\mathcal{R}}$ is projection valued (sharp). In the setting that
$\mathsf{E}_{\mathcal{R}}$ is sharp, the image $\rm{Im}(\yen^{\mathcal{R}})$
is a von Neumann algebra isomorphic to $B(\mathcal{H}_{\mathcal{S}})$, making
$\yen^{\mathcal{R}}$ a faithful representation of
$B(\mathcal{H}_{\mathcal{S}})$ in
$B(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})$. Crucially, the
operators in the image of $\yen^{\mathcal{R}}$ are framed (see Prop. 3.7.1. in
[31] for the proof), and invariant
$\yen^{\mathcal{R}}(B(\mathcal{H}_{\mathcal{S}}))\subset
B(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})^{G}$ (which can
be verified by direct computation).

The $\yen$ construction [7] was developed as a generalization, and making
rigorous, of the $\$$ map [5], and was used to construct particular
invariants; the physical difference between observables in the image of
$\yen^{\mathcal{R}}$ and general invariants was not discussed at that time but
will play an important role here. The twirl map arises as a special case of
this construction when $\mathcal{H}_{\mathcal{R}}$ is taken to be
$\mathbb{C}$, with (necessarily) trivial $G$ action. Indeed, the notion of a
covariant POVM then coincides with that of a normalized invariant measure
(Haar measure), and thus there is exactly one when $G$ is compact, and none
otherwise. Another simple example is when $\mathcal{R}$ is taken to be the
canonical irreducible system of imprimitivity for a finite group $G$, where
$\yen^{\mathcal{R}}$ reads

$\yen^{\mathcal{R}}(A_{\mathcal{S}})=\sum_{g\in G}\outerproduct{g}{g}\otimes
U_{\mathcal{S}}(g)A_{\mathcal{S}}U_{\mathcal{S}}(g)^{*}.$

The relativization in some sense replaces the naive ‘invariantization’ given
by the twirl and applies to the general case of locally compact groups and
arbitrary (strongly continuous, unitary) representations.

Physically, $\yen^{\mathcal{R}}$ is understood as providing an explicit
inclusion of the frame
$\mathcal{R}=(U_{\mathcal{R}},\mathsf{E}_{\mathcal{R}},\mathcal{H}_{\mathcal{R}})$
into the description; we view $\yen^{\mathcal{R}}(A_{\mathcal{S}})$ as a
_relativization_ of $A_{\mathcal{S}}$. This naturally extends to POVMs by
composition. For instance, for $G=\mathbb{R}$, and fixing the appropriate
frame observable, $\yen^{\mathcal{R}}$ takes the position observable
$Q_{\mathcal{S}}$ to
$Q_{\mathcal{S}}\otimes\mathbbm{1}_{\mathcal{R}}-\mathbbm{1}_{\mathcal{S}}\otimes
Q_{R}$ (which can be proven at the level of the respective spectral measures);
it also produces unsharp relative time observables, relative angle and phase
observables for $G=U(1)$ ([15, 7, 40]). We take the view that what is measured
is the relation between system and frame, or more precisely, the relative
observable, such as the relative position. This is in line with, but stronger
than, the requirement that observable quantities are (gauge-)invariant. The
covariant POVM of the frame is never directly measured (as a non-invariant
object), but as we shall see, can be justified as being a crucial ingredient
in the description.

######  Definition 3.6.

An operator $A\in
B(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})^{G}$ is called
$\mathcal{R}$-_relative_ (or just ’relative’ when the frame is clear from the
context) if it belongs to the ultraweak closure of the image of
$\yen^{\mathcal{R}}$. We write

$B(\mathcal{H}_{\mathcal{S}})^{\mathcal{R}}:=\yen^{\mathcal{R}}(B(\mathcal{H}_{\mathcal{S}}))^{\rm
cl}\subset B(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})^{G}$

for the set of $\mathcal{R}$-relative operators.

By definition, $B(\mathcal{H}_{\mathcal{S}})^{\mathcal{R}}$ is ultraweakly
closed, and hence also norm closed and therefore an _operator space_ (e.g.
[41]). The term ‘relative observables’ will usually be reserved for POVMs with
the image in the space of relative operators. They may arise by composing
POVMs on the system $\mathcal{S}$ with the relativization map. We note that
for a sharp frame, $\yen^{\mathcal{R}}$ is a normal homomorphism and therefore
the image is automatically closed. Relative operators are thus invariant but
do not exhaust the invariant operators in general. That relative operators are
also framed is explored in [31] (Prop. 3.7.1.).

For any $A_{\mathcal{S}}\in B(\mathcal{H}_{\mathcal{S}})^{G}$, and any
principal frame $\mathcal{R}$ we have
$\yen^{\mathcal{R}}(A_{\mathcal{S}})=\mathbb{1}_{\mathcal{R}}\otimes
A_{\mathcal{S}}$, and thus
$\yen^{\mathcal{R}}(B(\mathcal{H}_{\mathcal{S}})^{G})\cong
B(\mathcal{H}_{\mathcal{S}})^{G}$, whatever the frame. The invariant algebra
can thus be understood as a frame-independent description - this perspective
will be strengthened when we consider states. The definition of a relational
Dirac observable as in [3] is recovered for $g\in G$ by
$\yen^{\mathcal{R}}(g.A_{S})$ when $\mathsf{E}_{\mathcal{R}}$ is a covariant
POVM associated to a coherent state system. Thus in the case of coherent state
frames the set of relational Dirac observables and relativized self-adjoint
operators are the same.

An example of relative observables arises when one frame is relativized with
respect to another.

######  Definition 3.7.

For a pair of principal frames $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ define
the _observable of relative orientation_ (of $\mathcal{R}_{2}$ with respect to
$\mathcal{R}_{1}$), denoted by $\mathsf{E}_{2}*\mathsf{E}_{1}$, via the
relativization map777 The notation is chosen to reflect the observation that
when $\mathcal{H}_{\mathcal{S}}$ and $\mathcal{H}_{\mathcal{R}}$ are both
taken to be $\mathbb{C}$, the formula reproduces standard convolution of
measures.

$\mathsf{E}_{2}*\mathsf{E}_{1}:=\yen^{\mathcal{R}_{1}}\circ\mathsf{E}_{2}=\int_{G}d\mathsf{E}_{1}(g)\otimes
g.\mathsf{E}_{2}(\cdot).$ (10)

Indeed, many important examples of relative observables—position, time, phase
and angle ([15, 7])—are relative orientation observables. The name can be
further justified by the following.

######  Proposition 3.8.

For a pair of principal localizable frames $\mathcal{R}_{1}$ and
$\mathcal{R}_{2}$ and corresponding localizing sequences $(\omega_{n})$ and
$(\rho_{m})$ (centred at $e\in G$), writing
$\Omega_{n,m}(h):=\omega_{n}\otimes h^{-1}.\rho_{m}$ we have

$\lim_{n,m\to\infty}\mu_{\Omega_{n,m}(h)}^{\mathsf{E}_{2}*\mathsf{E}_{1}}=\delta_{h}.$

###### Proof.

We calculate

$\displaystyle\lim_{n,m\to\infty}\tr[(\omega_{n}\otimes
h^{-1}.\rho_{m})\int_{G}d\mathsf{E}_{1}(g)\otimes g.\mathsf{E}_{2}(X)]$
$\displaystyle=\lim_{m\to\infty}\tr[h^{-1}.\rho_{m}\mathsf{E}_{2}(X)]$
$\displaystyle=\lim_{m\to\infty}\tr[\rho_{m}\mathsf{E}_{2}(h^{-1}.X)]=\delta_{e}(h^{-1}.X)=\delta_{h}(X),$

where we have used Prop. 2.13 twice. ∎

Thus when the first frame is localized at the identity, and the second at
$h\in G$, the relative orientation observable gives a probability distribution
localized at $h$, which is understood as their relative orientation.888We have
$\lim_{m\to\infty}\mu^{\mathsf{E}_{2}}_{h^{-1}.\rho_{m}}=\delta_{h}$, so that
the states $h^{-1}.\rho_{m}$ approximate localization at $h$ for large $m$.
From here, we see that
$\lim_{n\to\infty}\mu_{\Omega_{n,m}(h)}^{\mathsf{E}_{1}\otimes\mathbbm{1}_{2}}=\delta_{e}$
and
$\lim_{m\to\infty}\mu_{\Omega_{n,m}(h)}^{\mathbbm{1}_{1}\otimes\mathsf{E}_{2}}=\delta_{h}$.
Note that since $\mathsf{E}_{2}*\mathsf{E}_{1}$ is invariant, we could just as
well evaluate it on e.g. $(h.\omega_{n}\otimes\rho_{m})$, approximating the
first frame localized at $h^{-1}\in G$, and the second at the identity, with
the same result. Notice also that if $\mathsf{E}_{1}$ is relativized with
respect to $\mathsf{E}_{2}$, the probability distribution of the relative
orientation observable evaluated on $\omega_{n}\otimes h^{-1}.\rho_{m}$ is
localized at $h^{-1}$ as one would expect. Indeed, a simple calculation gives

$\mathsf{E}_{2}*\mathsf{E}_{1}(X)={\rm
SWAP}\circ\mathsf{E}_{1}*\mathsf{E}_{2}(X^{-1}),$

where ${\rm SWAP}$ switches the tensor product factors as in [2], i.e., for
$A_{1}\otimes A_{2}\in
B(\mathcal{H}_{\mathcal{R}_{1}}\otimes\mathcal{H}_{\mathcal{R}_{2}})$ we have
${\rm SWAP}(A_{1}\otimes A_{2})=A_{2}\otimes A_{1}\in
B(\mathcal{H}_{\mathcal{R}_{2}}\otimes\mathcal{H}_{\mathcal{R}_{1}})$, and
$X^{-1}:=\\{g\in G\hskip 2.0pt|\hskip 2.0ptg^{-1}\in X\\}$.

####  3.3.2 Relative state spaces

Despite playing an essential role, the notion of ‘state relative to a frame’
is left implicit/taken as primary in [1], where the ‘modern’ notion of quantum
reference frame transformation was first introduced. One objective of that
work was, as we understand it, to provide frame changes without recourse to
any external or classical referent. A definition of relative state is provided
in [2] and the frame changes there are in line with [1] (though applied also
in more general settings); however, no operational justification is provided,
and strictly speaking, the procedure applies only to countable $G$, since the
frame state is assumed to be perfectly localized at $e\in G$. We now seek to
rectify these various shortcomings, by providing an operationally motivated
notion of relative state, which rigorously extends that of [2] to continuous
(locally compact) groups, and provides an alternative conceptual foundation
for the notion of relative state as it arises in the theory of QRFs; in
particular we show that the localization at/near $e$ is required to view
states as relative states.

######  Definition 3.9.

Given a frame $\mathcal{R}$ and a system $\mathcal{S}$, we will denote by
$\sim_{\mathcal{R}}$ the operational equivalence relation on
$\mathcal{T}(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})$ taken
with respect to the set $B(\mathcal{H}_{\mathcal{S}})^{\mathcal{R}}$. The
space of $\mathcal{R}$-relative (or just ’relative’ when the frame is clear
from the context) trace class operators is given by

$\mathcal{T}(\mathcal{H}_{\mathcal{S}})_{\mathcal{R}}:=\mathcal{T}(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})/\sim_{\mathcal{R}};$

the operational state space of $\mathcal{R}$-_relative states_ is defined to
be

$\mathcal{S}(\mathcal{H}_{\mathcal{S}})_{\mathcal{R}}:=\mathcal{S}(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})/\sim_{\mathcal{R}}.$

In standard quantum mechanics the duality $\mathcal{T}(\mathcal{H})^{*}\cong
B(\mathcal{H})$ (which is an isomorphism of Banach spaces) broadly establishes
the states and observables as dual objects: each is determined by the other.
This duality persists at the relative level:

######  Proposition 3.10.

The Banach space $B(\mathcal{H}_{\mathcal{S}})^{\mathcal{R}}$ has the uniquely
given predual

$B(\mathcal{H}_{\mathcal{S}})^{\mathcal{R}}_{*}\cong\mathcal{T}(\mathcal{H}_{\mathcal{S}})_{\mathcal{R}}.$

###### Proof.

Use Prop. 2.3. ∎

Denote the image of the predual map $\yen^{\mathcal{R}}_{*}$ restricted to
states by

$\mathcal{S}(\mathcal{H}_{\mathcal{S}})^{\mathcal{R}}:=\yen^{\mathcal{R}}_{*}(\mathcal{S}(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}}))\subseteq\mathcal{S}(\mathcal{H}_{\mathcal{S}}).$

We then have the following.

######  Proposition 3.11.

The set of $\mathcal{R}$-relative states is a total convex state space in
$\mathcal{T}(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})^{\rm{sa}}/\sim_{\mathcal{R}}$
and the map

$y_{\mathcal{R}}:S(\mathcal{H}_{\mathcal{S}})^{\mathcal{R}}\ni\Omega^{\mathcal{R}}\equiv\yen^{\mathcal{R}}_{*}(\Omega)\mapsto[\Omega]_{\mathcal{R}}\in\mathcal{S}(\mathcal{H}_{\mathcal{S}})_{\mathcal{R}}$

is well-defined and establishes the state space isomorphism
$S(\mathcal{H}_{\mathcal{S}})^{\mathcal{R}}\cong
S(\mathcal{H}_{\mathcal{S}})_{\mathcal{R}}$.

###### Proof.

Use Prop. 2.5 and 2.6 for $\Lambda=\yen^{\mathcal{R}}$. ∎

Since this characterization of relative states will be used frequently, it is
worth emphasising how we use the above notation. Given an arbitrary
$\Omega\in\mathcal{S}(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})$,
we write
$\Omega^{\mathcal{R}}=\yen^{\mathcal{R}}_{*}(\Omega)\in\mathcal{S}(\mathcal{H}_{\mathcal{S}})$,
and $[\Omega]_{\mathcal{R}}$ for the corresponding equivalence class on
$\mathcal{S}(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})$,
i.e., the element of
$\mathcal{S}(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})/{\sim_{\mathcal{R}}}$.

As a direct consequence of the invariance of the image of
$\yen^{\mathcal{R}}$, the predual $\yen^{\mathcal{R}}_{*}$ map factorizes
through
$\mathcal{T}(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})/\sim_{G}$
and thus

$\mathcal{S}(\mathcal{H}_{\mathcal{S}})^{\mathcal{R}}=\yen^{\mathcal{R}}_{*}(\mathcal{S}(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})/\sim_{G}).$

For $G$ compact we then have
$\yen^{\mathcal{R}}_{*}(\Omega)=\yen^{\mathcal{R}}_{*}(\mathcal{G}_{*}(\Omega))$;
$\mathcal{G}_{*}(\omega\otimes\rho)$ is sometimes referred to as the
’relational encoding’ of $\rho$ in the older quantum reference frames
literature (c.f. [6]). Note that $\yen^{\mathcal{R}}_{*}$ also factors through
the $G$-orbits; if the action of $G$ is understood as a gauge transformation,
applying $\yen_{*}^{\mathcal{R}}$ to the state on
$B(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})^{G}$ can thus be
viewed as providing a relative description whilst maintaining (gauge)
invariance.

###  3.4 Restriction and Conditioning

We now recall the restriction map (e.g. [16]) that allows for the conditioning
of observables of the system-plus-reference with a specified state of the
reference.

######  Definition 3.12.

Let $\omega\in\mathcal{S}(\mathcal{H}_{\mathcal{R}})$ be any (normal) state of
the reference. The _restriction map_
$\Gamma_{\omega}:B(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})\to
B(\mathcal{H}_{\mathcal{S}})$ is given by

$\tr[\rho\Gamma_{\omega}(A)]=\tr[(\omega\otimes\rho)A],$ (11)

holding for all $A\in
B(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})$ and
$\rho\in\mathcal{S}(\mathcal{H}_{\mathcal{S}})$. Equivalently, it is the
continuous linear extension to
$B(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})$ of

$\Gamma_{\omega}:A_{R}\otimes A_{\mathcal{S}}\mapsto\tr[\omega
A_{R}]A_{\mathcal{S}},$

or the dual of the isometric embedding
$\mathcal{V}_{\omega}:\rho\mapsto\omega\otimes\rho$, referred to as the
$\omega$-_product map_ , i.e.,
$\tr[\rho\Gamma_{\omega}(A)]=\tr[\mathcal{V}_{\omega}(\rho)A]$.999Versions of
this map have appeared elsewhere in the literature, often in a rather ill-
defined form. Usually the implicit desired map is
$B(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})\ni
A\mapsto\innerproduct{\varphi_{R}\otimes\cdot}{A\varphi_{R}\otimes\cdot}:\mathcal{H}_{\mathcal{S}}\times\mathcal{H}_{\mathcal{S}}\to\mathbb{R}$
for $A=A^{*}$, which as a bounded real quadratic form defines uniquely a
bounded self-adjoint $A_{\mathcal{S}}\in B(\mathcal{H}_{\mathcal{S}})$ which
for all $\varphi_{\mathcal{S}}$ satisfies
$\innerproduct{\varphi_{R}\otimes\varphi_{\mathcal{S}}}{A\varphi_{R}\otimes\varphi_{\mathcal{S}}}=\innerproduct{\varphi_{\mathcal{S}}}{A_{\mathcal{S}}\varphi_{\mathcal{S}}}=:\innerproduct{\varphi_{\mathcal{S}}}{\Phi_{\varphi_{R}}(A)\varphi_{\mathcal{S}}}$.
The generalisation of $\Phi_{\varphi_{R}}$ to mixtures is then given by
$\Gamma_{\omega}$ defined in (11).

The restriction map $\Gamma_{\omega}$ is normal, completely positive and
trace-preserving (hence trace norm continuous), and is a noncommutative
conditional expectation (e.g., [42]; see [43] for a comprehensive review). It
is equivariant (covariant) exactly when $\omega$ is invariant. It is
understood as providing a description of the system contingent on a particular
state of the frame.

####  3.4.1 Conditioned relative observables

######  Definition 3.13.

The map

$\yen^{\mathcal{R}}_{\omega}:=\Gamma_{\omega}\circ\yen^{\mathcal{R}}:B(\mathcal{H}_{\mathcal{S}})\to
B(\mathcal{H}_{\mathcal{S}}),$

will be called the $\omega$-_conditioned $\mathcal{R}$-relativization map_.

This map gives a (typically non-invariant) description inside
$B(\mathcal{H}_{\mathcal{S}})$ of the (invariant) relative description in
terms of $B(\mathcal{H}_{\mathcal{S}})^{\mathcal{R}}$, contingent on the state
$\omega$ of the reference. As a composition of such maps,
$\yen^{\mathcal{R}}_{\omega}$ is unital, normal and completely positive. The
$\omega$-conditioned relativized operators take the form:

$\yen^{\mathcal{R}}_{\omega}(A_{\mathcal{S}})=\int_{G}d\mu^{\mathsf{E}_{\mathcal{R}}}_{\omega}(g)U_{\mathcal{S}}(g)A_{\mathcal{S}}U_{\mathcal{S}}(g)^{*},$
(12)

representing a ‘weighted average’ of the operator $A_{\mathcal{S}}$ with
respect to the probability distribution given through the evaluation of
$\omega$ on the frame observable. Note that different $\omega$ may give rise
to the same measure $\mu^{\mathsf{E}_{\mathcal{R}}}_{\omega}$ which in turn
gives rise to the same $\omega$-conditioned relativization.

######  Definition 3.14.

The operator space given by the ultraweak closure of ${\rm
Im}\yen^{\mathcal{R}}_{\omega}$, i.e.,

$B(\mathcal{H}_{\mathcal{S}})^{\mathcal{R}}_{\omega}:=\yen^{\mathcal{R}}_{\omega}(B(\mathcal{H}_{\mathcal{S}}))^{\rm
cl}\subseteq B(\mathcal{H}_{\mathcal{S}}),$

is called the space of $\omega$-_conditioned_ $\mathcal{R}$-_relative
operators_.

####  3.4.2 Product relative states

######  Definition 3.15.

We will denote by $\sim_{(\mathcal{R},\omega)}$ the operational equivalence
relation on $\mathcal{T}(\mathcal{H}_{\mathcal{S}})$ taken with respect to
$B(\mathcal{H}_{\mathcal{S}})^{\mathcal{R}}_{\omega}$. The space of
$\omega$-_product_ $\mathcal{R}$-_relative trace class operators_ is given by

$\mathcal{T}(\mathcal{H}_{\mathcal{S}})_{\mathcal{R}}^{\omega}:=\mathcal{T}(\mathcal{H}_{\mathcal{S}})/\sim_{(\mathcal{R},\omega)},$

while the operational state space of $\omega$-_product_
$\mathcal{R}$-_relative states_ is defined to be

$\mathcal{S}(\mathcal{H}_{\mathcal{S}})_{\mathcal{R}}^{\omega}:=\mathcal{S}(\mathcal{H}_{\mathcal{S}})/\sim_{(\mathcal{R},\omega)}.$

######  Definition 3.16.

The predual of the $\omega$-conditioned $\mathcal{R}$-relativization map will
be denoted by

$\mathcal{P}^{\mathcal{R}}_{\omega}:=(\yen^{\mathcal{R}}_{\omega})_{*}=\yen^{\mathcal{R}}_{*}\circ\mathcal{V}_{\omega}:\mathcal{T}(\mathcal{H}_{\mathcal{S}})\to\mathcal{T}(\mathcal{H}_{\mathcal{S}}),$

and referred to as the $\mathcal{R}$-_relative_ $\omega$-_product map_.

######  Proposition 3.17.

For any (normal) state $\omega\in\mathcal{S}(\mathcal{H}_{\mathcal{R}})$ the
$\omega$-conditioned relative observables form a Banach space with the unique
Banach predual given by

$[B(\mathcal{H}_{\mathcal{S}})^{\mathcal{R}}_{\omega}]_{*}\cong\mathcal{T}(\mathcal{H}_{\mathcal{S}})_{\mathcal{R}}^{\omega}.$
(13)

Moreover, the set of $\omega$-_product relative states_
$S(\mathcal{H}_{\mathcal{S}})^{\omega}_{\mathcal{R}}$ is a total convex state
space in $\mathcal{T}(\mathcal{H}_{\mathcal{S}})_{\mathcal{R}}^{\omega}$ and
there is a state space isomorphism

$S(\mathcal{H}_{\mathcal{S}})^{\omega}_{\mathcal{R}}\cong\mathcal{P}^{\mathcal{R}}_{\omega}(\mathcal{S}(\mathcal{H}_{\mathcal{S}}))\subseteq\mathcal{S}(\mathcal{H}_{\mathcal{S}}).$

###### Proof.

Use Prop. 2.3, 2.5 and 2.6 with $\Lambda=\yen^{\mathcal{R}}_{\omega}$. ∎

The $\omega$-conditioned relative observables are understood as follows. If we
view the reference $\mathcal{R}$ as initially external to the system, meaning
that it is not explicitly realised in the theoretical description but
nevertheless plays some role in the description of the phenomena/observable
probability distributions, the relativization map $\yen^{\mathcal{R}}$
describes how some $A_{\mathcal{S}}\in B(\mathcal{H}_{\mathcal{S}})$ is
realized as an invariant $\yen^{\mathcal{R}}(A_{\mathcal{S}})\in
B(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})^{G}$ (naturally
extending to POVMs) with respect to the frame $\mathcal{R}$. The restriction
map then provides a description of $\mathcal{S}$ contingent on the given state
preparation of $\mathcal{R}$, after which $\mathcal{R}$ can be viewed as
external, arriving at the corresponding subset of observables of
$\mathcal{S}$. We note (and discuss further later) that not all of
$B(\mathcal{H}_{\mathcal{S}})$ is typically in the image of the
$\omega$-conditioned relativization, and therefore the definition is not
vacuous. Eq. (13) captures again a duality between states and observables, now
in the conditioned relativized setting.

Before we describe the properties of product relative states, we summarize
graphically the maps and spaces thus far encountered as a dual pair of
commutative diagrams. The "$i$" maps denote the inclusions of (ultraweakly
closed) operator spaces, while "$\pi$" maps denote the corresponding quotient
projections between the relevant predual spaces.

${B(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})}$${B(\mathcal{H}_{\mathcal{S}})^{\mathcal{R}}}$${B(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})^{G}}$${B(\mathcal{H}_{\mathcal{S}})}$${B(\mathcal{H}_{\mathcal{S}})^{\mathcal{R}}_{\omega}}$$\scriptstyle{\Gamma_{\omega}\circ
i^{\mathcal{R}}}$$\scriptstyle{i^{G}_{\mathcal{R}}}$$\scriptstyle{i^{\mathcal{R}}}$$\scriptstyle{i^{G}}$$\scriptstyle{\Gamma_{\omega}\circ
i^{G}}$$\scriptstyle{\yen^{\mathcal{R}}}$$\scriptstyle{\yen^{\mathcal{R}}_{\omega}}$

The predual of the above diagram, restricted to state spaces for the
convenient representation of
$\mathcal{S}(\mathcal{H}_{\mathcal{S}})_{\mathcal{R}}^{\omega}\subseteq\mathcal{S}(\mathcal{H}_{\mathcal{S}})$,
reads

${\mathcal{S}(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})}$${\mathcal{S}(\mathcal{H}_{\mathcal{S}})_{\mathcal{R}}}$${\mathcal{S}(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})_{G}}$${\mathcal{S}(\mathcal{H}_{\mathcal{S}})}$${\mathcal{S}(\mathcal{H}_{\mathcal{S}})_{\mathcal{R}}^{\omega}}$$\scriptstyle{\pi_{\mathcal{R}}}$$\scriptstyle{\pi_{G}}$$\scriptstyle{\yen^{\mathcal{R}}_{*}}$$\scriptstyle{\pi_{G}^{\mathcal{R}}}$$\scriptstyle{\pi_{\mathcal{R}}\circ\mathcal{V}_{\omega}}$$\scriptstyle{\mathcal{P}^{\mathcal{R}}_{\omega}}$$\scriptstyle{\pi_{G}\circ\mathcal{V}_{\omega}}$

####  3.4.3 Properties of product relative states

The $\omega$-product relative states take the form

$\rho^{(\omega)}:=\yen^{\mathcal{R}}_{*}(\omega\otimes\rho)=\int_{G}d\mu_{\omega}^{\mathsf{E}_{\mathcal{R}}}(g)U_{\mathcal{S}}(g)^{*}\rho
U_{\mathcal{S}}(g),$ (14)

where the measure $\mu_{\omega}^{\mathsf{E}_{\mathcal{R}}}$ is given as usual
by
$\mu_{\omega}^{\mathsf{E}_{\mathcal{R}}}(X):=\tr[\mathsf{E}_{\mathcal{R}}(X)\omega]$,
and we have introduced the notation $\rho^{(\omega)}$ to indicate the
particular frame state that is used for conditioning. The product relative
states arise from product states on the composite system; in the case of a
localizable frame in a localised state they generalize the alignable states of
[28] (we recall that $\yen^{\mathcal{R}}_{*}$ factors through the $G$-action).
Notice again, as in Eq. (12), that the state $\rho^{(\omega)}$ depends only on
the measure $\mu_{\omega}^{\mathsf{E}_{\mathcal{R}}}$ once $U_{\mathcal{S}}$
and $\rho$ are given.

######  Proposition 3.18.

Product relative states satisfy the following symmetry condition

$\rho^{(h.\omega)}=(h^{-1}.\rho)^{(\omega)}$ (15)

###### Proof.

We calculate:

$\rho^{(h.\omega)}=\int_{G}d\mu^{\mathsf{E}_{\mathcal{R}}}_{h.\omega}(g)g.\rho=\int_{G}d\mu^{\mathsf{E}_{\mathcal{R}}}_{\omega}(hg)g.\rho\\\
=\int_{G}d\mu^{\mathsf{E}_{\mathcal{R}}}_{\omega}(g^{\prime})(h^{-1}g^{\prime}).\rho=\int_{G}d\mu^{\mathsf{E}_{\mathcal{R}}}_{\omega}(g^{\prime})g^{\prime}.(h^{-1}.\rho)=(h^{-1}.\rho)^{(\omega)}.$

We have changed the integration variable $g^{\prime}=hg$ and used the fact
that
$\mu^{\mathsf{E}_{\mathcal{R}}}_{h.\omega}(g)=\mu^{\mathsf{E}_{\mathcal{R}}}_{\omega}(hg)$
which follows directly from the covariance of $\mathsf{E}_{\mathcal{R}}$. ∎

Thus reorienting the frame by $h\in G$ is equivalent to reorienting the system
by $h^{-1}\in G$. This represents a type of ‘active-versus-passive’
equivalence for quantum frame reorientation.

The next proposition demonstrates the plausible claim that invariant system
states are defined without reference to an external frame or, more precisely,
are independent from the chosen reference.

######  Proposition 3.19.

Let $\rho\in\mathcal{S}(\mathcal{H}_{\mathcal{S}})^{G}$. Then
$\rho^{(\omega)}=\rho$ for any $\omega$, and any choice of frame
$\mathcal{R}$.

###### Proof.

We calculate:

$\rho^{(\omega)}=\int_{G}d\mu_{\omega}^{\mathsf{E}_{\mathcal{R}}}(g)U_{\mathcal{S}}(g)^{*}\rho
U_{\mathcal{S}}(g)=\int_{G}d\mu_{\omega}^{\mathsf{E}_{\mathcal{R}}}(g)\rho=\rho,$

where we only needed to use invariance of $\rho$ and normalization of
$\mathsf{E}_{\mathcal{R}}$. ∎

We recall that the invariance of $\rho$ typically requires $G$ compact. It
follows directly from Prop. 3.18 that $\rho^{(h.\omega)}=\rho^{(\omega)}$ if
$\rho$ is invariant, showing directly how the relative description of
$\mathcal{S}$ in terms of product relative states is not sensitive to frame
reorientations if $\rho$ is invariant, as one would expect. We note that the
frame-independence of invariant states follows directly from our framework,
and does not need to be stated as an assumption. Observe that Prop. 3.19 can
be immediately generalized in order to pertain to non-compact $G$, or more
precisely, to the setting where we do not demand that $\rho$ is literally
invariant under the representation of $G$, but instead is identified with the
class of $G$-indistinguishable density operators.

######  Proposition 3.20.

Let $\rho\in\mathcal{S}(\mathcal{H}_{\mathcal{S}})$. Then
$[\rho^{(\omega)}]_{G}=[\rho]_{G}$ for any $\omega$, and any choice of frame
$\mathcal{R}$.

###### Proof.

Set $A\in B(\mathcal{H}_{\mathcal{S}})^{G}$. Then

$\tr[\rho^{(\omega)}A]=\tr[\int_{G}d\mu_{\omega}^{\mathsf{E}_{\mathcal{R}}}(g)U_{\mathcal{S}}(g)^{*}\rho
U_{\mathcal{S}}(g)A]=\int_{G}\tr[d\mu_{\omega}^{\mathsf{E}_{\mathcal{R}}}(g)\rho
U_{\mathcal{S}}(g)AU_{\mathcal{S}}(g)^{*}]=\tr[\rho A],$

using the cyclicity of the trace, the invariance of $A$ and again the
normalisation of the measure. ∎

Another plausible intuition—that a reference in an invariant state can only
give rise to invariant relative states—is confirmed by the following
proposition.

######  Proposition 3.21.

Suppose $\omega\in\mathcal{S}(\mathcal{H}_{\mathcal{R}})^{G}$. Then
$\rho^{(\omega)}=\mathcal{G}(\rho)$ for any $\rho$.

###### Proof.

If $\omega$ is invariant, then $\mu^{\mathsf{E}_{\mathcal{R}}}_{\omega}$ is
Haar measure, denoted $d\mu$ as before. We then have

$\rho^{(\omega)}=\int_{G}d\mu(g)U_{\mathcal{S}}(g)^{*}\rho
U_{\mathcal{S}}(g)=\mathcal{G}(\rho).$

∎

Thus if the reference is in an invariant state, the only relative states
defined with respect to it are also invariant.

Now consider $\mathcal{H}_{\mathcal{R}}=\mathbb{C}$. As already noted, any
unitary representation on $\mathbb{C}$ is trivial, the only state is trivially
invariant, and a covariant POVM $\mathsf{E}:\mathcal{B}(G)\to
B(\mathbb{C})\simeq\mathbb{C}$ is a measure $\mu_{\mathsf{E}}$ on $G$
satisfying $\mu_{\mathsf{E}}(g.X)=g.\mu_{\mathsf{E}}(X)=\mu_{\mathsf{E}}(X)$,
and is thus the normalized Haar measure. There is a unique such covariant POVM
iff $G$ is compact, and in that case, $\yen^{\mathcal{R}}_{*}=\mathcal{G}$.
Therefore, there will only be invariant states in the image. From this we
conclude that in general not all states are relative states, and cannot be
approximated by relative states.

###  3.5 Localizing the Reference

The structure of the $\omega$-conditioned $\mathcal{R}$-relative observables
is dictated by the frame $\mathcal{R}$ and the state $\omega$, as has been
investigated in e.g. [7, 18]. Those works analyzed the agreement between the
standard description of $\mathcal{S}$ and the relative one described by the
compound system $\mathcal{S}$ and $\mathcal{R}$, in which it was found that
the localization/delocalization of $\omega$ was the key ingredient (for
good/not good agreement). The main positive result in this vein (i.e.,
addressing the question of good agreement between relative/non-relative
quantities) is Thm. 1. in [7], which we now generalize to arbitrary locally
compact $G$:

######  Theorem 3.22.

Let
$\mathcal{R}=(U_{\mathcal{R}},\mathsf{E}_{\mathcal{R}},\mathcal{H}_{\mathcal{R}})$
be a localizable principal frame and $(\omega_{n})$ a localizing sequence
centered at $e\in G$. Then for any $A_{\mathcal{S}}\in
B(\mathcal{H}_{\mathcal{S}})$ we have

$\lim_{n\to\infty}(\Gamma_{\omega_{n}}\circ\yen^{\mathcal{R}})(A_{\mathcal{S}})=A_{\mathcal{S}},$
(16)

where the limit is understood as usual in the ultraweak sense.

###### Proof.

It is enough to check the agreement of expectation values of both sides of Eq.
(16). Thus take $\rho\in\mathcal{S}(\mathcal{H}_{\mathcal{S}})$ and calculate

$\tr[\rho(\Gamma_{\omega_{n}}\circ\yen)(A_{\mathcal{S}})]=\tr[\rho\int_{G}d\mu^{\mathsf{E}_{\mathcal{R}}}_{\omega_{n}}(g)g.A_{\mathcal{S}}]=\int_{G}d\mu^{\mathsf{E}_{\mathcal{R}}}_{\omega_{n}}(g)\tr[\rho(g.A_{\mathcal{S}})]$

The function $g\mapsto\tr[\rho(g.A_{\mathcal{S}})]$ is continuous and bounded,
and by Prop. 2.13 the sequence of measures
$(\mu^{\mathsf{E}_{\mathcal{R}}}_{\omega_{n}})$ converges weakly to
$\delta_{e}$, so by the portemanteau theorem we have:

$\lim_{n\to\infty}\int_{G}d\mu^{\mathsf{E}_{\mathcal{R}}}_{\omega_{n}}(g)\tr[\rho(g.A_{\mathcal{S}})]=\tr[\rho
A_{\mathcal{S}}];$

therefore the sequence of operators
$(\Gamma_{\omega_{n}}\circ\yen^{\mathcal{R}})(A_{\mathcal{S}})$ converges to
$A_{\mathcal{S}}$ in the ultraweak topology. ∎

Note that if $G$ is finite and the frame is ideal, no limiting procedure is
needed and the agreement is exact, i.e., we can choose
$\omega^{\mathcal{R}}=\outerproduct{e}{e}$ and it holds that
$(\Gamma_{\omega}\circ\yen^{\mathcal{R}})(A_{\mathcal{S}})=A_{\mathcal{S}}$.
Thm. 3.22 above allows us to prove the following.

######  Proposition 3.23.

The relativization maps $\yen^{\mathcal{R}}:B(\mathcal{H}_{\mathcal{S}})\to
B(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})^{G}$ for
localizable frames $\mathcal{R}$ are isometric.

###### Proof.

Given $A\in B(\mathcal{H}_{\mathcal{S}})$ it is shown in [7] that
$||\yen^{\mathcal{R}}(A_{\mathcal{S}})||\leq||A_{\mathcal{S}}||$, so we only
need to prove the converse inequality. By Th. 3.22 there is a localizing
sequence of states $(\omega_{n})$ such that for all
$\rho\in\mathcal{S}(\mathcal{H}_{\mathcal{S}})$

$|\tr[\rho
A_{\mathcal{S}}]|=\lim_{n\to\infty}|\tr[\rho(\Gamma_{\omega_{n}}\circ\yen^{\mathcal{R}})(A_{\mathcal{S}})]|=\lim_{n\to\infty}|\tr[(\omega^{n}\otimes\rho)\yen^{\mathcal{R}}(A_{\mathcal{S}})]|\leq\sup_{\Omega\in\mathcal{S}(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})}|\tr[\Omega\yen^{\mathcal{R}}(A_{\mathcal{S}})]|=||\yen^{\mathcal{R}}(A_{\mathcal{S}})||.$

We then have $||A_{\mathcal{S}}||=\sup_{\rho}|\tr[\rho
A_{\mathcal{S}}]|\leq||\yen^{\mathcal{R}}(A_{\mathcal{S}})||$. ∎

We may also consider product relative states. Indeed, for finite group $G$ and
the frame $\mathsf{E}_{\mathcal{R}}$ given by

$G\ni g\mapsto P(g)=\outerproduct{g}{g}\in B(L^{2}(G)),$

any state is an $\ket{e}$-product relative state since

$\yen^{\mathcal{R}}_{*}(\outerproduct{e}{e}\otimes\rho)=\sum_{g\in
G}\tr[P(g)\outerproduct{e}{e}]g.\rho=\sum_{g\in G}\delta_{ge}g.\rho_{S}=\rho.$
(17)

This is the setting considered in [2], and states of the form
$\outerproduct{e}{e}\otimes\rho$ are called “aligned" (to the identity) [28].
Since $\yen^{\mathcal{R}}_{*}$ is constant on orbits, the above calculation is
not sensitive to whether the state is aligned or only alignable [28]. In this
work, we can make the intuition considered in [1, 2] (of the frame state being
‘localized at the identity’) rigorous in the general context of locally
compact groups through the use of localizing sequences. Dualizing Th. 3.22 for
a localizing sequence $(\omega_{n})$ centred at $e\in G$ and an arbitrary
$\rho\in\mathcal{S}(\mathcal{H}_{\mathcal{S}})$, we have

$\lim_{n\to\infty}\yen^{\mathcal{R}}_{*}(\omega_{n}\otimes\rho)=\lim_{n\to\infty}\int_{G}d\mu^{\mathsf{E}_{\mathcal{R}}}_{\omega_{n}}(g)g.\rho=\rho.$
(18)

Note that the same result holds if we replace $\omega_{n}\otimes\rho$ with
$h.(\omega_{n}\otimes\rho)$.

Thus we arrive at the following proposition:

######  Proposition 3.24.

Let $\mathcal{R}$ be a localizable frame. Then
$\mathcal{S}(\mathcal{H}_{\mathcal{S}})^{\mathcal{R}}$ is dense in the
operational topology on $\mathcal{S}(\mathcal{H}_{\mathcal{S}})$.

Thus for localizable frames, any state of $\mathcal{S}$ can be arbitrarily
well approximated by a sequence of relative states of the form
$\yen^{\mathcal{R}}_{*}(\omega_{n}\otimes\rho)$, thus (to arbitrary
approximation) ‘all states are relative states’ when the frame is localizable.
This justifies the free use of statements of the form ‘the state relative to
the frame is…’ in other work, showing that the localizability, and the actual
localization of the frame state is needed, and that the understanding of
$\rho$ relative to $\outerproduct{e}{e}$ takes the form
$\outerproduct{e}{e}\otimes\rho$ can be justified on operational
considerations.

To summarise the story so far, we have seen that for localizable frames, the
standard kinematics of quantum mechanics is recovered in the operational sense
as a limiting procedure of localizing the state of the reference. Given that
typically a large Hilbert space dimension is required for localizability, this
points to a sort of classicality requirement on the frame. There is also a
kind of converse to this statement proven in [18]: if $\mathcal{R}$ is not
localizable, there is a lower bound on the difference between an effect
$\mathsf{F}_{\mathcal{S}}$ and
$(\Gamma_{\omega}\circ\yen^{\mathcal{R}})(\mathsf{F}_{\mathcal{S}})$. We
interpret this as follows (see also [7] for a more detailed account): in the
presence of a ‘good’ frame, the ordinary framework of quantum mechanics based
on $B(\mathcal{H}_{\mathcal{S}})$ (with its states and observables) captures
arbitrarily accurately the true, relative world described by
$B(\mathcal{H}_{\mathcal{S}})^{\mathcal{R}}$ upon localizing the frame state
$\omega$, thereby identifying
$B(\mathcal{H}_{\mathcal{S}})^{\mathcal{R}}_{\omega}$ with
$B(\mathcal{H}_{\mathcal{S}})$. This, we suspect, is the reason that
$B(\mathcal{H}_{\mathcal{S}})$ (with its states and observables) was
discovered to be the correct description of a quantum system
$\mathcal{S}$—sufficiently good frames are typically used in quantum
experiments. Therefore $B(\mathcal{H}_{\mathcal{S}})$ with its state space
_actually_ describes the relation between a quantum system and a ‘good’ frame.
By contrast, in the setting that the frame is not localizable, or the state is
not localized, there are elements of $B(\mathcal{H}_{\mathcal{S}})$ which can
never capture experimentally realisable measurement statistics for that given
experimental arrangement.

##  4 Quantum Reference Frame Transformations

In this section, we consider a system and two frames, and provide the
transformation rule mapping the state description relative to one frame to
that relative to the other. The starting point is the frame-independent
invariant algebra $B(\mathcal{H}_{\mathcal{T}})^{G}$, which is the ‘universe
of discourse’ for the _internal quantum reference frames programme_ as
envisioned in this work (though differs from e.g. [44, 3] where the algebra of
operators on the _physical Hilbert space_ is chosen). This is in
contradistinction to e.g. [6], which is concerned with recovering the state of
a quantum system described relative to one frame, given that same state is
initially described relative to another frame.

The idea is to choose some subsystem (the ‘initial frame’) with respect to
which the other subsystems can be described, and find a means by which to
change the description relative to that subsystem to a description relative to
another subsystem, also internal to the given setup. Therefore it is assumed
that $\mathcal{H}_{\mathcal{T}}$ decomposes as
$\mathcal{H}_{\mathcal{T}}\cong\mathcal{H}_{\mathcal{R}_{1}}\otimes\mathcal{H}_{\mathcal{R}_{2}}\otimes\mathcal{H}_{\mathcal{S}}$,
on which we have as usual a strongly continuous unitary representation
$U_{\mathcal{T}}:G\to B(\mathcal{H}_{\mathcal{T}})$, such that
$U_{\mathcal{T}}=U_{1}\otimes U_{2}\otimes U_{\mathcal{S}}$. Since
$\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ are understood to be quantum reference
frames, each comes equipped with a system of covariance based on $G$, denoted
$\mathsf{E}_{1}$ and $\mathsf{E}_{2}$ respectively.

In this paper we analyze the quantum reference frame transformations in the
case where the initial frame $\mathcal{R}_{1}$ is localizable. A central
observation of the perspective-neutral approach (see e.g. [3]) is that to
transform between relative descriptions, one must pass through a ‘global’
description, given as the physical Hilbert space comprising the system and all
frames; the framework here follows this point of view, but differs in the
implementation. We will also see that the frame change we provide captures the
intuition given in [2] \- that to pass from the relative state to the
corresponding ‘global’ state we need to ‘attach the identity state’. Whilst
the states $\ket{e}$ used in [2] are not available as normal states in the
case of continuous groups, we can make the idea precise in the context of
arbitrary locally compact topological groups using localizing sequences as we
have seen at various points already. Indeed, we may use Eq. (18), which we
recall states that for a localizing sequence $(\omega_{n})$ centred at the
identity

$\lim_{n\to\infty}\yen^{\mathcal{R}}_{*}\left(\omega_{n}\otimes\Omega^{\mathcal{R}}\right)=\lim_{n\to\infty}\int_{G}d\mu_{\omega_{n}}^{\mathsf{E}_{\mathcal{R}}}(g)g.\Omega^{\mathcal{R}}=\Omega^{\mathcal{R}},$
(19)

to see that in the case of a localizable principal frame $\mathcal{R}$, any
$\mathcal{R}$-relative state can be obtained as the limit of a sequence of
$\omega_{n}$-product $\mathcal{R}$-relative states as above. This is the idea
behind the lifting construction, which we introduce in the next subsection.
After that, we explain how the operational equivalence is employed in the
context of a pair of frames, defining the framed relative descriptions, to
finally be able to provide the localizable frame change transformations as
state space maps between the relevant operational state spaces.

###  4.1 Lifting

Consider a unitary (strongly continuous) representation $U_{\mathcal{T}}:G\to
B(\mathcal{H}_{\mathcal{T}})$ of a (locally compact) group $G$ on a
(separable) Hilbert space $\mathcal{H}_{\mathcal{T}}$, with two subsystems
$\mathcal{S}$ and $\mathcal{R}$ distinguished, i.e.
$\mathcal{H}_{\mathcal{T}}\cong\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}}$.
The invariants in $B(\mathcal{H}_{\mathcal{T}})^{G}$ may be conditioned upon
states of the reference system by applying the restriction maps. Again,
assuming that the factorization into
$\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}}$ respects the
$G$-action, i.e., $U_{\mathcal{T}}=U_{\mathcal{R}}\otimes U_{\mathcal{S}}$,
and $\mathcal{R}$ is a frame equipped with a covariant POVM
$\mathsf{E}_{\mathcal{R}}:\mathcal{B}(G)\to B(\mathcal{H}_{\mathcal{R}})$, we
can relativize the restricted operators, arriving at a _subset_ of the
relative ones.

Thus we introduce the map

$\Gamma^{\mathcal{R}}_{\omega}:=\yen^{\mathcal{R}}\circ\Gamma_{\omega}\circ
i^{G}:B(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})^{G}\to
B(\mathcal{H}_{\mathcal{S}})^{\mathcal{R}},$ (20)

where $i^{G}$ denotes the inclusion
$B(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})^{G}\hookrightarrow
B(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})$. As a
composition of such maps $\Gamma^{\mathcal{R}}_{\omega}$ is unital, normal,
and completely positive. It may be understood as providing a relative
‘version’ of a given invariant.

The discrepancy between an invariant effect $\mathsf{F}$ and its relative
version $\Gamma^{\mathcal{R}}_{\omega}(\mathsf{F})$, given a frame prepared in
the state $\omega$, is quantified as
$||\Gamma^{\mathcal{R}}_{\omega}(\mathsf{F})-\mathsf{F}||$, which is an
operational measure since for an effect
$\mathsf{F}\in\mathcal{E}(\mathcal{H}_{\mathcal{S}})$,
$||\mathsf{F}||=\sup_{\rho\in\mathcal{S}(\mathcal{H}_{\mathcal{S}}})|\tr[\rho\mathsf{F}]|$,
i.e., the discrepancy has a probabilistic interpretation. The best case is
estimated as
$\inf_{\omega}||\Gamma^{\mathcal{R}}_{\omega}(\mathsf{F})-\mathsf{F}||$. If
$\mathcal{R}$ is localizable and $\mathsf{F}$ is already relative, it is
readily seen that this discrepancy can be made arbitrarily small.

######  Definition 4.1.

The predual of $\Gamma^{\mathcal{R}}_{\omega}$ pre-composed with the state
space isomorphism
$\mathcal{S}(\mathcal{H}_{\mathcal{S}})^{\mathcal{R}}\cong\mathcal{S}(\mathcal{H}_{\mathcal{S}})_{\mathcal{R}}$
(c.f. 3.11) will be denoted by

$\mathcal{L}^{\mathcal{R}}_{\omega}:=(\Gamma^{\mathcal{R}}_{\omega})_{*}\circ
y_{\mathcal{R}}:\mathcal{S}(\mathcal{H}_{\mathcal{S}})^{\mathcal{R}}\to\mathcal{S}(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})_{G}$

and called the $\mathcal{R}$-_relative_ $\omega$-_lifting map_.

The lifting procedure allows for the ‘attachment’ of the state of a frame
state to the state of the system, whilst respecting the symmetry-induced
operational equivalence class structure. In the case of a localizable frame
$\mathcal{R}$, using $\mathcal{L}^{\mathcal{R}}_{\omega_{n}}$ we can lift an
arbitrary $\mathcal{R}$-relative state to a state on the invariant algebra (as
the $G$-equivalence classes), which gives back the initial relative state to
arbitrary precision, upon applying $\yen^{\mathcal{R}}_{*}$. Indeed, for
$\Omega\in\mathcal{S}(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})$
and $\Omega^{\mathcal{R}}=\yen^{\mathcal{R}}_{*}(\Omega)$ we have

$\mathcal{L}^{\mathcal{R}}_{\omega}:\mathcal{S}(\mathcal{H}_{\mathcal{S}})^{\mathcal{R}}\ni\Omega^{\mathcal{R}}\mapsto[\omega\otimes\Omega^{\mathcal{R}}]_{G}\in\mathcal{S}(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})_{G},$

and given a localizing sequence $(\omega_{n})$ centred at $e\in G$ we get (see
(19))

$\lim_{n\to\infty}\yen^{\mathcal{R}}_{*}\circ\mathcal{L}^{\mathcal{R}}_{\omega_{n}}(\Omega^{\mathcal{R}})=\lim_{n\to\infty}\yen^{\mathcal{R}}_{*}[\omega_{n}\otimes\Omega^{\mathcal{R}}]_{G}=\Omega^{\mathcal{R}}.$

Thus in the case of a localizable frame $\mathcal{R}$, the lifting maps taken
with respect to a localizing sequence of frame states provide an approximate
right inverse to the predual $\yen^{\mathcal{R}}_{*}$ map, which gives rise to
the relative states represented as states of the system. This is in analogy to
the inverse of the reduction map of the perspective-neutral approach [3].
Since
$y_{\mathcal{R}}:\mathcal{S}(\mathcal{H}_{\mathcal{S}})^{\mathcal{R}}\xrightarrow{\sim}\mathcal{S}(\mathcal{H}_{\mathcal{S}})_{\mathcal{R}}$
is a state space isomorphism, which in general does not lift to an isomorphism
at the level of Banach spaces (see 3.11), we are only permitted to lift states
(and not general trace class operators).

As usual, given a unital normal positive map we may define the corresponding
operator space—the ultraweak closure of the image of the map—and the
corresponding predual space with the (total convex) state space isomorphic to
the image of the predual map on states. Here the $\omega$-lifted
$\mathcal{R}$-relative states are separated by the image of
$\Gamma^{\mathcal{R}}_{\omega}$.

The channels $\Gamma^{\mathcal{R}}_{\omega}$ and
$\mathcal{L}^{\mathcal{R}}_{\omega}$ can be illustrated on the following pair
of commuting diagrams. Since $y_{\mathcal{R}}$ is a state space isomorphism,
we restrict the predualization of the first diagram (corresponding to the
bottom-left triangle of the second diagram) to the state spaces, in order to
also represent the lifting. The map (20) can be depicted on the following
diagram

${B(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})^{G}}$${B(\mathcal{H}_{\mathcal{S}})^{\mathcal{R}}}$${B(\mathcal{H}_{\mathcal{S}})}$$\scriptstyle{\Gamma^{\mathcal{R}}_{\omega}}$$\scriptstyle{\Gamma_{\omega}\circ
i^{G}}$$\scriptstyle{\yen^{\mathcal{R}}}$

,

with the predual, restricted to state spaces and including also the
isomorphism $y_{\mathcal{R}}$ and the lifting map, taking the form

${\mathcal{S}(\mathcal{H}_{\mathcal{R}}\otimes\mathcal{H}_{\mathcal{S}})_{G}}$${\mathcal{S}(\mathcal{H}_{\mathcal{S}})_{\mathcal{R}}}$${\mathcal{S}(\mathcal{H}_{\mathcal{S}})^{\mathcal{R}}}$${\mathcal{S}(\mathcal{H}_{\mathcal{S}})}$$\scriptstyle{(\Gamma_{\omega})_{*}}$$\scriptstyle{\yen^{\mathcal{R}}_{*}}$$\scriptstyle{y_{\mathcal{R}}}$$\scriptstyle{\mathcal{L}_{\omega}^{\mathcal{R}}}$$\scriptstyle{i_{\mathcal{R}}}$$\scriptstyle{\pi_{G}\circ\mathcal{V}_{\omega}}$

.

###  4.2 Framed Relative Observables and States

We now combine the framing with the relative state construction to give the
framed relative states, which are the final ingredient required for the
provision of the frame change map.

######  Definition 4.2.

Given a pair of frames, $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$, and a system
$\mathcal{S}$, we will denote by

$B(\mathcal{H}_{\mathcal{R}_{2}}\otimes\mathcal{H}_{\mathcal{S}})^{\mathcal{R}_{1},\mathsf{E}_{2}}={\rm
span}\\{\yen^{\mathcal{R}_{1}}(\mathsf{E}_{2}(X)\otimes\mathsf{F}_{\mathcal{S}})\hskip
3.0pt|\hskip 3.0ptX\in\mathcal{B}(G),\hskip
3.0pt\mathsf{F}_{\mathcal{S}}\in\mathcal{E}(\mathcal{H}_{\mathcal{S}})\\}^{\rm
cl}.$ (21)

the (ultraweakly closed) operator space of $\mathcal{R}_{1}$-_relativized_
$\mathsf{E}_{2}$-_framed operators_.

######  Definition 4.3.

The $(\mathcal{R}_{1},\mathsf{E}_{2})$-equivalence relation on
$\mathcal{T}(\mathcal{H}_{\mathcal{R}_{1}}\otimes\mathcal{H}_{\mathcal{R}_{2}}\otimes\mathcal{H}_{\mathcal{S}})$,
denoted $T\sim_{(\mathcal{R}_{1},\mathsf{E}_{2})}T^{\prime}$, is defined as
the operational equivalence with respect to
$B(\mathcal{H}_{\mathcal{R}_{2}}\otimes\mathcal{H}_{\mathcal{S}})^{\mathcal{R}_{1},\mathsf{E}_{2}}$.
Elements of the corresponding operational state space

$\mathcal{S}(\mathcal{H}_{\mathcal{R}_{2}}\otimes\mathcal{H}_{\mathcal{S}})_{\mathcal{R}_{1},\mathsf{E}_{2}}:=\mathcal{S}(\mathcal{H}_{\mathcal{R}_{1}}\otimes\mathcal{H}_{\mathcal{R}_{2}}\otimes\mathcal{H}_{\mathcal{S}})/\sim_{(\mathcal{R}_{1},\mathsf{E}_{2})}$

will be called $\mathsf{E}_{2}$-_framed_ $\mathcal{R}_{1}$-_relative states_.

We omit here the obvious definition of $\mathsf{E}_{2}$-framed
$\mathcal{R}_{1}$-relative trace class operators and the corresponding duality
result since they won’t be needed. As is easily confirmed, the
$\mathsf{E}_{2}$-framed $\mathcal{R}_{1}$-relative state space can
equivalently be seen as the $\sim_{\mathsf{E}_{2}}$-quotient of the
$\mathcal{R}_{1}$-relative state space, i.e.

$\mathcal{S}(\mathcal{H}_{\mathcal{R}_{2}}\otimes\mathcal{H}_{\mathcal{S}})^{\mathcal{R}_{1}}_{\mathsf{E}_{2}}:=\mathcal{S}(\mathcal{H}_{\mathcal{R}_{2}}\otimes\mathcal{H}_{\mathcal{S}})^{\mathcal{R}_{1}}/\sim_{\mathsf{E}_{2}}\cong\mathcal{S}(\mathcal{H}_{\mathcal{R}_{2}}\otimes\mathcal{H}_{\mathcal{S}})_{\mathcal{R}_{1},\mathsf{E}_{2}},$

where
$\mathcal{S}(\mathcal{H}_{\mathcal{R}_{2}}\otimes\mathcal{H}_{\mathcal{S}})^{\mathcal{R}_{1}}\subseteq\mathcal{S}(\mathcal{H}_{\mathcal{R}_{2}}\otimes\mathcal{H}_{\mathcal{S}})$;
this is the perspective that is most useful in the context of frame
transformations, since it allows for the application of the lifting procedure.
The corresponding affine quotient projection will be denoted by

$\pi^{\mathcal{R}_{1}}_{\mathsf{E}_{2}}:\mathcal{S}(\mathcal{H}_{\mathcal{R}_{2}}\otimes\mathcal{H}_{\mathcal{S}})^{\mathcal{R}_{1}}\twoheadrightarrow\mathcal{S}(\mathcal{H}_{\mathcal{R}_{2}}\otimes\mathcal{H}_{\mathcal{S}})^{\mathcal{R}_{1}}_{\mathsf{E}_{2}},$

while $[\Omega^{\mathcal{R}_{1}}]_{\mathsf{E}_{2}}$ refers to the
$\mathsf{E}_{2}$-equivalence classes of $\mathcal{R}_{1}$-relative states in
$\mathcal{S}(\mathcal{H}_{\mathcal{R}_{2}}\otimes\mathcal{H}_{\mathcal{S}})^{\mathcal{R}}\subseteq\mathcal{S}(\mathcal{H}_{\mathcal{R}_{2}}\otimes\mathcal{H}_{\mathcal{S}})$.

Note that in the absence of the system $\mathcal{S}$, we have

$\mathcal{S}(\mathcal{H}_{\mathcal{R}_{2}})^{\mathcal{R}_{1}}_{\mathsf{E}_{2}}=\mathcal{S}(\mathcal{H}_{\mathcal{R}_{1}}\otimes\mathcal{H}_{\mathcal{R}_{2}})/\sim_{\mathsf{E}_{1}*\mathsf{E}_{2}},$

where $\sim_{\mathsf{E}_{2}*\mathsf{E}_{1}}$ denotes operational equivalence
with respect to the relative orientation observable
$\mathsf{E}_{2}*\mathsf{E}_{1}=\yen^{\mathcal{R}_{1}}\circ\mathsf{E}_{2}$.
Thus in this situation, the $\mathsf{E}_{2}$-framed $\mathcal{R}_{1}$-relative
states only allow for the measurement of the relative orientation observable.
It is easy to see that in the general case, the framed relative states also
allow one to separate states with respect to the observable
$\mathsf{E}_{2}*\mathsf{E}_{2}\otimes\mathbb{1}_{\mathcal{S}}$. Indeed
$\yen^{\mathcal{R}}(\mathsf{E}_{2}(X)\otimes\mathbb{1}_{\mathcal{S}})=\mathsf{E}_{2}*\mathsf{E}_{2}(X)\otimes\mathbb{1}_{\mathcal{S}}\in
B(\mathcal{H}_{\mathcal{R}_{2}}\otimes\mathcal{H}_{\mathcal{S}})^{\mathcal{R}_{1},\mathsf{E}_{2}}$.

###  4.3 Changing Reference

We now turn to the frame transformations, given as maps between the
$\mathsf{E}_{2}$-framed $\mathcal{R}_{1}$-relative description and the
$\mathsf{E}_{1}$-framed $\mathcal{R}_{2}$-relative one:

$\Phi_{1\to
2}:\mathcal{S}(\mathcal{H}_{\mathcal{R}_{2}}\otimes\mathcal{H}_{\mathcal{S}})^{\mathcal{R}_{1}}_{\mathsf{E}_{2}}\to\mathcal{S}(\mathcal{H}_{\mathcal{R}_{1}}\otimes\mathcal{H}_{\mathcal{S}})^{\mathcal{R}_{2}}_{\mathsf{E}_{1}}.$
(22)

Since relative operators are framed, no operational distinction arises when
taking the $\mathsf{E}_{2}$-equivalence class on top of the relative state
space in the domain. As such, the frame change is given with domain and
codomain as in (22) and not the relative state spaces themselves—it is the
additional quotients that allow for the given map to be invertible in the high
localisation limit.

In analogy to the perspective-neutral approach [3], the frame change map is
constructed by passing through the ‘global’ description which involves the
system and both frames, whilst making sure to respect the symmetry, which is
given in terms of
$\mathcal{S}(\mathcal{H}_{\mathcal{R}_{1}}\otimes\mathcal{H}_{\mathcal{R}_{2}}\otimes\mathcal{H}_{\mathcal{S}})_{G}$.
We first employ the lift $\mathcal{L}_{\omega}^{\mathcal{R}_{1}}$, followed by
the application of $\yen^{\mathcal{R}_{2}}_{*}$, giving a state relative to
$\mathcal{R}_{2}$. Imposing the $\mathsf{E}_{1}$ and $\mathsf{E}_{2}$
equivalences on top of that and localizing the lifting gives a consistent
notion of invertible operational frame transformations for localizable frames.
Indeed, we obtain the following (for brevity whenever we write ‘localizing
sequence’ without further qualification, this is centred at $e\in G$):

######  Definition 4.4.

Consider a pair of principal frames $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$
with $\mathcal{R}_{1}$ localizable and denote by $\mathsf{E}_{1}$ and
$\mathsf{E}_{2}$ their frame observables. For a system $\mathcal{S}$ the map

$\Phi^{\rm loc}_{1\to
2}:=\lim_{n\to\infty}\pi^{\mathcal{R}_{2}}_{\mathsf{E}_{1}}\circ\yen^{\mathcal{R}_{2}}_{*}\circ\mathcal{L}^{\mathcal{R}_{1}}_{\omega_{n}}:\mathcal{S}(\mathcal{H}_{\mathcal{R}_{2}}\otimes\mathcal{H}_{\mathcal{S}})^{\mathcal{R}_{1}}_{\mathsf{E}_{2}}\to\mathcal{S}(\mathcal{H}_{\mathcal{R}_{1}}\otimes\mathcal{H}_{\mathcal{S}})^{\mathcal{R}_{2}}_{\mathsf{E}_{1}},$

where $(\omega_{n})$ is any localizing sequence for $\mathsf{E}_{1}$, will be
called an _(internal) localized frame transformation_.

Concretely, the localized frame transformation evaluated on an
$\mathsf{E}_{2}$-equivalence class of $\mathcal{R}_{1}$-relative states gives

$\Phi^{\rm loc}_{1\to
2}:[\Omega^{\mathcal{R}_{1}}]_{\mathsf{E}_{2}}\mapsto\lim_{n\to\infty}[\yen^{\mathcal{R}_{2}}_{*}\circ\mathcal{L}^{\mathcal{R}_{1}}_{\omega_{n}}(\Omega^{\mathcal{R}_{1}})]_{\mathsf{E}_{1}}=\lim_{n\to\infty}[\yen^{\mathcal{R}_{2}}_{*}(\omega_{n}\otimes\Omega^{\mathcal{R}_{1}})]_{\mathsf{E}_{1}}.$
(23)

######  Theorem 4.5.

Localized frame transformations are well-defined state space maps making the
following diagram commute

${\mathcal{S}(\mathcal{H}_{\mathcal{R}_{1}}\otimes\mathcal{H}_{\mathcal{R}_{2}}\otimes\mathcal{H}_{\mathcal{S}})_{G}}$${\mathcal{S}(\mathcal{H}_{\mathcal{R}_{2}}\otimes\mathcal{H}_{\mathcal{S}})^{R_{1}}_{\mathsf{E}_{2}}}$${\mathcal{S}(\mathcal{H}_{\mathcal{R}_{1}}\otimes\mathcal{H}_{\mathcal{S}})^{R_{2}}_{\mathsf{E}_{1}}.}$$\scriptstyle{\pi^{\mathcal{R}_{1}}_{\mathsf{E}_{2}}\circ\yen^{\mathcal{R}_{1}}_{*}}$$\scriptstyle{\pi^{\mathcal{R}_{2}}_{\mathsf{E}_{1}}\circ\yen^{\mathcal{R}_{2}}_{*}}$$\scriptstyle{\Phi_{1\to
2}^{\rm loc}}$

If $\mathcal{R}_{2}$ is also localizable, the analoguosly defined map
$\Phi_{2\to 1}^{\rm loc}$ provides the inverse, i.e.

$\Phi_{2\to 1}^{\rm loc}\circ\Phi_{1\to 2}^{\rm
loc}=\text{Id}_{\mathcal{S}(\mathcal{H}_{\mathcal{R}_{2}}\otimes\mathcal{H}_{\mathcal{S}})^{R_{1}}_{\mathsf{E}_{2}}}.$

###### Proof.

See Appendix C.1. ∎

The formula
$\lim_{n\to\infty}\pi^{\mathcal{R}_{2}}_{\mathsf{E}_{1}}\circ\yen^{\mathcal{R}_{2}}_{*}\circ\mathcal{L}^{\mathcal{R}_{1}}_{\omega_{n}}$
gives a well-defined map
$\mathcal{S}(\mathcal{H}_{\mathcal{R}_{2}}\otimes\mathcal{H}_{\mathcal{S}})^{\mathcal{R}_{1}}\to\mathcal{S}(\mathcal{H}_{\mathcal{R}_{1}}\otimes\mathcal{H}_{\mathcal{S}})^{\mathcal{R}_{2}}_{\mathsf{E}_{1}}$,
so that we could think about about transforming $\mathcal{R}_{1}$-relative
states to $\mathsf{E}_{1}$-framed $\mathcal{R}_{2}$-relative states. Moreover,
without assuming localizability of $\mathcal{R}_{1}$ we can define an obvious
non-localized but instead _state-dependent_ analogue of the localized frame
transformations

$\Phi_{1\to
2}^{\omega}:=\pi^{\mathcal{R}_{2}}_{\mathsf{E}_{1}}\circ\yen^{\mathcal{R}_{2}}_{*}\circ\mathcal{L}^{\mathcal{R}_{1}}_{\omega}:\mathcal{S}(\mathcal{H}_{\mathcal{R}_{2}}\otimes\mathcal{H}_{\mathcal{S}})^{\mathcal{R}_{1}}\to\mathcal{S}(\mathcal{H}_{\mathcal{R}_{1}}\otimes\mathcal{H}_{\mathcal{S}})^{\mathcal{R}_{2}}_{\mathsf{E}_{1}}.$
(24)

Analyzing properties of this construction is left for future work. In case
$\mathcal{R}_{2}$ is not localized, the invertibility is lost, pointing to a
form of decoherence.

With three frames, i.e., with
$\mathcal{H}_{\mathcal{T}}\cong\mathcal{H}_{\mathcal{R}_{1}}\otimes\mathcal{H}_{\mathcal{R}_{2}}\otimes\mathcal{H}_{\mathcal{R}_{3}}\otimes\mathcal{H}_{\mathcal{S}}$,
three covariant POVMs $\mathsf{E}_{1},\mathsf{E}_{2}$ and $\mathsf{E}_{3}$ as
frame observables, and diagonal $G$-action on $\mathcal{H}_{\mathcal{T}}$, the
frame transformations compose in the following sense.

######  Theorem 4.6.

Assume $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ to be localizable principal
frames and let $\mathcal{R}_{3}$ be an arbitrary principal frame. Then

$\pi^{\mathcal{R}_{3}}_{\mathsf{E}_{2}}\circ\Phi^{\rm loc}_{1\to 3}=\Phi^{\rm
loc}_{2\to 3}\circ\Phi^{\rm loc}_{1\to 2}.$ (25)

The map $\pi^{\mathcal{R}_{3}}_{\mathsf{E}_{2}}$ denotes the
$\mathsf{E}_{2}$-framing projection from
$\mathcal{S}(\mathcal{H}_{\mathcal{R}_{1}}\otimes\mathcal{H}_{\mathcal{R}_{2}}\otimes\mathcal{H}_{\mathcal{S}})^{\mathcal{R}_{3}}_{\mathsf{E}_{1}}$
and is required in order to account for the choice of the second frame
observable.

###### Proof.

See Appendix C.2. ∎

###  4.4 Comparison to other work

It is worth making a brief comparison to the works [1, 2] (which have since
been dubbed ‘purely perspectival’ approaches), and the perspective-neutral
approach (e.g. [3]). We start with the former.

In [1] three copies of $L^{2}(\mathbb{R})$ are considered, with the compound
system given as $L^{2}(\mathbb{R})_{\mathcal{R}_{1}}\otimes
L^{2}(\mathbb{R})_{\mathcal{R}_{2}}\otimes