# Why standard entanglement theory is inappropriate for the study of Bell
scenarios

David Schmid  Perimeter Institute for Theoretical Physics, 31 Caroline Street
North, Waterloo, Ontario Canada N2L 2Y5  Institute for Quantum Computing and
Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario
N2L 3G1, Canada  Thomas C. Fraser  Perimeter Institute for Theoretical
Physics, 31 Caroline Street North, Waterloo, Ontario Canada N2L 2Y5  Institute
for Quantum Computing and Department of Physics and Astronomy, University of
Waterloo, Waterloo, Ontario N2L 3G1, Canada  Ravi Kunjwal  Centre for Quantum
Information and Communication, Ecole polytechnique de Bruxelles, CP 165,
Université libre de Bruxelles, 1050 Brussels, Belgium  Ana Belén Sainz
International Centre for Theory of Quantum Technologies, University of Gdańsk,
80-308 Gdańsk, Poland  Elie Wolfe  Perimeter Institute for Theoretical
Physics, 31 Caroline Street North, Waterloo, Ontario Canada N2L 2Y5  Robert W.
Spekkens  Perimeter Institute for Theoretical Physics, 31 Caroline Street
North, Waterloo, Ontario Canada N2L 2Y5

###### Abstract

A standard approach to quantifying resources is to determine which operations
on the resources are freely available and to deduce the ordering relation
among the resources that these operations induce. If the resource of interest
is the nonclassicality of the correlations embodied in a quantum state, that
is, entanglement, then it is typically presumed that the appropriate choice of
free operations is local operations and classical communication (LOCC). We
here argue that, in spite of the near-universal endorsement of the LOCC
paradigm by the quantum information community, this is the wrong choice for
one of the most prominent applications of entanglement theory, namely, the
study of Bell scenarios. The nonclassicality of correlations in such
scenarios, we argue, should be quantified instead by local operations and
shared randomness (LOSR). We support this thesis by showing that various
perverse features of the interplay between entanglement and nonlocality are
merely an artifact of the use of LOCC-entanglement and that the interplay
between LOSR-entanglement and nonlocality is natural and intuitive.
Specifically, we show that the LOSR paradigm (i) provides a resolution of the
“anomaly of nonlocality”, wherein partially entangled states exhibit more
nonlocality than maximally entangled states, (ii) entails a notion of genuine
multipartite entanglement that is distinct from the conventional one and which
is free of several of its pathological features, and (iii) makes possible a
resource-theoretic account of the self-testing of entangled states which
simplifies and generalizes prior results. Along the way, we derive some
fundamental results concerning the necessary and sufficient conditions for
convertibility between pure entangled states under LOSR and highlight some of
their consequences, such as the impossibility of catalysis for bipartite pure
states.

The term “entangled” was first used only for pure states of a composite
system, and meant simply that the state was not a tensor product of states of
the components Schrödinger (1935). Thus, for pure states, entanglement is
synonymous with correlation. When the quantum information community turned its
attention to mixed states, the term “entangled” obtained a broader meaning,
aimed at capturing the nonclassicality of correlations. Specifically, a
quantum state was taken to exhibit purely classical correlations if it could
be expressed as a mixture of product states Werner (1989), in which case it
was called separable. Around the same time, it was realized that entangled
states (both pure and mixed) could be used to implement useful information-
processing tasks, and they began to be studied as a resource. Because the
tasks being considered at the time mainly concerned the resourcefulness of
entangled states in circumstances wherein the separated parties had access to
classical communication channels (for instance, their use in simulating
quantum channels via the teleportation protocol Bennett  _et al._ (1993) and
in enhancing communication via the dense coding protocol Bennett and Wiesner
(1992)), it was natural to define the interconvertibility preorder of
entangled states relative to local operations and classical communication
(LOCC) Bennett  _et al._ (1996a). This choice was consistent with the previous
definition of the boundary between entangled and unentangled states, since the
states one can prepare freely by LOCC are precisely the separable states.

However, LOCC is not the only choice of free operations that could have been
used to formalize the notion of entanglement as a resource. Consider the set
of local operations and shared randomness (LOSR), wherein the parties have
access to a common source of classical randomness, but no classical channels
among them. If one chooses LOSR as the set of free operations, one also
reproduces the standard definition of entangled states as nonseparable states,
since the free states relative to LOSR are also the separable ones. The
ordering induced over entangled states by LOSR, however, is different from the
one induced by LOCC, even in the case of pure states. Consequently,
quantification of entanglement relative to LOSR leads to quite different
results than one obtains by quantifying it relative to LOCC.

To distinguish these two notions of entanglement, we will henceforth use the
terms LOCC-entanglement and LOSR-entanglement.

In this work, we argue that for the study of Bell scenarios as well as those
information-processing tasks that are built on Bell inequality violations—such
as nonlocal games Broadbent and Méthot (2006); Palazuelos and Vidick (2016);
Johnston  _et al._ (2016), key distribution Barrett  _et al._ (2005); Acín
_et al._ (2006); Scarani  _et al._ (2006); Acín et al. (2007); Vazirani and
Vidick (2014); Kaniewski and Wehner (2016) and randomness generation Colbeck
(2009); Colbeck and Renner (2012); Pironio et al. (2010); Dhara  _et al._
(2013)—the appropriate set of free operations for defining the relevant
resource of entanglement is LOSR rather than LOCC.111In scenarios (such as
teleportation) wherein the parties have access to classical (but not quantum)
communication resources, LOCC-entanglement remains the most relevant notion.

Our first argument in favor of LOSR-entanglement being the right choice for
quantifying entanglement in Bell scenarios is given in Section I. It begins by
noting that LOSR operations are the right choice for quantifying
nonlocality222We here mean the nonclassicality properties of the correlations
that the box describes, and will follow the standard convention of referring
to these as “nonlocality” without endorsing the view that the correct
explanation of Bell inequality violations is to acknowledge the existence of
superluminal causes (see the discussion in Sec. 2.3.1 of Ref. Wolfe  _et al._
(2019)). of boxes—that is, multipartite processes with only classical inputs
and classical outputs. This was first argued in Ref. Wolfe  _et al._ (2019),
and we provide independent arguments here. We then proceed to show that if one
hopes to define a resource theory that encompasses both entangled states and
boxes as different types of resources of nonclassicality, then it follows that
LOSR is also the correct choice for quantifying the nonclassicality of
correlations in quantum states, i.e., for quantifying entanglement.
Furthermore, one has no choice but to work within such a mixed-type resource
theory, because conversions from entangled states to nonlocal boxes are
precisely the focus of any study of the interplay of entanglement and
nonlocality in Bell scenarios.

The remainder of the article supports this argument by showing that the use of
LOSR-entanglement resolves and clarifies a number of long-standing problems
from the Bell literature.

In Section II, we consider various conceptual confusions surrounding so-called
“anomalies of nonlocality” Methot and Scarani (2007), that is, situations
wherein features of nonlocal boxes are found to be realizable by a partially
entangled state but not by a maximally entangled state. The lesson that has
until now been drawn from these purported anomalies is that, in spite of prior
intuitions to the contrary, there are measures of the nonlocal yield of a
state (i.e., the nonlocality of boxes that can be realized from the state)
that are not monotonically related to the state’s entanglement. We show,
however, that this is the wrong lesson to draw and that the anomaly is merely
an artifact of mistakenly presuming that the notion of entanglement that is
appropriate for Bell scenarios is the one based on LOCC rather than the one
based on LOSR. Furthermore, we vindicate the prior intuitions by showing that
every measure of the nonlocal yield of an entangled state is indeed a valid
measure of a state’s LOSR-entanglement.

In Section III, we show that by taking LOSR-entanglement as the appropriate
notion for Bell scenarios, one can resolve an analogous (but previously
unrecognized) anomaly concerning the interconversion between genuinely 3-way
entangled states and genuinely 3-way nonlocal boxes. The resolution highlights
the fact that the notion of genuine multipartite entanglement changes when
entanglement is judged relative to LOSR rather than LOCC. Furthermore, our
notion of genuine multipartite entanglement does not have a pathological
property that the traditional notion exhibits, namely that the to set of
states which are not genuinely multipartite entangled fails to be closed under
tensor product Contreras-Tejada  _et al._ (2020).

In Section IV, we demonstrate that well-known results concerning self-testing
of entangled states can be understood naturally within a resource theory that
allows both entangled states and nonlocal boxes as resources, as long as the
resource of entanglement is judged relative to LOSR rather than LOCC. Not only
does our approach provide a more elegant characterization of self-testing, it
also extends the realm of applicability of self-testing. For example, our
definition can be applied to mixed states. Indeed, we prove that some mixed
states can be self-tested, overturning the prior claims Coladangelo  _et al._
(2017); Šupić  _et al._ (2018); Šupić and Bowles (2019) that this is
impossible.

We close by deriving some general results about the preorder of pure entangled
states under LOSR. These results will justify some of the critical steps in
our arguments, so we will be referencing forward to them throughout the text.

##  I Defining nonclassicality of states and boxes within a single resource
theory

Understanding the interplay of entanglement and nonlocality means
understanding whether particular types and measures of entanglement are
required to realize particular types and measures of nonlocality. In order to
so, one must articulate precisely what it means for an entangled state to be
converted to a nonlocal box. But in addition to this, one must specify the
conversion relations that hold among boxes, because these determine measures
of nonlocality (via order-preserving functions, i.e., monotones), as well as
the conversion relations that hold among states, because these determine
measures of entanglement. Consequently, there are three conversion relations
of interest—box-to-box, state-to-box, and state-to-state—and the free
operations governing each of them must be understood as the free operations
induced by some general constraint that is then particularized to these cases.

We will argue that all three types of conversion relations must be judged
relative to LOSR. Our argument, developed below, invokes the closure under
composition of the set of free operations in a resource theory and consists of
three parts: (i) firstly, we establish that LOSR is the right choice of free
operations for box-to-box conversions, (ii) secondly, we establish that LOSR
is the right choice of free operations for state-to-box conversions given that
it is the right choice for box-to-box conversions, and (iii) lastly, we
establish that LOSR is the right choice of free operations for state-to-state
conversions given that it is the right choice for state-to-box conversions.

(i) LOSR is the right choice of free operations for box-to-box conversions.
In a Bell scenario, one imagines that a multi-partite system is prepared in
some joint state, the different parts are sent to different wings of the
experiment, and several different local measurements are performed on each.

Achieving violations of a Bell inequality in such an experiment is generally
only thought to be surprising if the settings and outcomes at one wing are
space-like separated from those at the other wing.333Another school of thought
Wood and Spekkens (2015) is that Bell inequality violations are surprising
whenever no cause-effect influences can propagate between the wings. In this
view, space-like separation of wings is sufficient but not necessary for
achieving this (where the sufficiency follows from an assumption that
relativity theory restricts not only signals to be sublumimal, but all causal
influences as well). This is because if there can be a causal influence from
the setting or outcome at one wing to the outcome at the other, then it is
straightforward to achieve a Bell inequality violation even if all resources
are classical. Hence, the only boxes we consider in this article are those
whose wings are space-like separated.

Every operation that takes boxes having space-like separated wings to other
such boxes necessarily cannot make use of any communication between the wings,
because all information sent from one wing necessarily propagates to a time-
like separated region at any other wing. It follows that the choice of the set
of free operations is restricted to include only operations that do not make
use of any communication between the wings.444A similar comment applies for
what are called common-cause boxes in Ref. Wolfe  _et al._ (2019) even in
those cases wherein the restriction to common causes is not by virtue of
space-like separation.  The set of free operations, therefore, is clearly not
LOCC. The local operation implemented on one wing can still be correlated with
that implemented on another wing, but only by virtue of a common cause
(quantum or classical). Therefore, if one is interested in the nonclassicality
of such correlations, one should take as free all and only the local
operations that are correlated by virtue of a classical common cause, which is
simply another way of saying that one should take the free operations to be
LOSR Wolfe  _et al._ (2019).

See also the independent arguments from Refs. Wolfe  _et al._ (2019); De
Vicente (2014); Geller and Piani (2014) in favour of LOSR as the right choice
of free operations for a resource theory of nonclassicality of boxes.

(ii) LOSR is the right choice of free operations for state-to-box conversions.
Traditionally, the question of whether a given entangled state can realize a
given nonlocal box is usually interpreted as whether there exists some set of
quantum measurements on each wing that can be implemented on the given state
to yield the conditional probability distribution of outcomes given settings
associated to the given nonlocal box. In other words, whether a given state-
to-box conversion relation holds or not is traditionally evaluated relative to
local operations (LO).

However, if one considers how state-to-box conversion relations fit within a
broader resource theory including box-to-box conversions, one comes to the
conclusion that it is LOSR, rather than LO, that must be considered free. The
argument for this makes use of the fact that LOSR is the set of free
operations that naturally quantifies conversion relations between boxes (as
argued for before) as well as the fact that the set of free operations is
presumed to be closed under composition. It proceeds as follows: if one
considers a state-to-box conversion under LO as free and one also considers a
box-to-box conversion under LOSR as free, then the composition of these, which
is a state-to-box conversion under LOSR, must also be considered free. The
closure under composition of the set of free operations can also be seen to
imply our conclusion in another way. A particular type of operation that
converts a state to a box is one that is a composition of discarding the state
and preparing a box. The operations of this sort that must be deemed free are
those wherein the box that is prepared is a free box. But in order to prepare
all boxes which are considered free in the resource theory, local operations
are insufficient; one needs to have access to shared randomness. In other
words, a subset of the state-to-box conversions are compositions of a state-
to-nothing conversion and a nothing-to-box conversion, but the fact that a
free nothing-to-box conversion may require LOSR implies that certain free
state-to-box conversions also require LOSR.

There are many known results demonstrating that certain state-to-box
conversions are impossible under LO. Allowing additional access to shared
randomness will enable some such conversions to become possible. However, a
class of such nonconversions go through in the LOSR paradigm as well, and we
will find these useful in what follows.

######  Lemma 1.

A box which is convexly extremal in the set of quantumly realizable boxes can
be generated from a given state under LOSR if and only if it can be generated
from that state under LO.

This is because by virtue of the convex-extremality of the target box, if a
mixture of different LO operations can take the state to this box, then every
LO operation in the mixture must do so as well. Finally, we note that LOCC is
obviously not a sensible choice for the set of free operations for state-to-
box conversion, as it would overturn results about the interplay between
entanglement and nonlocality which are neither problematic nor controversial.
For instance, relative to this choice, one would have to grant that a
maximally entangled state was capable of realizing a Hardy-type proof Hardy
(1993) of Bell’s theorem.

(iii) LOSR is the right choice of free operations for state-to-state
conversions.
We now use the closure under composition of the set of free operations to
argue that LOSR, rather than LOCC, is the appropriate choice of free
operations for state-to-state conversions as well. The reasoning is as
follows: if one considers a state-to-state conversion under LOCC as free, and
a state-to-box conversion under LOSR as free, then the composition of these,
which is a state-to-box conversion under LOCC, should also be considered free.
Given that LOCC is not the right way of assessing state-to-box conversions,
one can conclude that it is not the right way of assessing state-to-state
conversions either.

Each of the next three sections supports our argument by giving an independent
motivation for taking LOSR as the free operations.

##  II Resolving the anomaly of nonlocality

As summarized in the introduction and in Ref. Methot and Scarani (2007), the
“anomaly of nonlocality” refers to the fact that there are situations wherein
features of nonlocal boxes are found to be realizable by a partially entangled
state but not by a maximally entangled state. Because entanglement and
nonlocality are often conceptualized as resources, it is natural to frame the
anomaly of nonlocality within a rigorous resource-theoretic framework. Here we
follow the approach of the framework developed in Ref. Coecke  _et al._
(2016).555Recall that a set of free operations defines an ordering relation
(formally, a preorder) on resources, where one resource is at least as
resourceful as a second if it can be freely converted to the second. Two
resources are equivalently resourceful (or in the same equivalence class) if
each can be freely converted into the other, and two resources are
incomparable if neither can be freely converted into the other.  By viewing
entanglement through this lens, we show that each instance of the anomaly of
nonlocality can be recast as a set of claims that are not merely
counterintuitive but contradictory, thereby signalling a flaw in the
conceptual scheme within which they arose.

To begin, we note that it is possible to find a nonlocal box $B$ which can be
realized from some partially entangled pure state of a given Schmidt rank,

$\displaystyle\ket{\psi_{\rm partial}}\mapsto B,$ (1)

but which cannot be realized from any maximally entangled pure state of the
same Schmidt rank,666Formally, $\ket{\psi_{\rm max}}$ is any state for which
the squared Schmidt coefficients describe a uniform distribution for the given
Schmidt rank, while $\ket{\psi_{\rm partial}}$ is any state for which they
describe a nonuniform distribution.

$\displaystyle\ket{\psi_{\rm max}}\not\mapsto B.$ (2)

The following list provides a number of concrete examples of this phenomenon.
For each example, we specify the box $B$ appearing therein by reference to a
convex function that witnesses its nonlocality; that is, for each such box
$B$, one can find a $\ket{\psi_{\rm max}}$ and a $\ket{\psi_{\rm partial}}$ of
the same Schmidt rank such that Eqs. (1) and (2) hold:

  * •

a box that achieves a probability greater than zero of running Hardy’s version
of Bell’s theorem Hardy (1993).

  * •

a box that maximally violates a tilted Bell inequality Acín  _et al._ (2002);
Liang  _et al._ (2011); Vidick and Wehner (2011), thereby offering more noise
resistance for that inequality Acín  _et al._ (2002).

  * •

a box that has extractable secret key rate higher than $\approx 0.144$ Scarani
_et al._ (2006); Acín  _et al._ (2006).

  * •

a box that has Kullback-Leibler divergence (i.e., relative entropy distance)
from the set of local boxes larger than $\approx 0.058$ Acín  _et al._ (2005).

On the other hand, standard entanglement theory tells us that any partially
entangled pure state can be realized, using local operations and classical
communication (LOCC), starting from a maximally entangled state of the same
Schmidt rank Nielsen (1999):

$\displaystyle\ket{\psi_{\rm max}}\mapsto\ket{\psi_{\rm partial}}.$ (3)

It is now evident what is puzzling about these three claims: if the conversion
relations in Eqs. (3) and (1) hold in a resource theory, then given that
resource conversion relations are necessarily777Transitivity of resource
conversions is necessary in the framework of Ref. Coecke  _et al._ (2016)
because the free operations are required to be closed under sequential
composition. transitive in any such theory—i.e., if $R_{1}\mapsto R_{2}$ and
$R_{2}\mapsto R_{3}$ then $R_{1}\mapsto R_{3}$—it follows that we should have
$\ket{\psi_{\rm max}}\mapsto B$, which contradicts Eq. (2).

We now identify the flaw in the conceptual scheme that led to this
contradiction: the implicit idea that the three conversion relations all hold
relative to a single notion of resourcefulness, that is, that they all hold
relative to the same set of free operations and thus can be considered as
relations holding in one and the same resource theory. In the description of
the anomaly, the claim about the convertibility of one entangled state to
another, Eq. (3), is implicitly evaluated relative to LOCC, while claims about
the convertibility or nonconvertibility of an entangled state to a box, Eqs.
(1) and (2) respectively, are implicitly evaluated relative to LO.

In Section I, however, we argued that both conversions from states to states
and from states to boxes must be evaluated relative to LOSR. As we now show,
this resolves the contradiction. The standard claims about the state-to-box
conversions are not modified when one replaces LO by LOSR: Eq. (1) holds with
respect to LOSR because $\textup{LO}\subset\textup{LOSR}$, and Eq. (2) holds
with respect to LOSR by Lemma 1 and the fact that, for each of the examples
given, the box in question is convexly-extremal in the quantum set. (This
follows from the fact that the functions which the boxes maximize in each
example are convex-linear.) On the other hand, the standard claim about the
state-to-state conversion is modified when one replaces LOCC by LOSR. If one
judges conversion between entangled states relative to LOSR, rather than LOCC,
then it is the negation of Eq. (3) that holds, namely,

$\displaystyle\ket{\psi_{\rm max}}\not\mapsto\ket{\psi_{\rm partial}}.$ (4)

This is because $\ket{\psi_{\rm max}}$ and $\ket{\psi_{\rm partial}}$ are
incomparable in the resource theory of LOSR-entanglement; that is, neither can
be converted into the other under LOSR, as shown below (see Corollary 5). But
Eq. (4), unlike Eq. (3), is consistent with Eqs. (2) and (1), and therefore
there is no contradiction (and hence no anomaly).

The terms “partially entangled” and “maximally entangled” are apt descriptions
of $\ket{\psi_{\rm partial}}$ and $\ket{\psi_{\rm max}}$ when one is
considering their LOCC-entanglement properties because for any LOCC-
entanglement monotone $M_{\rm LOCC}$, $M_{\rm LOCC}(\psi_{\rm partial})\leq
M_{\rm LOCC}(\psi_{\rm max})$ and because furthermore, there exists some LOCC-
entanglement monotone $M^{*}_{\rm LOCC}$ for which $M^{*}_{\rm LOCC}(\psi_{\rm
partial})<M^{*}_{\rm LOCC}(\psi_{\rm max})$. When considering their LOSR-
entanglement properties, however, the terminology is no longer appropriate. In
accordance with Eq. (4), there necessarily exist LOSR-entanglement monotones,
$M_{\rm LOSR}$, relative to which $M_{\rm LOSR}(\psi_{\rm partial})>M_{\rm
LOSR}(\psi_{\rm max})$. From this perspective, it is natural (rather than
anomalous) that there exist tasks—such as realizing the sorts of nonlocal
boxes that appear in the list presented earlier—for which the type of LOSR-
entanglement required to realize the task is present in $|\psi_{\rm
partial}\rangle$ but not in $|\psi_{\rm max}\rangle$. Indeed, one can define
such an LOSR-entanglement monotone (i.e., one that is not also an LOCC-
entanglement monotone) from each example of an “anomaly of nonlocality”: given
a function over boxes that witnesses the type of nonlocality described in the
example, the LOSR monotone over states is simply the maximum value of that
function among boxes achievable under LOSR operations from the given state. We
provide the details in Appendix A.

The best known of the anomalies of nonlocality is the one concerning Hardy’s
version of Bell’s theorem, so it is useful to reiterate our conclusion for it
specifically. The fact that the Hardy-type correlations cannot be achieved by
a maximally entangled state but can be achieved by a partially entangled state
surprises almost everyone who encounters the topic. Presumably this is
because—based on their familiarity with LOCC-entanglement—they expect that
whatever resource of nonclassicality is present in a partially entangled
state, it ought to be less than the resource of nonclassicality that is
present in a maximally entangled state. The resolution of the puzzle is that
the notion of nonclassicality that is relevant for Bell scenarios is LOSR-
entanglement, not LOCC-entanglement, and that there are measures of LOSR-
entanglement relative to which a partially entangled state is more
nonclassical than a maximally entangled state.

The LOSR-incomparability of $\ket{\psi_{\rm max}}$ and $\ket{\psi_{\rm
partial}}$ also harmonizes with the recently demonstrated Wolfe  _et al._
(2019) LOSR-incomparability of a Tsirelson box (that which provides the
maximal possible quantum violation of the Clauser-Horne-Shimony-Holt
inequality Clauser  _et al._ (1969)), denoted by $B_{\rm Tsirelson}$, and a
Hardy box (that which achieves a nonzero probability of running Hardy’s
version of Bell’s theorem), denoted by $B_{\rm Hardy}$. Indeed, both instances
of incomparability can be inferred directly from the transitivity of resource
conversions within an LOSR resource theory incorporating both states and boxes
together with known facts about the possible and impossible state-to-box
conversions under LOSR, namely, that $\ket{\psi_{\rm max}}\mapsto B_{\rm
Tsirelson}$ while $\ket{\psi_{\rm max}}\not\mapsto B_{\rm Hardy}$, and that
$\ket{\psi_{\rm partial}}\mapsto B_{\rm Hardy}$ while $\ket{\psi_{\rm
partial}}\not\mapsto B_{\rm Tsirelson}$.888These facts about state-to-box
conversions are well-known when the operations are LO, and can be inferred to
hold also for LOSR by appealing to Lemma 1 and the convex extremality of
$B_{\rm Tsirelson}$ and $B_{\rm Hardy}$. For instance, to see that these
state-to-box conversion relations imply that $\ket{\psi_{\rm
max}}\not\mapsto\ket{\psi_{\rm partial}}$, it suffices to note that if it were
the case that $\ket{\psi_{\rm max}}\mapsto\ket{\psi_{\rm partial}}$, then we
could follow this conversion with $\ket{\psi_{\rm partial}}\mapsto B_{\rm
Hardy}$ in order to have a means of converting $\ket{\psi_{\rm max}}$ to
$B_{\rm Hardy}$, thereby yielding a contradiction. Similarly, to see that
these relations imply that $B_{\rm Tsirelson}\not\mapsto B_{\rm Hardy}$, one
merely notes that if it were the case that $B_{\rm Tsirelson}\mapsto B_{\rm
Hardy}$, then by implementing $\ket{\psi_{\rm max}}\mapsto B_{\rm Tsirelson}$
followed by $B_{\rm Tsirelson}\mapsto B_{\rm Hardy}$, one would have a means
of converting $\ket{\psi_{\rm max}}$ to $B_{\rm Hardy}$, thereby yielding a
contradiction. Similar arguments can be given to establish that
$\ket{\psi_{\rm partial}}\not\mapsto\ket{\psi_{\rm max}}$ and $B_{\rm
Hardy}\not\mapsto B_{\rm Tsirelson}$. As an aside, this argument provides an
alternative proof of the LOSR-incomparability of $B_{\rm Tsirelson}$ and
$B_{\rm Hardy}$ to the one presented in Ref. Wolfe  _et al._ (2019).

##  III Genuine multipartite entanglement

There is also some tension between results concerning genuine multipartite
entanglement and those concerning genuine multipartite nonlocality when the
definitions of these concepts are motivated by the LOCC paradigm for
entanglement. We again begin by reframing this tension as an outright
inconsistency by formulating a genuine multipartite “anomaly of nonlocality”.
We consider the case of three parties for concreteness, although our analysis
can be easily generalized to cases with more parties.

Denote the entangled state ${\frac{1}{\sqrt{2}}(\ket{00}+\ket{11})}$ by
$\ket{\phi^{+}}$. In the pre-order of tripartite entangled states relative to
LOCC-convertibility, the state ${\ket{\psi_{\rm
2Bell}}\equiv|\phi^{+}\rangle_{A_{1}B}\otimes|\phi^{+}\rangle_{A_{2}C}}$ is
above the state ${\ket{\psi_{\rm
GHZ}}\equiv\tfrac{1}{\sqrt{2}}(|000\rangle_{ABC}+|111\rangle_{ABC})}$ because
the former can be deterministically converted to the latter,

$\displaystyle\ket{\psi_{\rm 2Bell}}\mapsto\ket{\psi_{\rm GHZ}}.$ (5)

It suffices for one party to prepare three systems in their lab in the state
$\ket{\psi_{\rm GHZ}}$, and then to use $\ket{\psi_{\rm 2Bell}}$ to teleport
one part to each of the other two parties (which requires classical
communication). And yet, there are tripartite boxes, such as the Mermin box
Mermin (1990); Brassard  _et al._ (2005), that can be realized from
$\ket{\psi_{\rm GHZ}}$ by local measurements,

$\displaystyle\ket{\psi_{\rm GHZ}}\mapsto B_{\rm Mermin},$ (6)

but that cannot be so realized from $\ket{\psi_{\rm 2Bell}}$,

$\displaystyle\ket{\psi_{\rm 2Bell}}\not\mapsto B_{\rm Mermin},$ (7)

as follows from a result in Ref. Wolfe  _et al._ (2019).999The relevant result
from Ref. Wolfe  _et al._ (2019) is that the Mermin box cannot be achieved by
LOSR processing from any state, $\rho_{\rm triangle}$, that is a tensor
product of states having entanglement between pairs of parties only (i.e.,
generated from entangled sources consistent with the so-called “triangle
scenario” network), $\rho_{\rm triangle}\not\mapsto B_{\rm Mermin}$. Eq. (7)
follows because $\ket{\psi_{\rm 2Bell}}$ is an instance of such a state. As
before, Eqs. (5), (6), and (7) seem to imply a contradiction given the
transitivity of resource conversion relations.

The resolution of the puzzle proceeds as in the case of the bipartite
anomalies. The reason for the seeming contradiction is that the conversion
relations of Eqs. (6) and (7) are implicitly evaluated relative to LO (which
here is equivalent to LOSR, given Lemma 1 and the fact that the box under
consideration is convexly extremal), while that of Eq. (5) is evaluated
relative to LOCC. If, however, one also evaluates conversions between
entangled states relative to LOSR, then $\ket{\psi_{\rm 2Bell}}$ and
$\ket{\psi_{\rm GHZ}}$ are found to be incomparable, so that the negation of
Eq. (5) holds, that is,

$\ket{\psi_{\rm 2Bell}}\not\mapsto\ket{\psi_{\rm GHZ}},$ (8)

and the contradiction is blocked. The proof of incomparability follows from a
condition for LOSR-convertibility we derive further on (see Corollary 4), as
is made explicit in Appendix B.

We have presented this anomaly and its resolution because it sheds light on
how one ought to define notions of genuine multipartite entanglement and
nonlocality, specifically, what it means for these to be “genuinely 3-way”.

We begin by presenting a slightly different perspective on the anomaly.
Consider the conventional definition of “genuinely 3-way entanglement” Gühne
_et al._ (2005); Amico  _et al._ (2008). Recalling that a tripartite state is
termed biseparable if there is a partitioning of the parties into two groups
such that the state is separable relative to this bipartition, the definition
can be simply stated as follows: A tripartite state is genuinely 3-way
entangled if and only if it is not a mixture of biseparable states.

Now note that $\ket{\psi_{\rm 2Bell}}$ counts as genuinely 3-way entangled by
this definition. This is somewhat counterintuitive a priori, given that
$\ket{\psi_{\rm 2Bell}}$ is composed of states that contain only bipartite
entanglement. One can glean some insight into what led to the adoption of the
conventional definition, in spite of its counterintutive features, from the
fact that $\ket{\psi_{\rm 2Bell}}$ is above $\ket{\psi_{\rm GHZ}}$ in the LOCC
order (Eq. (5)). Because it is generally agreed that the definition of
genuinely 3-way entanglement should be such that $\ket{\psi_{\rm GHZ}}$ counts
as genuinely 3-way entangled, it follows that because $\ket{\psi_{\rm 2Bell}}$
is above $\ket{\psi_{\rm GHZ}}$ in the order, it too must qualify as having
such entanglement.

With the conventional notion of genuinely 3-way entanglement in mind, we are
now in a position to describe the new perspective on the tripartite anomaly.
First, note that it is generally agreed that any definition of genuinely 3-way
nonlocality should be such that $B_{\rm Mermin}$ counts as genuinely 3-way
nonlocal. But because $\ket{\psi_{\rm 2Bell}}$ is above $\ket{\psi_{\rm GHZ}}$
in the LOCC order, one would expect that whatever resource of genuinely 3-way
entanglement is required to generate the genuinely 3-way nonlocality inherent
in $B_{\rm Mermin}$, $\ket{\psi_{\rm 2Bell}}$ would have it if $\ket{\psi_{\rm
GHZ}}$ does. And yet this intuitive conclusion is in conflict with Eq. (7).

The thesis of this article, that entanglement in Bell scenarios should be
defined relative to LOSR rather than LOCC, implies that the conventional
notion of genuinely 3-way entanglement—which, as we argued above, is motivated
by LOCC—need not be the relevant one for Bell scenarios. We propose an
alternative definition here, motivated by LOSR. We also show that there is a
corresponding alternative definition for genuinely 3-way nonlocality which
mirrors our alternative definition of genuinely 3-way entanglement. The
tripartite anomaly is shown to admit of a natural resolution relative to these
two notions, demonstrating that these have a more natural interplay than
exists between the conventional notions.

In the LOSR paradigm, the distinction between a classical and a quantum
resource shared among some parties is the distinction between sharing
classical randomness and sharing entanglement. This can be fruitfully
understood in a causal manner: the distinction concerns whether the systems in
the possession of the parties are influenced by a common cause that is
classical or by one that is quantum. Consequently, the natural manner of
defining a resource of 3-way nonclassicality is to consider resourcefulness
relative to a set of operations wherein all 2-way nonclassicality is
considered free—that is, where 2-way common causes are allowed to be
quantum—while the 3-way common cause is required to be classical. Thus, we
define genuinely 3-way nonclassicality for quantum states and for nonlocal
boxes (as well as for other types of multipartite processes, such as quantum
measurements, quantum channels, and multi-time processes) as those that are
nonfree relative to LOSR supplemented by 2-way shared entanglement
(LOSR2WSE).101010One can define genuinely $k$-way nonclassicality among $n$
parties (where $k\leq n$) in an analogous manner: via LOSR supplemented by
$(k-1)$-way shared entanglement Navascués  _et al._ (2020).

######  Definition 1.

A nonlocal box is genuinely 3-way nonlocal and an entangled state is genuinely
3-way entangled if and only if they cannot be obtained by local operations
together with a source of correlations consisting of shared entanglement
between each pair of parties and shared randomness among all three.

These notions are illustrated in Fig. 1 and Fig. 2. Ref. Wolfe  _et al._
(2019) discusses boxes that are genuinely 3-way nonlocal in this sense, and
Ref. Navascués  _et al._ (2020) discusses states that are genuinely 3-way
entangled in this sense. The associated causal structure has been termed the
“quantum triangle scenario with shared randomness” Wolfe  _et al._ (2019);
Kraft  _et al._ (2020).

Figure 1:  A tripartite entangled state is not genuinely 3-way entangled if it
can be decomposed as shown; that is, if it is achievable by LOSR supplemented
by bipartite shared entanglement. Throughout, double wires represent quantum
systems and single wires represent classical systems.  Figure 2:  A tripartite
box is not genuinely 3-way nonlocal if it can be decomposed as shown; that is,
if it is achievable by LOSR supplemented by bipartite shared entanglement.

It is clear that $\ket{\psi_{\rm 2Bell}}$ can be realized via LOSR2WSE, while
$\ket{\psi_{\rm GHZ}}$ cannot, as shown in Ref. Navascués  _et al._ (2020),
and so of the two, only $\ket{\psi_{\rm GHZ}}$ is genuinely 3-way entangled
according to this definition. In this approach, therefore, the intuitions we
noted earlier regarding genuinely 3-way entanglement are vindicated: if a
state can be obtained from one where all of the entanglement is of the
bipartite variety, then it is not genuinely 3-way entangled.

It is also the case that $B_{\rm Mermin}$ cannot be realized by LOSR2WSE, as
follows from results in Ref. Wolfe  _et al._ (2019)111111Ref. Wolfe  _et al._
(2019) primarily concerns the quantum triangle scenario _without_ shared
randomness (i.e., without a 3-way classical common cause), though the
technique there is readily adaptable to allow for it. A forthcoming revision
of Ref. Wolfe  _et al._ (2019) addresses this latter case more explicitly., so
that $B_{\rm Mermin}$ is genuinely 3-way nonlocal according to this definition
(just as it was relative to previous definitions).

In the description of the tripartite anomaly, the fact that $\ket{\psi_{\rm
GHZ}}$ can be converted into $B_{\rm Mermin}$ while $\ket{\psi_{\rm 2Bell}}$
cannot (Eqs. (6) and (7)) was only surprising relative to the belief that
$\ket{\psi_{\rm 2Bell}}$ must have more genuinely 3-way nonclassicality than
$\ket{\psi_{\rm GHZ}}$ does, on the grounds that it is above $\ket{\psi_{\rm
GHZ}}$ in the LOCC order.121212This is in precise analogy to how the fact that
$\ket{\psi_{\rm partial}}$ can be converted to a box manifesting Hardy-type
correlations while $\ket{\psi_{\rm max}}$ cannot (Eqs. (1) and (2)) is only
surprising relative to the belief that $\ket{\psi_{\rm max}}$ has more
nonclassicality than $\ket{\psi_{\rm partial}}$ does. But this is overturned
in the approach just described, since $\ket{\psi_{\rm 2Bell}}$ explicitly has
less genuinely 3-way nonclassicality than $\ket{\psi_{\rm GHZ}}$ does, since
$\ket{\psi_{\rm 2Bell}}$ has none while $\ket{\psi_{\rm GHZ}}$ has some. Given
that $B_{\rm Mermin}$ has genuinely 3-way nonclassicality according to our
definition, it becomes intuitively clear why only $\ket{\psi_{\rm GHZ}}$ and
not $\ket{\psi_{\rm 2Bell}}$ can be converted into $B_{\rm Mermin}$.

Finally, we note that whereas our definition of genuinely 3-way
nonclassicality fits within the mathematical framework for resource theories
Coecke  _et al._ (2016) (as the property of being nonfree relative to
LOSR2WSE), the conventional definition does not. The latter fact is most
easily seen by recalling an awkward feature of the conventional definition,
namely, that the set of states that are not genuinely multipartite entangled
is not closed under tensor products. For example, consider the tripartite
states $|\phi^{+}\rangle_{A_{1}B}\otimes|0\rangle_{C}$ and
$|\phi^{+}\rangle_{A_{2}C}\otimes|0\rangle_{B}$. Both are biseparable
(relative to different partitions of the tripartite system). Their tensor
product, however, is essentially equivalent to $|\psi_{\rm
2Bell}\rangle=|\phi^{+}\rangle_{A_{1}B}\otimes|\phi^{+}\rangle_{A_{2}C}$,
which, as noted previously, is not biseparable and therefore is genuinely
3-way entangled according to the conventional definition.

Although this feature of the conventional definition has been acknowledged by
some as counterintuitive and somewhat perverse Contreras-Tejada  _et al._
(2020) we wish to draw attention here to the fact that it is inconsistent with
the framework of resource theories131313Just as the conversion relations in
each of the anomalies were inconsistent with the transitivity of resource
conversion relations., because the latter stipulates that the set of free
resources must be closed under parallel composition (see Definition 2.1 of
Ref. Coecke  _et al._ (2016)). From the resource-theoretic perspective,
therefore, the fact that the property of biseparability is not preserved under
parallel composition implies that biseparability is simply not a valid
candidate for the property that defines the set of free resources in a
resource theory. Hence, the failure of biseparability is not a valid candidate
for the property that defines the resource of genuinely multipartite
nonclassicality.

##  IV Self-testing of entangled states

A final reason why the resource theory of LOSR-entanglement is the one that is
relevant for Bell scenarios is that (unlike LOCC-entanglement) it allows one
to understand and extend results in self-testing Mayers and Yao (1998, 2003);
Šupić and Bowles (2019); Scarani (2012).

Colloquially, a state is said to be self-tested by a box if this state is “the
only one” from which the box can be obtained by implementing local
measurements. However, there is never just a single state that can yield a
given box, and so the standard definition allows for some freedom in the
specification of the state. For the bipartite case, Refs. Popescu and Rohrlich
(1992); Braunstein  _et al._ (1992) and subsequent works specify this freedom
as follows: to say that $|\psi\rangle$ is self-tested by $B$ is to say that
$B$ can be obtained by local measurements on a state $|\phi\rangle$ only if
$\ket{\phi}$ can be mapped to $|\psi\rangle|\zeta\rangle$ for some
$|\zeta\rangle$ by a local isometry.141414We here take $|\phi\rangle$ and
$|\zeta\rangle$ to be pure, even though in some treatments, such as Ref. Šupić
and Bowles (2019), they are taken to be mixed. The assumption of purity
involves no loss of generality because of the Stinespring dilation theorem
Stinespring (1955); Paulsen (2003). We note that it is often asserted
Coladangelo  _et al._ (2017); Šupić  _et al._ (2018); Šupić and Bowles (2019)
that mixed states cannot be self-tested and it is perhaps for this reason that
self-testing has conventionally been defined only for pure states.

As we did with the anomalies of nonlocality, we will begin by reframing self-
testing from a resource-theoretic perspective. We will see that the above
condition for pure states is elegantly reproduced in the resource theory of
LOSR-entanglement and that standard self-testing results harmonize with the
LOSR order over entangled states. Moreover, our resource-theoretic reframing
of the notion of self-testing implies that it can be applied to a much broader
range of resources; for example, we prove that mixed states can be self-
tested, in spite of previous assertions to the contrary Coladangelo  _et al._
(2017); Šupić  _et al._ (2018); Šupić and Bowles (2019).

Our resource-theoretic definition of self-testing of states is as follows:

######  Definition 2.

We say that a density operator $\rho$ is self-tested by a box $B$ if it holds
that

$\displaystyle\rho\mapsto B$

and

$\displaystyle\forall\sigma:\textrm{if}\;\sigma\mapsto
B\;\textrm{then}\;\sigma\mapsto\rho,$

where all conversions are evaluated relative to LOSR.

We give an equivalent but more abstract definition in Appendix C; namely:
$\rho$ is self-tested by $B$ if the upward closure of $B$ contains only those
states which are also in the upward closure of $\rho$. (See Appendix C for a
definition of upward closure.)

An advantage of this formulation of self-testing is that it generalizes
immediately to other sorts of objects and even to other resource theories. We
return to this point in the discussion section.

Our definition of self-testability of a state coincides with the conventional
one when the states are pure; that is, when $\rho=\ket{\psi}\bra{\psi}$ and
$\sigma=\ket{\phi}\bra{\phi}$. The proof is simple: as per conditions (i) and
(iii) of Lemma 3, the most general $|\phi\rangle$ that can be converted to
$\ket{\psi}$ by LOSR is one that can be mapped to $|\psi\rangle|\zeta\rangle$
for some $|\zeta\rangle$ by a local isometry.

So we see that the articulation of the freedom of a state in the conventional
definition of self-testing is nothing more than the condition for LOSR-
convertibility for pure entangled states.

However, our definition also applies to mixed states, and it is not hard to
see that some mixed states can be self-tested. In fact, we now show that for
every pure state that is self-tested by some box, there is a family of mixed
states that is self-tested by the same box. This follows from the simple
observation that for any bipartite state $\ket{\psi}_{AB}$, any mixed state
$\rho_{AB}$ defined as follows is in its LOSR-equivalence class:

$\displaystyle\rho_{AB}:=\sum_{ij}p(ij)U_{A}^{(i)}\otimes U_{B}^{(j)}$
$\displaystyle\ket{\psi}\\!\bra{\psi}_{AB}U_{A}^{(i)\dagger}\otimes
U_{B}^{(j)\dagger}$ (9)
$\displaystyle\otimes\ket{i}\\!\bra{i}_{A}\otimes\ket{j}\\!\bra{j}_{B}$

where $p(ij)$ is a probability distribution, $U_{A}^{(i)}$ and $U_{B}^{(j)}$
are arbitrary unitaries, and $\\{\ket{i}_{A}\\}_{i}$ and
$\\{\ket{j}_{B}\\}_{j}$ are arbitrary orthonormal bases for a local flag
degree of freedom for Alice and for Bob, respectively. Indeed, to convert
$\ket{\psi}$ to $\rho$, the parties prepare their flag systems in the
separable state $\sum_{ij}p(ij)\ket{i}\bra{i}_{A}\otimes\ket{j}\bra{j}_{B}$,
and Alice applies a controlled unitary
$\sum_{i}U_{A}^{(i)}\otimes\ket{i}_{A}\bra{i}$ with her local flag as the
control and her half of $\ket{\psi}$ as the target, and Bob acts similarly. To
convert $\rho$ to $\ket{\psi}$, Alice applies the controlled unitary
$\sum_{i}U_{A}^{(i)\dagger}\otimes\ket{i}\bra{i}_{A}$ with her local flag as
the control and her half of $\rho$ as the target, Bob acts similarly, and then
each traces out their local flag system. If $\ket{\psi}$ can be self-tested,
the LOSR-equivalence of $\ket{\psi}$ and $\rho$ implies that $\rho$ can also
be self-tested. This construction also generalizes immediately to the
multipartite case. Note that shared randomness is not required for the
argument, as this construction generates a mixed state even if
$p(ij)=p(i)p(j)$.

Such mixed states have a very specific form, and they are of higher dimension
than the pure state from which it is generated. It remains an open question
whether more generic mixed states can also be self-tested. A positive answer
would constitute major progress for the field of robust self-testing, since
every state prepared in a real experiment is somewhat mixed.

Our definition also clarifies results concerning self-testing in the
multipartite scenario. For example, chiral pure states—states which are
inequivalent under local unitary transformations to their complex conjugate—
are often proposed as examples of the failure of self-testing, since a chiral
state and its complex conjugate can always realize the same set of
boxes.151515This was noted by Jed Kaniewski. A similar failure of self-
testing, but for measurements rather than states, was previously noted in Ref.
McKague and Mosca (2010). There exist chiral pure states even for three-qubit
systems Acín  _et al._ (2000, 2001). Prior to our work, the local isometric
freedom in the traditional definition of self-testing did not have an a priori
justification, and so it was not obvious that one should consider chiral
states as an example of the failure of self-testing as opposed to considering
them to demonstrate the need for modifying the freedom in the specification of
the self-tested state (to include complex conjugation as well as local
isometries). However, our resource-theoretic characterization, together with
our arguments for LOSR as the appropriate set of free operations, provide such
an a priori justification, and so eliminate the ambiguity. If $\ket{\psi}$ is
chiral and $\ket{\psi}\mapsto B$, then the complex conjugate $\ket{\psi^{*}}$
also maps to $B$, that is, $\ket{\psi^{*}}\mapsto B$, and yet
$\ket{\psi^{*}}\not\mapsto\ket{\psi}$. Hence $\ket{\psi}$ cannot be self-
tested. A similar argument can also be run for states which are not LOSR-
equivalent to their partial transposes, using the results of Ref. Hirsch and
Huber (2020).

There are two common conceptualizations of self-testing that have not been
differentiated in previous work, but which our resource-theoretic
characterization of self-testing shows to be distinct.

The conceptualization that comes closest to our approach is that based on
‘extraction’, wherein self-testing is understood as a certification that every
state $\sigma$ that could possibly have generated the observed box $B$ by some
processing is such that the state $\rho$ can be ‘extracted’ from it by some
processing. Our definition of self-testing is of this form, but where it is
stipulated that “by some processing” must be interpreted as “by an operation
that is free relative to the resource theory that captures the notion of
nonclassicality of correlations.” Once it is realized that this is how one
justifies a given choice of processings, it becomes clear that the first order
of business in defining self-testing is to determine which set of free
operations is appropriate. This is precisely what the present article has
done, settling on LOSR as the right choice.

The second conceptualization is that the task of self-testing is to uniquely
certify up to some notion of equivalence the state $\rho$ which was used to
generate the observed box $B$. Without any further caveats, this is not
possible, as our approach clarifies: the set of states which could possibly
have generated a given box always belongs to an unbounded number of distinct
equivalence classes, namely, all of those that are strictly above that of the
self-tested state in the preorder. If one wishes to identify a unique LOSR-
equivalence class of states which could have been used to generate a given box
$B$, one must restrict attention to a particular subset of states, namely, one
wherein no two states are strictly ordered. Only then does satisfaction of the
condition for self-testability of $\rho$ imply that the state must have been
LOSR-equivalent to $\rho$, rather than implying only that it must be above
$\rho$.

So we see that these two conceptualizations of self-testing are not
equivalent.

This second view of self-testing motivates the task of finding subsets of
states that contain no strictly ordered elements, so that every pair of states
in the set is either equivalent or incomparable. But that is precisely the
sort of problem that is solved by determining the preorder of entangled states
under LOSR. We begin the task of characterizing this preorder in the next
section.

We prove in particular that all LOSR-equivalence classes of bipartite pure
states of a given Schmidt rank are LOSR-incomparable (see Corollary 5). This
explains why a previous result concerning unique certification of bipartite
pure entangled states within a set was required to restrict attention to a set
containing states of a fixed Schmidt rank Coladangelo  _et al._ (2017)
(although this restriction was not highlighted in the statement of the
result). Indeed, it is only relative to this caveat of fixed Schmidt rank that
the slogan of “all pure bipartite entangled states can be self-tested” (in the
sense of unique certification within a set) holds true.161616We note that the
existence of a continuum of states that can be uniquely certified in the sense
just described is only possible because of two features of the LOSR preorder:
it contains anti-chains—that is, sets of pairwise incomparable elements—of
infinite cardinality (i.e., “infinite width”) both among states (as follows
from Corollary 5 and the continuum of possibilities for vectors of squared
Schmidt coefficients of a given Schmidt rank) and among boxes (as proven in
Ref. Wolfe  _et al._ (2019)).

For what other sets of pure states might one be able to prove the possibility
of unique certification of states within that set (i.e., up to LOSR-
equivalence)? Our characterization of the LOSR pre-order of pure states, to be
presented in the next section, sheds light on this question. For instance, in
the case of bipartite states, where a complete characterization of the pre-
order is possible (see Eq. (14)), a sufficient condition for the
incomparability of two LOSR-inequivalent states is that the ratio of their
Schmidt ranks is not an integer (see Corollary 6 in Appendix E). Consequently,
if we let $S_{x}$ be the set of all pure bipartite states of Schmidt rank $x$,
then for any set of integers $\mathcal{X}$ for which every ratio is noninteger
(so that $\forall x,x^{\prime}\in\mathcal{X}$ it holds that
$x/x^{\prime}\not\in\mathbb{Z}^{+}$), the set of states
$\cup_{x\in\mathcal{X}}S_{x}$ is one relative to which unique certification of
every state (up to LOSR equivalence) might be achievable, while for any set of
integers $\tilde{\mathcal{X}}$ for which there exists a pair with an integer
ratio (so that $\exists x,x^{\prime}\in\tilde{\mathcal{X}}$ such that
$x/x^{\prime}\in\mathbb{Z}^{+}$), such unique certification is impossible.

##  V The resource theory of LOSR-entanglement

Having motivated the need for understanding LOSR-entanglement, we now prove
some basic results about the pre-order of bipartite and multipartite entangled
states under LOSR.

Because LOCC allows for shared randomness to be generated at one wing and then
shared with the other parties by classical communication, it strictly includes
LOSR,

$\displaystyle\textup{LOSR}\subset\textup{LOCC}.$ (10)

It follows, therefore, that if a conversion from a state $\rho$ to a state
$\sigma$ (mixed or pure) is not possible by LOCC, then it is also not possible
by LOSR either:

$\displaystyle\rho\not\mapsto\sigma\;{\rm by}\;\textup{LOCC}\ \ $
$\displaystyle\implies\ \ \rho\not\mapsto\sigma\;{\rm by}\;\textup{LOSR}.$
(11)

Consequently, any witness of LOCC-nonconvertibility is a witness of LOSR-
nonconvertibility as well, any monotone relative to LOCC Horodecki  _et al._
(2000); Vidal (2000); Gour (2005); Datta (2009) is also a monotone relative to
LOSR, and the LOCC-equivalence classes contain the LOSR-equivalence classes.

However, there are instances of LOSR-nonconvertibility that are not instances
of LOCC-nonconvertibility, or equivalently, cases where LOCC allows a
conversion that is not allowed under LOSR. Consequently, there are witnesses
of LOSR-nonconvertibility that are not witnesses of LOCC-nonconvertibility. It
also follows that there are LOSR monotones that are not LOCC monotones (some
explicit examples are provided in Appendix A), and that a complete set of
LOCC-entanglement monotones does _not_ constitute a complete set of LOSR-
entanglement monotones. The distinction between convertibility relations in
LOSR and in LOCC allows in principle for the equivalence classes of states
under LOSR to be strict subsets of those under LOCC. The following lemma,
however, precludes that possibility for pure states. Here, we make reference
to the set of local unitary operations (LU).

######  Lemma 2.

Consider the equivalence relation associated to reversible
interconvertibility.171717One resource is said to be reversibly
interconvertible with another if conversions in both directions are possible
under the free operations. Note that the free operation achieving the
conversion in one direction need not be the inverse of the free operation
achieving the conversion in the opposite direction.  The following statements
about equivalence between $n$-partite pure states $|\psi\rangle$ and
$|\phi\rangle$ (denoted $|\psi\rangle\rightleftharpoons|\phi\rangle$) are
equivalent

  1. (i)

$|\psi\rangle\rightleftharpoons|\phi\rangle$ relative to LOSR.

  2. (ii)

$|\psi\rangle\rightleftharpoons|\phi\rangle$ relative to LO.

  3. (iii)

$|\psi\rangle\rightleftharpoons|\phi\rangle$ relative to LU.

  4. (iv)

$|\psi\rangle\rightleftharpoons|\phi\rangle$ relative to LOCC.

###### Proof.

We note that the different sets of operations form a chain of strict
inclusions:

$\displaystyle\textup{LU}\subset\textup{LO}\subset\textup{LOSR}\subset\textup{LOCC}$
(12)

Next, we appeal to the known result that two pure states $|\psi\rangle$ and
$|\phi\rangle$ are LOCC-equivalent if and only if they are LU-equivalent
Bennett  _et al._ (2000). Since the set-theoretic inclusion relations are such
that LO and LOSR are both between LU and LOCC in Eq. (12), the result follows.
∎

It follows from Lemma 2 and the discussion above that all of the differences
between the LOCC and LOSR preorders correspond to pairs of equivalence classes
which are strictly ordered under LOCC but are incomparable under LOSR.

The LOCC preorder over pure states has been completely characterized in the
bipartite case Nielsen (1999), and in the following we obtain the analogous
result for the LOSR preorder. We also express some results concerning the
$n$-partite case for $n>2$.

######  Lemma 3.

The following statements about interconversion between $n$-partite pure states
$|\psi\rangle$ and $|\phi\rangle$ are equivalent

  1. (i)

$|\psi\rangle\mapsto|\phi\rangle$ by LOSR.

  2. (ii)

$|\psi\rangle\mapsto|\phi\rangle$ by LO.

  3. (iii)

$\exists|\eta_{1}\rangle\dots|\eta_{n}\rangle,|\zeta\rangle:|\psi\rangle|\eta_{1}\rangle\dots|\eta_{n}\rangle\rightleftharpoons|\phi\rangle|\zeta\rangle$
by LU, as illustrated in Fig. 3.

Figure 3: A depiction of condition (iii) of Lemma 3 for $n=3$ parties.

###### Proof.

(ii) $\implies$ (i) holds trivially, because every LO operation is an LOSR
operation. (i) $\implies$ (ii) follows, because every LOSR operation is a
mixture of LO operations, and given that the state $\ket{\phi}$ is pure, it
follows that the states resulting from each local operation of this mixture
must be proportional to $|\phi\rangle$. Consequently, any one of the LO
operations in the mixture can convert $|\psi\rangle$ to $|\phi\rangle$ on its
own.

(iii) $\implies$ (ii) trivially, because preparing local auxiliary systems,
applying local unitaries, and taking partial traces are free LO operations.
That (ii) $\implies$ (iii) follows from the Stinespring dilation theorem
Stinespring (1955); Paulsen (2003): any trace-preserving local operation can
be realized by introducing pure state auxiliary systems prepared in a product
state across the $n$-partition, $|\eta_{1}\rangle\dots|\eta_{n}\rangle$,
applying a local unitary to each composite of a local subsystem and a local
auxiliary system, and finally tracing out some subsystem of this composite.
Given that the final reduced state on the output $n$-partite system must be
the pure state $|\phi\rangle$, the final joint state must be of the form
$|\phi\rangle\otimes|\zeta\rangle$ for some (possibly entangled) state
$|\zeta\rangle$. ∎

Evidently, Lemmas 2 and 3 provide avenues for understanding the LOSR-
entanglement preorder by utilizing known results about LU-equivalence Kraus
(2010); Liu  _et al._ (2012); Acín  _et al._ (2001); Barnum and Linden (2001);
Biamonte  _et al._ (2013).

For instance, in the bipartite case, it is well known that local unitary
equivalence between pure states has a particularly simple form: if
$\lambda_{\psi}$ denotes the vector of squared Schmidt coefficients of a state
$|\psi\rangle$, and $v^{\downarrow}$ denotes the permutation of a vector $v$’s
components such that they are ordered from largest to smallest, then
$\ket{\psi}$ and $\ket{\phi}$ are LU-equivalent if and only if

$\displaystyle\lambda_{\psi}^{\downarrow}=\lambda_{\phi}^{\downarrow},$ (13)

that is, if and only if their vectors of squared Schmidt coefficients are
equal up to permutation. Lemma 2 then implies that this is also the condition
for LOSR-equivalence between pure states. This characterization of the LU-
equivalence of bipartite pure states also implies that condition (iii) in
Lemma 3 has a particularly simple form181818In the bipartite case, the
equivalence of condition (ii) and this form of condition (iii) is stated as
Exercise 12.22 of Ref. Nielsen and Chuang (2010).:

$\displaystyle\exists\ket{\zeta}:\lambda_{\psi}^{\downarrow}=(\lambda_{\phi}\otimes\lambda_{\zeta})^{\downarrow}.$
(14)

It follows that the pair of states $|\psi_{\rm max}\rangle$ and $|\psi_{\rm
partial}\rangle$ from the main text, which have the same Schmidt rank but
different vectors of Schmidt coefficients, are such that neither converts to
the other under LOSR, i.e., they are LOSR-incomparable. This justifies Eq.
(4).

In the case of $n$-partite systems for $n>2$, a necessary condition for LU-
equivalence between pure states is the equality of squared Schmidt
coefficients for each bipartition $\beta$ of the $n$-partite system, but
(unlike the bipartite case) this is not a sufficient condition Kraus (2010);
Acín  _et al._ (2001). Nevertheless, combining this condition with Lemma 3
yields the following corollary concerning LOSR-convertibility.

######  Corollary 4.

An $n$-partite pure state $\ket{\psi}$ can be converted to an $n$-partite pure
state $\ket{\phi}$ by LOSR only if

$\displaystyle\exists\ket{\zeta},\forall\beta:(\lambda^{(\beta)}_{\psi})^{\downarrow}=(\lambda^{(\beta)}_{\phi}\otimes\lambda^{(\beta)}_{\zeta})^{\downarrow},$
(15)

where for a pure state $\ket{\omega}$, $\lambda^{(\beta)}_{\omega}$ denotes
the vector of its squared Schmidt coefficients with respect to bipartition
$\beta$ of the $n$-partite system.

Despite the insufficiency of Corollary 4 for determining LOSR-convertibility
(which we demonstrate with an explicit example in Appendix D), it provides a
necessary condition for LOSR-convertibility which is not a necessary condition
for LOCC-convertibility. As an example, by appealing to Corollary 4, we show
in Appendix B that $\ket{\psi_{\rm 2Bell}}$ and $\ket{\psi_{\rm GHZ}}$ (which
are strictly ordered relative to LOCC) are incomparable relative to LOSR,
thereby justifying Eq. (8).

The condition stipulated by Corollary 4, that there exists a pure state
$\ket{\zeta}$ satisfying Eq. (15), is a non-trivial requirement. The problem
of deciding whether there exists an $n$-partite pure state $\ket{\zeta}$
admitting a given set of vectors of squared Schmidt coefficients
$\lambda^{(\beta)}_{\zeta}$ for a given family of bipartitions $\beta$ is
known as the spectral quantum marginals problem Klyachko (2006); Walter  _et
al._ (2013). Additionally, Appendix E describes a simple method for computing
the $\lambda_{\zeta}^{(\beta)}$ satisfying Eq. (15), given
$\lambda_{\psi}^{(\beta)}$ and $\lambda_{\phi}^{(\beta)}$, if a solution
exists.

We also note a second corollary of Lemma 3. Recall that the Schmidt rank of a
state $\ket{\psi}$ with respect to a bipartition $\beta$, henceforth denoted
$\mathrm{SR}_{\psi}^{(\beta)}$, is defined as the number of non-zero entries
of the vector of squared Schmidt coefficients $\lambda_{\psi}^{(\beta)}$.

######  Corollary 5.

If two pure states $\ket{\psi}$ and $\ket{\phi}$ have the same Schmidt ranks
for all bipartitions, i.e.,

$\displaystyle\forall\beta:\mathrm{SR}_{\psi}^{(\beta)}=\mathrm{SR}_{\phi}^{(\beta)},$
(16)

then they are either LOSR-equivalent (equivalently, LU-equivalent) or else
LOSR-incomparable.

###### Proof.

Suppose that $\ket{\psi}\mapsto\ket{\phi}$ under LOSR, so that condition (iii)
from Lemma 3 is satisfied. Under the assumption that Eq. (16) holds, it
follows that the state $\ket{\zeta}$ in this condition must be such that
$\forall\beta:\mathrm{SR}_{\zeta}^{(\beta)}=1$. But this implies that
$\ket{\zeta}$ is a product state,
$\ket{\zeta}=\ket{\zeta_{1}}\cdots\ket{\zeta_{n}}$, in which case condition
(iii) from Lemma 3 reduces to
$\exists|\eta_{1}\rangle\dots|\eta_{n}\rangle,|\zeta_{1}\rangle\dots|\zeta_{n}\rangle:|\psi\rangle|\eta_{1}\rangle\dots|\eta_{n}\rangle\rightleftharpoons|\phi\rangle|\zeta_{1}\rangle\dots|\zeta_{n}\rangle$
by LU, which is a condition that is symmetric in $\ket{\psi}$ and
$\ket{\phi}$. The same logic holds if one supposes that
$\ket{\phi}\mapsto\ket{\psi}$ under LOSR. Consequently, either both conversion
relations hold or neither does. In other words, either $\ket{\psi}$ and
$\ket{\phi}$ are LOSR-equivalent or they are LOSR-incomparable. Finally, LOSR-
equivalence is the same as LU-equivalence by Lemma 2.191919By the structure of
this proof, it is clear that one can replace the assumption of sameness of
Schmidt ranks for all bipartitions with any other condition that is symmetric
in $\ket{\psi}$ and $\ket{\phi}$ and which establishes that $\ket{\zeta}$ must
be a product state. For instance, if two pure states $\ket{\psi}$ and
$\ket{\phi}$ have the same marginal entropies for each subsystem (i.e., are
“marginally isentropic”) then one can also conclude that $\ket{\zeta}$ is a
product state and consequently that $\ket{\psi}$ and $\ket{\phi}$ are either
LOSR-equivalent or LOSR-incomparable. This result is similar to Theorem 1 of
Ref. Bennett  _et al._ (2000), which concerns LOCC rather than LOSR, and
establishes that if $\ket{\psi}$ and $\ket{\phi}$ are marginally isentropic,
then they are either LU-equivalent or LOCC-incomparable. Because LOCC-
incomparability implies LOSR-incomparability, and LU-equivalence is the same
as LOSR-equivalence (by Lemma 2), the analogue of Corollary 5 for marginally
isentropic states can also be considered a corollary of Theorem 1 of Ref.
Bennett  _et al._ (2000). ∎

Applying Corollary 5 to the case of bipartite states, it follows that any pair
of pure bipartite states having equal Schmidt rank but different squared
Schmidt coefficients are incomparable under LOSR. Thus, for example, it
follows that all two-qubit entangled pure states are LOSR-
incomparable.202020This provides a second way of seeing that the pair of
bipartite pure states $|\psi_{\rm max}\rangle$ and $|\psi_{\rm
partial}\rangle$ are LOSR-incomparable.

Another result about LOSR-entanglement for pure bipartite states that follows
from Lemma 3 concerns entanglement catalysis. Specifically, whereas there is
nontrivial catalysis for pairs of bipartite pure states in the theory of LOCC-
entanglement Jonathan and Plenio (1999), this is not the case in the theory of
LOSR-entanglement. If $\ket{\psi}$ cannot be converted to $\ket{\phi}$ under
LOSR, i.e., $\ket{\psi}\not\mapsto\ket{\phi}$, then $\ket{\psi}$ also cannot
be catalytically converted to $\ket{\phi}$ under LOSR, i.e., there is no pure
state $\ket{\chi}$ such that
$\ket{\psi}\otimes\ket{\chi}\mapsto\ket{\phi}\otimes\ket{\chi}$. The proof is
provided in Appendix F. Consequently, whether one uses LOCC or LOSR as the
free operations also makes a difference for the catalytic order over entangled
states.

##  VI Discussion

The fact that researchers have previously taken LOCC-entanglement, rather than
LOSR-entanglement, as the appropriate notion in Bell scenarios has led to the
view that the interplay between entanglement and nonlocality is rather
perverse. We have shown, however, that this perversity is merely an artifact
of considering the wrong notion of entanglement. Once one focusses on LOSR-
entanglement, the interplay with nonlocality is found to be quite natural.

In particular, we have shown that this perspective resolves the longstanding
‘anomaly of nonlocality’. In addition to its foundational dividends on this
front, the paradigm of LOSR-entanglement is also likely to be significant for
certain more practical problems. For instance, the fact that partially
entangled states provide an advantage over maximally entangled states for the
tasks of certified randomness generation Acín  _et al._ (2012) and for the
rate of secret key extraction Scarani  _et al._ (2006); Acín  _et al._ (2006)
suggests that one should seek to identify the LOSR-entanglement monotones that
quantify the degree of success one can achieve in such tasks.

Our suggestion to use LOSR supplemented by 2-way shared entanglement
(LOSR2WSE) to define genuinely 3-way entanglement and genuinely 3-way
nonlocality has not only conceptual merit—providing a satisfactory resolution
to a tripartite ‘anomaly of nonlocality’ for instance—but also the potential
for practical applications. For example, it opens the door to defining
quantitative measures of these types of entanglement and nonlocality. This is
in contrast to the LOCC-motivated notions of genuinely 3-way entanglement and
nonlocality, which we showed violate basic principles of a resource theory.
Our approach is also easily adapted to the study of types of nonclassicality
in causal structures beyond that of the standard Bell scenario Henson  _et
al._ (2014); Fritz (2012); Wolfe  _et al._ (2016), which has relevance for the
study of quantum communication networks (e.g., the so-called ‘quantum
internet’).

Self-testing of entangled states is another topic that has practical
significance; for instance, it provides a device-independent means of
certifying quantum devices. It is crucial, therefore, that we properly
understand what, precisely, is certified in a self-testing experiment. Our
resource-theoretic reframing of self-testing brings clarity to this question
(by establishing the freedom up to which a state may be identified by self-
testing), and has allowed us to identify and explain certain assumptions that
have been present in the literature (e.g., the restriction to a set of pure
bipartite entangled states of a fixed Schmidt rank). In addition, our new
approach to self-testing proves to be applicable also to mixed states.
Contending with mixed states is important for experimental implementations of
self-testing, because the ideal of purity is never achieved in the laboratory.
Hence, our results may shed light on how to achieve noise-robustness in self-
testing protocols. Finally, our approach to self-testing also opens the door
to exploring relaxations of the notion of certification of quantum devices:
consider the case of a box that does not self-test a state, i.e., the case
where there is no unique least equivalence class of states that can yield the
box under LOSR processing. In the conventional approach, such a box would be
disregarded since it is deemed not to be useful for device certification.
However, in our approach, it is natural to tolerate such nonuniqueness and to
focus on characterizing the set of states that can yield the given box under
LOSR processing. Cases of nonuniqueness will of course imply more uncertainty
about the identity of the device than if self-testing were possible, but they
can nonetheless provide a kind of noisy certification: the more constrained
the set one identifies, the less uncertainty one has about the quantum device.

Our resource-theoretic framing of self-testing also implies that the concept
of self-testing can be extended beyond quantum states and boxes, to arbitrary
pairs of resource types (hereafter referred to as “type I” and “type II”), and
to arbitrary resource theories. Suppose that resources of both types are
described in a single resource theory, so that one can specify the partial
order over all resources in terms of interconvertibility under the free
operations. Leveraging our abstract characterization of self-testing (given
explicitly in Appendix C), self-testing of a type I resource, $R_{I}$, by a
type II resource, $R_{II}$, simply means that the upward closure of $R_{II}$
among resources of type I is equal to the upward closure of $R_{I}$ among
resources of type I. Equivalently, it means that the upward closure of the
equivalence class of $R_{II}$ among equivalence classes of resources of type I
has a unique least element, namely, the equivalence class of $R_{I}$. Such a
notion of self-testing could be applied, for example, to the resource types
considered in the type-independent LOSR resource theory laid out in Refs.
Schmid  _et al._ (2019); Rosset  _et al._ (2019), which are relevant for semi-
device-independent certification protocols. It could also be applied to
completely different resource theories in other contexts.

Our results also call for a re-evaluation of other fundamental results
regarding the interplay between entanglement and nonlocality that we did not
explore here. For example, the fact that certain mixed entangled states seem
to be unable to realize any nonlocal box using projective measurements Werner
(1989) or POVM measurements Barrett (2002) is a feature of this interplay that
has generally been considered surprising. This observation led many to
consider to what extent having multiple copies of such entangled states
allowed one to realize a nonlocal box Bennett  _et al._ (1996b); Navascués and
Vértesi (2011); Palazuelos (2012). This in turn led many to study the
interplay of the distillable entanglement properties of a state (i.e., what
can be achieved with arbitrarily many copies) and its ability to realize
nonlocal boxes (e.g., Peres’ conjecture Peres (1999) and results pertinent to
it Vertesi and Brunner (2014)). The extent to which entanglement properties
are witnessed by Bell-inequality violations, hence also the extent to which
they can be device-independently certified, is one of the central questions
regarding the interplay of entanglement and nonlocality. Given that the sort
of entanglement being considered in past such works was LOCC-entanglement, and
given that, as we have argued here, this is not the appropriate sort to
consider for Bell scenarios, it will be interesting to reconsider all these
topics from the perspective of LOSR-entanglement.

In terms of developing the resource theory of LOSR-entanglement, we have
demonstrated the significance of LU-equivalence for the problem of
characterizing the necessary and sufficient conditions for convertibility of
pure entangled states under LOSR. Leveraging this result, we have solved the
problem completely in the bipartite case, and proven a useful necessary
condition for LOSR-convertibility in the multi-partite case.

Our reanalysis of entanglement in Bell scenarios has provided a strong
motivation, we believe, for developing the resource theory of LOSR-
entanglement to the same level of sophistication as has been achieved for
LOCC-entanglement. Another motivation for doing so is that the resource theory
of LOSR-entanglement has applications beyond the study of states and boxes in
Bell scenarios. Ref. Schmid  _et al._ (2019) showed that LOSR is the
appropriate set of free operations for evaluating the inconvertibility of many
other types of resources besides boxes, including steering assemblages
Einstein  _et al._ (1935); Schrodinger (1935); Wiseman  _et al._ (2007) and
teleportages Cavalcanti  _et al._ (2017); Šupić  _et al._ (2019); Hoban and
Sainz (2018), and that doing so unifies and extends a variety of seminal
results regarding interconversions between these distinct forms of
nonclassicality. Consequently, for the study of the interplay of entanglement
to these other types of nonclassical resources, it is also LOSR-entanglement
that is the appropriate notion.

We note, finally, that entanglement theory has also recently found
applications beyond quantum information processing, such as in many-body
physics and in the study of holography. Given that the notion of classical
communication does not seem to be pertinent in any of these applications,
there is reason to suspect that LOCC-entanglement might be ill-suited to these
applications. Our results, therefore, call out for a reassessment of how to
formalize the notion of entanglement in these fields of study, and a
consideration of whether the paradigm of LOSR-entanglement might be more
suitable.

## Acknowledgments

We acknowledge Denis Rosset and Jed Kaniewski for valuable discussions. D.S.
is supported by a Vanier Canada Graduate Scholarship. T.C.F. is supported by
the Natural Sciences and Engineering Research Council of Canada (NSERC), grant
411301803. R.K. is supported by the Program of Concerted Research Actions
(ARC) of the Université libre de Bruxelles. ABS acknowledges support by the
Foundation for Polish Science (IRAP project, ICTQT, contract no. 2018/MAB/5,
co-financed by EU within Smart Growth Operational Programme). This research
was supported by Perimeter Institute for Theoretical Physics. Research at
Perimeter Institute is supported in part by the Government of Canada through
the Department of Innovation, Science and Economic Development Canada and by
the Province of Ontario through the Ministry of Colleges and Universities.

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##  Appendix A Extracting a monotone from each instance of the anomaly

As mentioned in the main text, one can define LOSR monotones which capture
features of the LOSR preorder that are not captured by LOCC monotones simply
by repurposing each of the known examples of “anomalies of nonlocality”. We
now define a few such monotones explicitly. Each example of the anomaly
involves a particular nonlocality-witnessing function, that is, a real-valued
function over boxes for which there is a threshold value that attests to the
box being nonlocal. The natural monotone over quantum states corresponding to
that function is obtained by a yield-construction Gonda and Spekkens (2019),
in which one computes the maximum value of that function over all boxes that
can be generated using LOSR operations from the given state. The resulting
function from states to the real numbers is an LOSR-entanglement monotone, due
to the explicit maximization over LOSR operations in the definition.

In the first anomaly we listed above, the function which was used to witness
the nonlocality of a given box was the maximum probability with which that box
could run Hardy’s version of Bell’s theorem. Denoting this function $f_{\rm
probHardy}$, one can define an LOSR monotone for states as follows:

$M_{\rm probHardy}(\rho):=\max_{\tau\in\textup{LOSR}}\\{f_{\rm
probHardy}(\tau(\rho))\\},$ (17)

where $\tau$ is any LOSR operation taking states to boxes. As an aside, it is
worth noting that the monotone $M_{\rm NPR}$ from Ref. Wolfe  _et al._ (2019),
which is defined for boxes, is maximized by a particular Hardy box (defined in
Table 4 of Ref. Wolfe  _et al._ (2019)). Hence, another relevant LOSR monotone
for states can be defined as follows:

$M_{\rm NPR}(\rho):=\max_{\tau\in\textup{LOSR}}\\{M_{\rm NPR}(\tau(\rho))\\},$
(18)

where $\tau$ is any LOSR operation taking states to boxes.

In the second anomaly we listed above, the function which was used to witness
nonlocality was a tilted Bell inequality (viewed as a function from boxes to
the reals). Denoting the function defined by the tilted Bell inequality with
tilt $\alpha$ Yang and Navascués (2013); Bamps and Pironio (2015) as
$f_{\alpha-{\rm tilted}}$, an LOSR monotone for states can be defined as
follows:

$M_{\alpha-{\rm tilted}}(\rho):=\max_{\tau\in\textup{LOSR}}\\{f_{\alpha-{\rm
tilted}}(\tau(\rho))\\},$ (19)

where $\tau$ is any LOSR operation taking states to boxes.

In the third anomaly we listed above, the function which was used to witness
nonlocality was the secret key rate extractable from the given box. Denoting
this function $f_{\rm KR}$, an LOSR monotone for states can be defined as
follows:

$M_{\rm KR}(\rho):=\max_{\tau\in\textup{LOSR}}\\{f_{\rm KR}(\tau(\rho))\\},$
(20)

where $\tau$ is any LOSR operation taking states to boxes.

By now, the pattern is clear. One can define an analogous yield-based monotone
for each of the anomalies.

We leave the task of using these monotones to glean insights into the LOSR
preorder over quantum states for future work. A deficiency of the above
monotones (as presented) is that it may be difficult to perform the
optimization in their definitions (especially since the convex set which is
optimized over does not have a finite set of extreme points). Those monotones
which are defined in terms of an abstract function (such as $f_{\rm KR}$) will
also be at least as difficult to compute as the functions themselves. In most
cases, finding a closed form expression for a given monotone is paramount for
it to be practically useful.

##  Appendix B Proving LOSR incomparability between two Bell-pairs and the GHZ
state

The purpose of this section is to prove Eq. (8), i.e. the LOSR incomparability
of the tripartite pure states

$\displaystyle\ket{\psi_{\rm 2Bell}}$
$\displaystyle\equiv|\phi^{+}\rangle_{A_{1}B}\otimes|\phi^{+}\rangle_{A_{2}C},\text{
and }$ (21) $\displaystyle|\psi_{\rm GHZ}\rangle$
$\displaystyle\equiv\tfrac{1}{\sqrt{2}}(|000\rangle_{ABC}+|111\rangle_{ABC}),$
(22)

as was required for resolving the anomaly regarding genuinely $3$-way
entanglement.

To prove incomparability, we must show that neither state can be converted
into the other. The impossibility of each of these two conversions is proved
by showing that the necessary condition given in Eq. (15) is not satisfied.
For tripartite systems, there are exactly three bipartitions among the three
parties, henceforth denoted $\\{A|BC,B|AC,C|AB\\}$. The associated squared
Schmidt coefficients for $\ket{\psi_{\rm 2Bell}}$ and $|\psi_{\rm GHZ}\rangle$
are as follows:

$\displaystyle\lambda^{(A|BC)}_{\rm 2Bell}$
$\displaystyle=(\tfrac{1}{4},\tfrac{1}{4},\tfrac{1}{4},\tfrac{1}{4}),\qquad$
$\displaystyle\lambda^{(A|BC)}_{\rm GHZ}$
$\displaystyle=(\tfrac{1}{2},\tfrac{1}{2}),$ (23)
$\displaystyle\lambda^{(B|AC)}_{\rm 2Bell}$
$\displaystyle=(\tfrac{1}{2},\tfrac{1}{2}),$
$\displaystyle\lambda^{(B|AC)}_{\rm GHZ}$
$\displaystyle=(\tfrac{1}{2},\tfrac{1}{2}),$
$\displaystyle\lambda^{(C|AB)}_{\rm 2Bell}$
$\displaystyle=(\tfrac{1}{2},\tfrac{1}{2}),$
$\displaystyle\lambda^{(C|AB)}_{\rm GHZ}$
$\displaystyle=(\tfrac{1}{2},\tfrac{1}{2}).$

First, when considering the $A|BC$ partition, it becomes clear that
$|\psi_{\rm GHZ}\rangle\not\mapsto|\psi_{\rm 2Bell}\rangle$ under LOSR as
there is no vector $\lambda^{(A|BC)}_{\zeta}$ such that
$(\lambda^{(A|BC)}_{\rm GHZ})^{\downarrow}=(\lambda^{(A|BC)}_{\rm
2Bell}\otimes\lambda^{(A|BC)}_{\zeta})^{\downarrow}$, and thus Eq. (15) cannot
be satisfied. (One can also see this through the failure of the condition in
Corollary 6 in Appendix E, since the ratio of the Schmidt rank of $|\psi_{\rm
GHZ}\rangle$ to that of $|\psi_{\rm 2Bell}\rangle$ is not an integer.) Second,
Eq. (15) implies that $|\psi_{\rm 2Bell}\rangle\mapsto|\psi_{\rm GHZ}\rangle$
only if there exists an auxiliary state $\ket{\zeta}$ such that

$\displaystyle\lambda^{(A|BC)}_{\zeta}$
$\displaystyle=(\tfrac{1}{2},\tfrac{1}{2}),$ (24)
$\displaystyle\lambda^{(B|AC)}_{\zeta}$ $\displaystyle=(1),$ (25)
$\displaystyle\lambda^{(C|AB)}_{\zeta}$ $\displaystyle=(1).$ (26)

However, these squared Schmidt coefficients are not consistent with any
tripartite state $\ket{\zeta}$. This can be seen by the fact that Eq. (25)
implies that $\ket{\zeta}$ factorizes across the $B|AC$ partition, and Eq.
(26) implies that $\ket{\zeta}$ factorizes across the $C|AB$ partition, and
together these two facts imply that $\ket{\zeta}$ must factorize across the
$A|BC$ partition. But this contradicts Eq. (24), since the latter can only
hold if $\ket{\zeta}$ is entangled across the $A|BC$ partition. Therefore,
$|\psi_{\rm 2Bell}\rangle\not\mapsto|\psi_{\rm GHZ}\rangle$ under LOSR.

##  Appendix C An equivalent characterization of self-testing

We now elaborate on a more abstract resource-theoretic characterization of
self-testing of states which is equivalent to the one given in the main text.

Consider the preorder of resources, including both states and boxes, induced
by convertibility relations under LOSR operations.

######  Definition 3.

The _upward closure_ of a resource $R$, denoted here by ${\rm UC}(R)$, is the
set of all resources that can be converted to $R$ via the free operations.
Formally,

$\displaystyle{\rm UC}(R):=\\{R^{\prime}:R^{\prime}\mapsto R\\}.$

Note that $R$ may be either a state or a box, as may be the resources
$R^{\prime}$ in its upward closure. Clearly, the upward closure of a resource
$R$ is the same as that of any resource within the equivalence class of $R$.
Hence, one could also think of ${\rm UC}(R)$ as “the upward closure of the
equivalence class of $R$”.

From our definition in the main text, $\rho$ is self-tested by $B$ if all
states which can generate $B$ can also generate $\rho$. In other words, $\rho$
is self-tested by $B$ if the upward closure of $B$ contains only those states
which are also in the upward closure of $\rho$.

######  Definition 4.

A state $\rho$ can be self-tested if there exists a box $B$ such that

${\rm UC}(B)\cap{\rm States}={\rm UC}({\rho})\cap{\rm States},$

where ${\rm States}$ is the set of quantum states.

The definition can also be usefully expressed in terms of the partial order of
equivalence classes of resources: $\rho$ is self-tested by $B$ if the upward
closure of the equivalence class of $B$ among equivalence classes of states
has a unique least element, namely, the equivalence class of $\rho$.

As pointed out in the main text and elaborated on in the discussion section,
this definition generalizes easily to more general types of resources and even
to other resource theories.

##  Appendix D On the insufficiency of the condition in Corollary 4 for LOSR-
convertibility

One consequence of Eq. (11) is that each pair of LOCC-incomparable states is
also LOSR-incomparable. We can leverage this observation, together with known
results on LOCC-incomparability, to show that Eq. (15) in Corollary 4 fails to
be a sufficient condition for all $n>2$.

Specifically, it is known that there exists chiral states for each $n>2$ which
are LOCC-incomparable to their complex conjugate (in a particular basis)
Bennett  _et al._ (2000); Kraus (2010). Eq. (11) implies that these chiral
states are also LOSR-incomparable to their complex conjugate. These instances
of LOSR-incomparability are not witnessed by a failure of Eq. (15) because the
squared Schmidt coefficients of any state are unchanged after complex
conjugation. Consequently, satisfaction of Eq. (15) is not a sufficient
condition for LOSR-convertibility.

For a concrete example with $n=3$, consider the tripartite state
$\ket{\psi}=\ket{+++}_{ABC}+\frac{i-1}{2\sqrt{2}}\ket{111}_{ABC}$ and its
complex conjugate (relative to the computational basis), $\ket{\psi^{\ast}}$.
Evidently, the squared Schmidt coefficients of $\ket{\psi}$ and
$\ket{\psi^{\ast}}$ coincide for all bipartitions:

$\displaystyle\forall\beta:\lambda^{(\beta)}_{\psi}=\lambda^{(\beta)}_{\psi^{\ast}}=\left(\tfrac{1}{2}+\sqrt{\tfrac{5}{32}},\;\tfrac{1}{2}-\sqrt{\tfrac{5}{32}}\right).$
(27)

It follows that the necessary condition for
$\ket{\psi}\mapsto\ket{\psi^{\ast}}$ expressed in Eq. (15) is seen to be
satisfied because one can take $\forall\beta:\lambda_{\zeta}^{(\beta)}=(1)$,
which corresponds to $\ket{\zeta}$ being a product state, i.e.,
$\ket{\zeta}=\ket{\zeta_{1}}_{A}\ket{\zeta_{2}}_{B}\ket{\zeta_{3}}_{C}$.

##  Appendix E A method for computing $\lambda_{\zeta}^{(\beta)}$

This appendix delineates a method for computing $\lambda_{\zeta}^{(\beta)}$ in
Eq. (15) (if it exists) for a fixed bipartition $\beta$ when $\ket{\psi}$ and
$\ket{\phi}$ are known. This method is useful because, as previously
mentioned, Eq. (15) constitutes a necessary constraint for LOSR (or LO)
convertibility between $n$-partite pure states. Moreover, in the case of
bipartite systems, Eq. (15) reduces to the necessary and sufficient condition
given by Eq. (14), for which the following procedure also applies.

Importantly, Eq. (15) already constrains the Schmidt ranks of $\ket{\zeta}$
relative to all bipartitions,

$\forall\beta:\mathrm{SR}_{\psi}^{(\beta)}=\mathrm{SR}_{\phi}^{(\beta)}\mathrm{SR}_{\zeta}^{(\beta)}.$

######  Corollary 6.

A pure state $\ket{\psi}$ can be converted to pure state $\ket{\phi}$ by LOSR
(or by LO) only if for each bipartition $\beta$, the ratio of Schmidt ranks
between $\ket{\psi}$ and $\ket{\phi}$ are positive integers, i.e.,

$\displaystyle\forall\beta:k^{(\beta)}\coloneqq\frac{\mathrm{SR}_{\psi}^{(\beta)}}{\mathrm{SR}_{\phi}^{(\beta)}}\in\mathbb{Z}^{+}.$
(28)

Therefore, one should first compute $\lambda_{\psi}^{(\beta)}$ and
$\lambda_{\phi}^{(\beta)}$ and thereby infer
$k^{(\beta)}\coloneqq\mathrm{SR}_{\psi}^{(\beta)}/\mathrm{SR}_{\phi}^{(\beta)}$;
if $k^{(\beta)}$ fails to be an integer for any bipartition $\beta$, then
there is no solution to Eq. (15) for any $\lambda_{\zeta}^{(\beta)}$.

For a fixed bipartition $\beta$, Eq. (15) can equivalently be expressed
through the existence of a permutation $\pi$:

$\displaystyle\exists\pi,\exists\zeta:\pi(\lambda_{\psi}^{(\beta)})=\lambda_{\phi}^{(\beta)}\otimes\lambda_{\zeta}^{(\beta)}.$
(29)

Consequently, by iterating through the permutations of
$\lambda_{\psi}^{(\beta)}$ (there are $(\mathrm{SR}_{\psi}^{(\beta)})!$ of
these) in search of a solution for $\lambda_{\zeta}^{(\beta)}$ to the linear
system given in Eq. (29), one can determine $\lambda_{\zeta}^{(\beta)}$, if it
exists. For illustrative purposes, consider the linear system when
$\mathrm{SR}_{\psi}^{(\beta)}=6$ and $\mathrm{SR}_{\phi}^{(\beta)}=2$. Letting
$\lambda_{\psi,i}^{(\beta)}$ denote the $i$-th component of the vector
$\lambda_{\psi}^{(\beta)}$, the linear system can be expressed as:

$\displaystyle\exists\pi:\begin{pmatrix}\lambda_{\psi,\pi(1)}^{(\beta)}\\\
\lambda_{\psi,\pi(2)}^{(\beta)}\\\ \lambda_{\psi,\pi(3)}^{(\beta)}\\\
\lambda_{\psi,\pi(4)}^{(\beta)}\\\ \lambda_{\psi,\pi(5)}^{(\beta)}\\\
\lambda_{\psi,\pi(6)}^{(\beta)}\end{pmatrix}=\begin{pmatrix}\lambda_{\phi,1}^{(\beta)}&0&0\\\
\lambda_{\phi,2}^{(\beta)}&0&0\\\ 0&\lambda_{\phi,1}^{(\beta)}&0\\\
0&\lambda_{\phi,2}^{(\beta)}&0\\\ 0&0&\lambda_{\phi,1}^{(\beta)}\\\
0&0&\lambda_{\phi,2}^{(\beta)}\end{pmatrix}\begin{pmatrix}\lambda_{\zeta,1}^{(\beta)}\\\
\lambda_{\zeta,2}^{(\beta)}\\\ \lambda_{\zeta,3}^{(\beta)}\end{pmatrix}.$ (30)

Alternatively, the normalization constraint for $\lambda_{\zeta}^{(\beta)}$,
$\sum_{i=1}\lambda_{\zeta,i}^{(\beta)}=1$, as well as the positivity
constraints, $\forall j:\lambda_{\zeta,j}^{(\beta)}\geq 0$, permit one to
regard the problem expressed by Eq. (29) as a polytope inclusion problem,
namely,

$\displaystyle\exists\pi:\pi(\lambda_{\psi}^{(\beta)})\in\mathrm{conv}\left(\\{\lambda_{\phi}^{(\beta)}\otimes
e_{j}\\}_{j=1}^{k^{(\beta)}}\right),$ (31)

where the set of vectors $\\{e_{j}\\}_{j=1}^{k^{(\beta)}}$ denote the vertices
of a $k^{(\beta)}$-simplex, i.e., $(e_{j})_{i}=\delta_{j,i}$, and
$\mathrm{conv}(\cdot)$ denotes the operation of taking the convex hull of a
set of vectors.

##  Appendix F Bipartite entanglement catalysis under LOSR

Recall that for bipartite pure states, the condition for
$\ket{\psi}\mapsto\ket{\phi}$ under LOSR can be expressed in terms of vectors
of squared Schmidt coefficients as

$\exists\ket{\zeta}:\lambda_{\psi}^{\downarrow}=(\lambda_{\phi}\otimes\lambda_{\zeta})^{\downarrow}.$
(32)

(See Eq. 14.) Consequently, the condition for
$\ket{\psi}\otimes\ket{\chi}\mapsto\ket{\phi}\otimes\ket{\chi}$ under LOSR is

$\exists\ket{\zeta}:(\lambda_{\psi}\otimes\lambda_{\chi})^{\downarrow}=(\lambda_{\phi}\otimes\lambda_{\chi}\otimes\lambda_{\zeta})^{\downarrow}.$
(33)

Now note that if
$(\lambda_{\psi}\otimes\lambda_{\chi})^{\downarrow}=(\lambda_{\phi}\otimes\lambda_{\chi}\otimes\lambda_{\zeta})^{\downarrow}$
holds for some $\ket{\zeta}$, then
$\lambda_{\psi}^{\downarrow}=(\lambda_{\phi}\otimes\lambda_{\zeta})^{\downarrow}$
holds for the same $\ket{\zeta}$. (To see this, take any entry in
$\lambda_{\chi}$, say $\lambda_{\chi,j}$. Then consider the subset of
equalities pertaining to $\lambda_{\chi,j}$, i.e.
$\exists\ket{\zeta}:(\lambda_{\chi,j}\lambda_{\psi})^{\downarrow}=(\lambda_{\chi,j}(\lambda_{\phi}\otimes\lambda_{\zeta}))^{\downarrow}$.
Dividing entry-wise by $\lambda_{\chi,j}$ yields the result.) But this says
that if the conversion can be achieved with a catalyst $\ket{\chi}$ then it
can also be achieved without the catalyst. Hence, there is no nontrivial
catalysis for pure bipartite states in the theory of LOSR entanglement.