Figure B.2 shows that under targeted subsidies, the most impoverished children
from the 1970 cohort experience a significant increase in heights, with median
heights increasing by 1.3, 0.6, 0.4, and 0.3 cm when the most impoverished
children receive 55, 30, 21, and 16 percent targeted discounts. With a 55
percent price discount, the most impoverished children’s median height reaches
77.9 cm and is 0.5 cm higher than the median height for the wealthiest
children. When both the richest and the most impoverished children receive 13
percent price discounts under the universal policy, median height increases
more for the poorest (+0.25 cm) than the richest (+0.17 cm).

a $\sigma_{R_{y,v}}$ = 0.5 cm

b $\sigma_{R_{y,v}}$ = 3.5 cm

Figure B.3: Effects of targeted policies on poorest households

Figure B.3 shows similar results in panel a under $\sigma_{R_{y,v}}=0.5$ and
in panel b under $\sigma_{R_{y,v}}=3.5$. Panel b of Figure B.3 shows that with
55 percent price discounts under the most-targeted policy, due to endogenous
shifts in reference-point distributions, median heights increase significantly
by 1.3, 1.9, 2.4, and 2.7 cm for the 1970, 1972, 1974, and 1976 cohorts of the
poorest children, respectively.B.6B.6B.6Correspondingly, median protein intake
increases by 23, 38, 50, and 58 percent for these cohorts. On the other hand,
the universal policy provides small height increases to the poorest initially,
but effects amplify significantly across cohorts. Specifically, for the 1970
cohort of the poorest children, the most-targeted policy (55 percent discount)
increases the median height by 1.3 cm, which is 4.8 times larger than the 0.26
cm median height increase induced by the universal policy (13 percent price
discount). For the 1976 cohort of the poorest children, however, the effect of
the most-targeted policy is only 1.8 times greater than the effect of the
universal policy, each of which increases median heights by 2.7 and 1.5 cm,
respectively.B.7B.7B.7Comparing the poorest children across cohorts, the
increase in median height is 5.8 times larger for the 1976 cohort compared to
the 1970 cohort (1.5 vs. 0.26 cm). The increase in median height for the
richest children across cohorts is 2.2 times (2.7 vs. 1.25 cm). Additionally,
for the 1970 cohort of the poorest children, 100 percent of the increases in
height are due to first-period price-discount effects, but for the 1976 cohort
of the poorest children, under 55, 30, 21, 15, and 13 percent price discounts,
first-period price-discount effects only account for 46, 31, 24, 19, and 16
percent of the total effect of each policy.

Figure B.2 presents results for the 1976 cohort. Here we compare heights for
the 1976 cohort to the heights at month 24 for the 1970 poorest and richest
cohorts without subsidies. Across the five policies from most-targeted to
universal, heights for the richest children increase by 1.3, 1.3, 1.3, 1.3,
and 1.6 cm, equivalent to 48, 65, 76, 81, and 103 percent of the increases in
heights for the poorest children, which are 2.7, 2.0, 1.7, 1.6, and 1.5 cm.
The 1.3 cm median height increases for the richest children under the four
targeted policies are due to the reference-point externality of the treatments
on non-targeted individuals. The poorest children reach a higher median height
(79.4 cm) than the richest non-targeted children (78.8 cm) only under the 55
percent price discount policy.

####  B.1.3 Distributional Effects of Different Levels of Targeted Policies

This section focuses on illustrating the potential distributional effects of
targeting given reference points. A planner might be interested in maximizing
some joint village welfare function that considers the entire distribution of
height or moments of this distribution. We do not assume which percentiles or
levels of height interest the planner. Given the specifics of our empirical
setting, we present the full predicted distribution that contains information
that might be of interest to a planner interested in early childhood height
and health outcomes.

Figure B.4 plots various percentile levels of overall – including both
targeted and non-targeted children – heights of month-24
distributionsB.8B.8B.8The distributional outcomes are presented without
drawing from the measurement error distributions. under each policy
experiment. As shown in both panel a and panel b of Figure B.4, given our
sample of individuals, mean heights are relatively constant across targeting
levels, and variance is minimized when 70 percent of the poorest children are
targeted to receive subsidies. Specifically, we analyze the effects of
targeting from 20 to 90 percent of the poorest children at 10 percent
intervals along with a universal policy. Price discounts at 55, 39, 30, 25,
21, 18, 16, 14, and 13 percent for each of the nine policies are calculated to
be approximately budgetary constant.

a $\sigma_{R_{y,v}}$ = 0.5 cm]

b $\sigma_{R_{y,v}}$ = 3.5 cm

Figure B.4: Height Distribution from Targeted to Universal Subsidies

The dashed black line in both panels of Figure B.4 shows that the mean heights
across policies differ by at most only 0.054 cm.B.9B.9B.9The highest mean in
panel b, 78.58 cm, is achieved under targeting 20 percent of the poorest
households, and the lowest mean, 78.52 cm, is achieved under the universal
policy. Our policy experiments shift a fixed amount of subsidies from one
subset of individuals to another in the form of price discounts. Our findings
suggest that these policies have linear effects on height.B.10B.10B.10One
might suspect that the variations in means across policies would be large
given the concavity of the production function: all else equal, one gram of
protein transferred from those with high intake to those with lower intake
should lead to a net gain in overall height. Here, however, given that
households re-optimize with new subsidies, the increases in protein intakes
are less than the amounts of protein transferred to the poor. Who receives the
transfers and how much they receive changes the relative heights among
individuals, but not the overall average height significantly.

As policies shift price-discount intensity and recipients, there are
significant variations in height distributions across policies. One measure of
variation is the difference between the 10th and 90th percentiles of heights
across policies, which is the difference between the dashed blue lines at the
top and bottom of both panels of Figure B.4. In panel b, these gaps are 1.5,
1.1, 0.9, and 0.75 cm when 20, 30, 40, and 50 percent of the most impoverished
children receive price discounts. The gaps are the tightest at 0.69, 0.66, and
0.66 cm under policies that provide 60, 70, and 80 percent of most
impoverished children with price discounts;B.11B.11B.11In panel b, the s.d. of
heights, at 0.40 cm, is also the smallest for the policy that targets 70
percent of most impoverished children. the gaps widen again to 0.8 and 0.9 cm
under the two most universal policies.

In our setting, the tightening of the height distribution’s higher and lower
percentiles drives the minimization of height variation when 70 percent of the
most impoverished children are targeted. Under the most-targeted policies,
children from the poorest households receive huge discounts and become the
tallest children, driving the 90th percentile of height at the month-24
distribution up.B.12B.12B.12For the most-targeted policy that provides 55
percent price discounts to the children in the lowest quintile of income,
these most impoverished children’s heights increase significantly, and they
move to the highest quintile of realized height in the month-24 distribution.
As shown in Figure B.2, the median heights of these most impoverished children
under the 55 percent discount policy exceed the median heights of children
from the wealthiest quintile of income. Hence this pushes the 80 and 90th
percentile of the overall height distribution up under the most-targeted
policies. Concurrently, many children with below-median income do not receive
subsidies under the most-targeted policies, and they push down the 10th and
20th percentiles of the height distribution. Second, under the universal
policies, children from the wealthiest households receive price discounts, and
they drive the 90th percentile height distribution up.B.13B.13B.13As shown in
Figure B.2, the smaller discount (13 percent discount under the universal
policy) given to the wealthiest children allows them to achieve higher heights
at month 24 than a more significant discount (16 to 21 percent under slightly
more-targeted policies) given to children in the lower portions of the income
distribution.

Concurrently, because subsidies for the poorest households are much lower than
under more-targeted policies, they push the 10th and 20th percentiles of the
height distribution lower. Consequently, we observe the broader distributions
of heights under the most-universal and most-targeted policies, but a tighter
distribution in the middle of Figure B.4.


# Operational locality in global theories

Lea Krämer  Institute for Theoretical Physics, ETH Zurich, Switzerland  Lídia
del Rio  Institute for Theoretical Physics, ETH Zurich, Switzerland  School of
Physics, University of Bristol, United Kingdom<EMAIL_ADDRESS>
###### Abstract

Within a global physical theory, a notion of locality allows us to find and
justify information-processing primitives, like non-signalling between distant
agents. Here we propose exploring the opposite direction: to take agents as
the basic building blocks through which we test a physical theory, and recover
operational notions of locality from signalling conditions. First we introduce
an operational model for the effective state spaces of individual agents, as
well as the range of their actions. We then formulate natural secrecy
conditions between agents and identify the aspects of locality relevant for
signalling. We discuss the possibility of taking commutation of
transformations as a primitive of physical theories, as well as applications
to quantum theory and generalized probability frameworks. This “it from bit”
approach establishes an operational connection between local action and local
observations, and gives a global interpretation to concepts like discarding a
subsystem or composing local functions.

Figure 1: There and back again: physical locality and signalling. On the left,
the spatial location of several agents (dark dots) and their range of
communication (overlapping circles) are depicted; on the right, the
corresponding signalling graph. a. In physical theories, the notion of a
space-time background where agents are positioned, together with principles
about the range of signalling (e.g. the finite speed of light) allow us to
derive information-processing concepts like non-signalling agents. b. Reverse
direction: starting from the notion of agents that may or not be able to
communicate and minimal assumptions on the nature and range of signalling, it
may be possible to deduce both the space-time structure of the theory and the
position of agents in it, to a good approximation. We can take inspiration
from a simple example in the field of localization in wireless sensor networks
(for a review see e.g. Ref. [1]). Forest fire prevention mechanisms can be
implemented by dropping a large number of smoke-detecting sensors from a plane
over the forest. The sensors (our agents) are equipped with short-range
communication systems, and land at random positions. One then collects the
data of which sensors can signal to each other. From the signalling graph, it
is possible to reconstruct the relative positions of the sensors on the ground
to high accuracy — that way, when the smoke alarm goes off in a sensor, the
fire-response team can quickly locate it.

In modelling local agents acting within a global theory, the intuitive
assumption is that both their actions and their knowledge are restricted to a
bounded region. The canonical example is a scientist who has full control of
her lab and can perform local tomography. In reality though, the breadth of
knowledge and the range of action of agents may be decoupled. For example,
prisoners can acquire global knowledge by reading the news, but their actions
are limited to small subsystems. Conversely, someone locked in a control room
may only have local knowledge of the shapes of different buttons, but pressing
one may have global consequences. The observation that the knowledge and
action do not always go hand in hand implies that in order to model agents we
have to specify both (Section 1). This naturally leads us to search for
minimal operational constraints needed to ensure that agents are truly local.

Here we motivate a notion of _secrecy_ between agents, which captures whether
actions performed by an agent (like writing a message, choosing a bit or
preparing a quantum state) can be perceived by another (Section 2);
traditional notions of non-signalling correspond to an extended secrecy
between space-like separated regions (Section 4). This work brings together
and clarifies concepts of locality used in quantum theory, generalized
probabilistic theories and field theories. It highlights that the _state
space_ and _transformations_ of a theory are but a subjective choice of
representation of the underlying physical theory from a viewpoint that is
convenient to a given agent, as argued by Spekkens [2]. Here, we tentatively
suggest commutation of transformations as a primitive of physical theories. In
particular, we show how to derive local agents (and effective descriptions of
local subsystems) from commutation relations on global transformations
(Section 3).

This work draws from our “Resource theories of knowledge” [3], and has natural
applications in multi-player settings, like cryptographic scenarios, games or
resource theories. There is yet a more exciting possible application: to
recover the space-time structure of a physical theory from the primitive
notion of test agents, in the spirit of Hardy’s operational GR [4] and to the
task of localization in wireless sensor networks [1]. The idea is to send out
agents (or probes) to unknown positions, see if they can communicate with each
other, and use the signalling graph to define distances between agents,
reconstruct their relative positions, and infer properties of space-time
(Figure 1). For this we must first find appropriate, theory-independent
notions of agents and signalling.

##  1 Modelling agents

We start with a top-down approach, where we first describe a global theory (as
seen by a global agent), and then model restricted agents acting within that
theory.

###  1.1 Global theory

From the point of view of a given global agent, a global theory may be
represented via a state space $\Omega$ and a set of transformations
$\mathcal{T}$ that are available to the agent [5, 6, 7, 8]. We can think of
the state space as the “language” chosen by this global observer to describe
nature. For example, $\Omega$ could be the set of coordinates and momenta of
all celestial bodies; in quantum theory, it could be the set of valid density
matrices over a global Hilbert space. It need not be a static picture: in
astronomy, an alternative state space $\Omega^{\prime}$ could be the set of
possible trajectories of celestial bodies, and in quantum theory it could
include all global Hamiltonians that determine the free evolution of density
matrices. Note that 1) $\Omega$ is not the ultimate description of reality,
just a convenient representation from the point of view of a global agent; 2)
different pictures, like $\Omega$ and $\Omega^{\prime}$, may be related and
mapped to one another [3, 2]; and 3) $\Omega$ need not have any special
structure _a priori_ besides being a set — indeed, the approach laid out here
will allow us to find an operational subsystem structure in the set of states.

The transformations in $\mathcal{T}$ represent all actions that the theory
allows the global agent to implement. We can think of them as the ways in
which the agent may test a theory, by applying actions that change state
parameters. For example, an explicit theory of a quantum universe may allow
only for unitary operations, while a more generous theory could equip the
agent with implicit large ancillas, and allow her to implement general quantum
channels, state preparations and even tomography. Again the two views can be
related: the latter is an _effective theory_ derived from the unitary quantum
theory, by internalizing part of the global space as belonging to the agent
and her instruments, and not to the object of study (the rest of the universe)
[3]. In the context of field theories, this is discussed as _emerging agency_
[4]. In a superdeterministic theory, there is only one possible course of
evolution for the universe, and $\mathcal{T}$ consists only of functions that
apply it (for example $\mathcal{T}\cong\\{e^{-iHt}\\}_{t}$ where the global
agent is given some choice of time). Formally, $\mathcal{T}$ is a monoid of
functions $f:\Omega\to\Omega$: it contains the identity transformation and is
closed under concatenation (an associative binary operation), such that
performing two actions subsequently, $f\circ g$, is still an allowed
operation. We discuss the monoidal assumption and possible relaxations in
Section 4.

###  1.2 Local agents

Figure 2: Building an agent’s effective state space. The different states of a
global space $\Omega$ are shown to an agent, who finds equivalence classes of
(subjectively) indistinguishable states. Their effective state space is then
the quotient space. a. The global state space $\Omega$ consists of three bits,
in the eight possible states depicted. An agent Alice can only see the first
bit, therefore she cannot distinguish the states in each vertical box (her
equivalence classes $[000]_{A}$ and $[100]_{A}$). Her effective state space
$\Omega_{A}=\Omega/\\!\sim_{A}$ has only two states, which can be relabelled
as $0_{A}$ and $1_{A}$ for convenience. Another agent Bob identifies the
equivalent classes $[000]_{B}$ and $[010]_{B}$, which leads us to conclude
that he can only see the second bit. Note that for example if Alice were able
to apply transformations that only change the first bit, she could not signal
to Bob (because he could not detect the change). b. Here $\Omega$ is the space
of colours, which were shown to a partly colourblind agent Marvin. Marvin
identified the colours that he could not distinguish, which allowed us to
build his reduced state space of colours $\Omega_{M}$.

Local agents are characterized by limited knowledge: their inability to
distinguish global states that appear identical in their eyes. We can
formalize this by building equivalence classes of states that are
indistinguishable from the perspective of an agent. For example, in quantum
theory, we could have an agent Bob who only has access to a Hilbert space
$\mathcal{H}_{B}$; two global states are indistinguishable (or equivalent)
from Bob’s perspective if they have the same marginal in $\mathcal{H}_{B}$.
This defines an equivalence relation
$\sigma\sim_{B}\rho:\operatorname{Tr}_{\overline{B}\,}\sigma=\operatorname{Tr}_{\overline{B}\,}\rho$,
where $\operatorname{Tr}_{\overline{B}\,}$ denotes a partial trace over all
systems except $B$. The corresponding equivalence classes are

$\displaystyle[\rho]_{B}:=\\{\sigma\in\Omega:\operatorname{Tr}_{\overline{B}\,}\sigma=\rho_{B}\\}.$

Taking the quotient over this equivalence relation gives us a new space state
$\Omega/\\!{\sim_{B}}$, which is in one-to-one correspondence with the set of
all reduced density matrices in $\mathcal{H}_{B}$. This is Bob’s _effective
state space_ , sufficient to encode all the information that he can observe
about any global state (Figure 2). In this case, the map from the global to
the local spaces (the _canonical map_) is given by the partial trace:

$\displaystyle{\bf h}_{B}:\Omega$ $\displaystyle\to\Omega/\\!{\sim_{B}}$
$\displaystyle\rho$
$\displaystyle\mapsto[\rho]_{B}\quad\cong\operatorname{Tr}_{\overline{B}\,}\rho=\rho_{B}.$

More generally, we can always build the effective state space of an agent in
this way, even if we do not know anything about the structure of the global
space (for instance whether it can be split into a convenient tensor form
$\mathcal{H}_{A}\otimes\mathcal{H}_{B}$). The construction of an agent’s
effective state space $\Omega_{B}:=\Omega/\\!{\sim_{B}}$ is in the spirit of
Leibniz principle of identity of indiscernibles [9]. Yet this operational
procedure emphasizes that both discernibility and identity are subjective
concepts (Figure 2). Limitations on Bob’s perspective may have nothing to do
with spatial locality. Bob might only have access to crude measurement
instruments unable to distinguish microscopic details of states, or he may not
be able to distinguish a global phase or gauge [4]. In generalized probability
frameworks, Bob’s perspective can correspond to a grouping of individual
global outcomes into events (Appendix E). In algebraic quantum field theory,
these equivalence classes could emerge from algebras of local observables (see
e.g. Ref. [10] for a review).

The other ingredient needed to define an agent, as we saw in the introduction,
is a description of the actions available to him. As his actions may have a
global impact, a minimal approach is to take them to be a submonoid
$\mathcal{T}_{B}\subseteq\mathcal{T}$ of the globally allowed transformations.
We discuss relaxations of this definition in Section 4. Generalizations of
this approach can be found in Ref. [3]. There, we also study explicit ways to
move between global and local views (technically, related by _Galois
insertions_), effective theories and other properties of local agents.

######  Definition 1 (Global theory and restricted agents).

A _global theory of agents_ is defined by a pair $(\Omega,\mathcal{T})$, where
$\Omega$ (the _state space_) is a set, and $\mathcal{T}$ is a monoid of
transformations $f:\Omega\to\Omega$, with the concatenation operation $\circ$.

A _restricted agent_ $B$ acting within the theory is defined by a pair
$(\sim_{B},\mathcal{T}_{B})$ , where $\sim_{B}$ is an equivalence relation in
$\Omega$ and $\mathcal{T}_{B}$ is a submonoid of $\mathcal{T}$ called the set
of _local operations_ of the agent. The quotient space
$\Omega_{B}:=\Omega/\\!\sim_{B}$ is called the _effective space_ of agent $B$.
The _reduction_ to the effective space is given by the canonical map

$\displaystyle{\bf h}_{B}:\Omega$ $\displaystyle\to\Omega_{B}$
$\displaystyle\rho$ $\displaystyle\mapsto[\rho]_{B}.$

We can always further coarse-grain the effective state space of a given agent
$B$ in order to obtain a more restricted agent $C$. For example, in
renormalization group flow, lowering the cutoff corresponds to coarse-graining
over more and more observables [11, 12]. The following proposition formalizes
this idea [3, Prop. III.5]. All proofs can be found in Appendix A.

######  Proposition 1 (Nested agents).

Let $(\Omega,\mathcal{T})$ be a global theory, and $B$, $C$ two restricted
agents. Then the following are equivalent:

  1. 1.

$C$ has more restricted knowledge than $B$, that is
$[\rho]_{B}\subseteq[\rho]_{C},\quad\forall\ \rho\in\Omega$,

  2. 2.

There exists an equivalence relation $\sim_{B\to C}$ in $B$’s effective state
space $\Omega_{B}$ such that $\Omega_{C}\cong\Omega_{B}/\sim_{B\to C}$.

##  2 Secrecy between agents

###  2.1 Secrecy

Having defined agents, we may study conditions for secrecy and non-signalling
between them. Consider a setup of two agents Alice and Bob, represented by
$A=(\sim_{A},\mathcal{T}_{A})$ and $B=(\sim_{B},\mathcal{T}_{B})$. Imagine
that Alice wants to keep her actions (like writing a message or preparing a
state) secret from Bob. This is achieved if Bob cannot tell whether she
applied them, even after post-processing.111Bob’s effective space may include
his local processing (“states that I can distinguish after applying all my
accessible operations”) or not (“states that I distinguish immediately, before
further processing”). For the sake of generality, we leave the freedom in this
decision up to the agent, and account for post-processing in the definition of
secrecy.

######  Definition 2 (Secrecy).

We say that an agent $A$ has access to _secret operations_
$\mathcal{T}_{A}^{S}\subseteq\mathcal{T}_{A}$ towards another agent $B$ if

$f_{B}\circ g_{A}(\rho)\ \sim_{B}\ f_{B}(\rho)$

for all $\rho\in\Omega,\ g_{A}\in\mathcal{T}_{A}^{S},\
f_{B}\in\mathcal{T}_{B}$. If all actions in $\mathcal{T}_{A}$ are secret
towards $B$ and $\mathcal{T}_{B}$ are secret towards $A$ we say that the two
agents are _mutually secret_.

We may ask if this definition is robust enough, that is, whether further pre-
or post-processing by Alice and Bob could destroy the secrecy of a choice of
action $g_{A}\in\mathcal{T}_{A}^{S}$. The next proposition shows that no
matter how many ‘secret’ transformations in $\mathcal{T}_{A}^{S}$ Alice
implements, or how Bob acts in between to try and recover information, he will
not detect any of the effects of Alice’s actions. In addition, it is easy to
see that pre-processing with a global function (such as distributing
entanglement between the two parties) cannot lift secrecy, since Definition 2
requires it to hold for all initial states.

######  Proposition 2 (Robustness of secrecy).

If $A$ has secret operations $\mathcal{T}_{A}^{S}$ with respect to $B$
(according to Definition 2), then pre- and post-processing cannot lift the
secrecy, that is

$\displaystyle
f_{B}^{N}\circ{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}g_{A}^{N}}\circ\dots\circ
f_{B}^{2}\circ{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}g_{A}^{2}}\circ
f_{B}^{1}\circ{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}g_{A}^{1}}\circ
f(\rho)$ $\displaystyle\sim_{B}\ f_{B}^{N}\circ\dots\circ f_{B}^{2}\circ
f_{B}^{1}\circ f(\rho),$

for all states $\rho\in\Omega$, secret operations
$\\{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}g_{A}^{i}}\\}_{i}\subseteq\mathcal{T}_{A}^{S}$
and $\\{f_{B}^{i}\\}_{i}\subseteq\mathcal{T}_{B}$, global operations
$f\in\mathcal{T}$ and $N\in\mathbb{N}$.

###  2.2 Extended secrecy

We may also ask whether Alice’s actions stay secret to Bob in the presence of
an additional global transformation $f\in\mathcal{T}$. Transformations such as
a subsystem swap or a communication channel may break secrecy; others, like
the use of a PR box, do not.222In generalized probability theories, PR boxes
can be seen as transformations that take classical inputs and return outputs
(Appendix E). For this situation, we define an extended notion of secrecy in
the spirit of Definition 2, which reduces to Definition 2 in the case
$f=\text{id}$. Here, Bob may try to post-process information before and after
the global transformation.

######  Definition 3 (Extended secrecy).

Let $A$ be an agent with access to secret operations towards an agent $B$,
$\mathcal{T}_{A}^{S}\subseteq\mathcal{T}_{A}$ . We say that
$\mathcal{T}_{A}^{S}$ is in addition secret (towards $B$) _in the presence of
a global transformation_ $f\in\mathcal{T}$ if

$f_{B}\circ f\circ f^{\prime}_{B}\circ g_{A}(\rho)\ \sim_{B}\ f_{B}\circ
f\circ f^{\prime}_{B}(\rho),$

for all $\rho\in\Omega,\ g_{A}\in\mathcal{T}_{A}^{S},\
f_{B},f^{\prime}_{B}\in\mathcal{T}_{B}$. We say that the agents are _mutually
secret in the presence of_ $f$ if all actions in $\mathcal{T}_{A}$ are secret
towards $B$ in the presence of $f$ and vice-versa.

We can now show that, analogously to Proposition 2, further pre- and post-
processing by Alice and Bob cannot lift the secrecy.

######  Proposition 3 (Robustness of extended secrecy).

If an agent $A$ only uses secret operations $g_{A}\in\mathcal{T}_{A}^{S}$ with
respect to the agent $B$ in the presence of $f\in\mathcal{T}$, then further
pre- and post-processing cannot lift the secrecy, that is

$\displaystyle\left(\bigcirc_{i=1}^{N}f_{B}^{i}\circ{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}g_{A}^{i}}\right)\circ
f\circ\left(\bigcirc_{i=1}^{N}{f^{\prime}}_{B}^{i}\circ{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{g^{\prime}}_{A}^{i}}\right)\circ
g(\rho)$ $\displaystyle\sim_{B}\ \left(\bigcirc_{i=1}^{N}f_{B}^{i}\right)\circ
f\circ\left(\bigcirc_{i=1}^{N}{f^{\prime}}_{B}^{i}\right)\circ g(\rho)$

for all states $\rho\in\Omega$, local operations
$\\{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}g_{A}^{i}}\\}_{i}\subseteq\mathcal{T}_{A}$
and $\\{f_{B}^{i}\\}_{i}\subseteq\mathcal{T}_{B}$, global operations
$g\in\mathcal{T}$ and $N\in\mathbb{N}$.

In particular, for the case in which Bob only implements post-processing at
the very end, Proposition 3 implies that $\mathcal{T}_{A}^{S}$ forms a monoid.

######  Corollary 4 (Secret monoid).

The set $\mathcal{T}_{A}^{S}$ of secret operations in the presence of a global
function $f\in\mathcal{T}$ forms a monoid, i.e.
$\text{id}\in\mathcal{T}_{A}^{S}$ and

$f_{A},g_{A}\in\mathcal{T}_{A}^{S}\implies f_{A}\circ
g_{A}\in\mathcal{T}_{A}^{S}.$

Naturally, if we further restrict the actions and knowledge of one of the
agents (as in Proposition 1), secrecy is maintained.

######  Corollary 5 (Restricted agents and secrecy).

Let $A$, $B$ and $C$ be three agents, such that $C$ is more restricted than
$B$, that is $\mathcal{T}_{C}\subseteq\mathcal{T}_{B}$ and
$[\rho]_{B}\subseteq[\rho]_{C}$, for all $\rho\in\Omega$.

If $\mathcal{T}_{B}$ was secret towards $A$ (in the presence of
$f\in\mathcal{T}$), the same is true of $\mathcal{T}_{C}$. If
$\mathcal{T}_{A}$ was secret towards $B$ (in the presence of $f$), it is still
secret towards $C$ (idem).

##  3 Commuting agents

a. b. c.

Figure 3: Three-bit example. Consider again the theory described by the state
space $\Omega$ of 3 bits, and all transformations on those bits. a. All
operations $\mathcal{T}_{C}$ that change the third bit (of which id and
$\textsc{not}_{C}$ are labeled). b. Equivalence classes $[x]_{\cancel{C}}$
built according to Definition 5. These correspond to the view of an agent who
can only distinguish the first two bits. The equivalence relation
$\sim_{\cancel{C}}$, which coarse-grains over the functions applied to the
third bit, gives us the largest effective state space relative to which
functions in $\mathcal{T}$ are secret. c. More coarse-grained equivalence
classes $[x]_{A}$ (vertical, yellow) and $[x]_{B}$ (horizontal, blue),
corresponding to an agent $A$ who can only distinguish the first bit and an
agent $B$ who only sees the second bit, respectively. Operations in
$\mathcal{T}_{C}$ are still secret relative to these two agents. In addition,
operations on the first bit are secret towards $B$ and vice-versa. These
smaller effective state spaces correspond to equivalence relations on the
effective state space $\Omega_{\cancel{C}}$ (as in the nested agents of
Proposition 1). The two-bit space $\Omega_{\cancel{C}}$ is a _common state
space_ of $A$ and $B$, including states that could be distinguished if the two
agents could work together, with $[x]_{\cancel{C}}=[x]_{A}\cap[x]_{B}$.

Now we explore how secrecy is affected when the actions of two agents $A$ and
$B$ commute. This is particularly relevant in the context of the non-
signalling principle, since actions at space-like separation naturally
commute.

######  Definition 4 (Commuting agents).

We say that two agents $A$ and $B$ commute if

$f_{B}\circ g_{A}(\rho)=g_{A}\circ f_{B}(\rho),$

for all $\rho\in\Omega,\ g_{A}\in\mathcal{T}_{A},\ f_{B}\in\mathcal{T}_{B}$.

For example, in field theory commutativity holds for measurements or field
interactions at space-like separation, and this is in general how causality is
recovered there [13]333The simplest illustration of this is the commutation of
the Klein-Gordon field operators $\phi(x)$ and $\phi(y)$ at space-like
separated $x$ and $y$, $[\phi(x),\phi(y)]=0.$ Such a commmutation condition is
also referred to in field theory as the _locality postulate_ [14].. Motivated
by this, we here take the commutation of actions in space-like separated
regions as a fundamental building block in deriving agents that are secret
relative to each other. Note that in particular, finding commuting sets of
transformations in $\mathcal{T}$ is something that can be done prior to
definitions of local agents; this is shown explicitly in Ref. [3]. Commutation
relations result in a nice algebraic structure — a lattice — in the space of
transformations [3]. This is also the case for the von Neumann bicommutant in
operator algebras [15]. Commutation is also an operational property of the
theory: for example, commutation is independent of the choice of reference
frames in relativity and quantum field theory [13, 14]. If two agents commute,
secrecy follows from simpler conditions.

######  Proposition 6 (Secrecy for commuting agents).

If $A$ and $B$ commute, then if there exists a subset of actions
$\mathcal{T}_{A}^{S}\subseteq\mathcal{T}_{A}$ such that,
$\forall\rho\in\Omega,\ g_{A}\in\mathcal{T}_{A}^{S},f_{B}\in\mathcal{T}_{B}$,

$\displaystyle f_{B}\circ f\circ g_{A}(\rho)\ \sim_{B}\ f_{B}\circ f(\rho),$

then $\mathcal{T}_{A}^{S}$ is secret towards $B$ in the presence of $f$. In
particular, $g_{A}(\rho)\sim_{B}\rho$ for all
$g_{A}\in\mathcal{T}_{A},\rho\in\Omega$ implies secrecy of $A$ towards $B$.

###  3.1 Secrecy from commutation

Starting only from commutation relations on the global transformations, we can
construct descriptions of local agents that have secret actions with respect
to each other. More specifically, given any two commuting submonoids
$\mathcal{T}_{A},\mathcal{T}_{B}\subseteq\mathcal{T}$, we can construct
equivalence relations $\sim_{A},\sim_{B}$ so that two agents Alice
$(\sim_{A},\mathcal{T}_{A})$ and Bob $(\sim_{B},\mathcal{T}_{B})$ have secret
actions with respect to each other.

The first step is to start with transformations $\mathcal{T}_{A}$ (“Alice’s
transformations"), and look for the most generous effective state space
$\Omega_{\cancel{A}}$ that is insensitive to transformations in
$\mathcal{T}_{A}$. This will model the perspective of an agent, Bob, who
cannot detect Alice’s actions. Essentially, this perspective identifies sets
of global states that Alice can locally make “converge” to the same state.

######  Definition 5 (Perspective insensitive to transformations).

Let $\mathcal{T}_{A}\subseteq\mathcal{T}$ be a submonoid of transformations.
First we define a binary relation $\sim^{\prime}_{\cancel{A}}$ in $\Omega$
called _convergence through $\mathcal{T}_{A}$_ as

$\rho\sim^{\prime}_{\cancel{A}}\sigma\iff\exists\
f_{A},g_{A}\in\mathcal{T}_{A}\text{ s.t. }f_{A}(\rho)=g_{A}(\sigma),$

We take the transitive closure $\sim_{\cancel{A}}$ of
$\sim^{\prime}_{\cancel{A}}$ to define the _perspective insensitive to
transformations_ $\mathcal{T}_{A}$,

$\displaystyle\rho\sim_{\cancel{A}}\sigma\iff\exists\ n\in\mathbb{N},\
\\{\tau_{i}\\}_{i=1}^{n}\subseteq\Omega:$
$\displaystyle\rho\sim^{\prime}_{\cancel{A}}\tau_{1}\sim^{\prime}_{\cancel{A}}\tau_{2}\sim^{\prime}_{\cancel{A}}\dots\sim^{\prime}_{\cancel{A}}\tau_{n}\sim^{\prime}_{\cancel{A}}\sigma.$

The above construction gives us minimal restrictions for independent agents.
The following theorem is adapted from [3].

######  Theorem 7 (Deriving secret agents).

Commuting submonoids $\mathcal{T}_{A},\mathcal{T}_{B}\subseteq\mathcal{T}$
give rise to descriptions of mutually secret agents

$A=(\sim_{\cancel{B}},\mathcal{T}_{A}),\qquad
B=(\sim_{\cancel{A}},\mathcal{T}_{B}).$

Indeed, all agents whose actions commute with $\mathcal{T}_{A}$ and for whom
transformations in $\mathcal{T}_{A}$ are secret must be described by a coarse-
graining of $\sim_{\cancel{A}}$ (Figure 3). This and related minor results can
be found in Appendix D. In Appendix F we generalize Theorem 7 to extended
secrecy in the presence of global functions. There, we also extend the
construction of the effective spaces of two agents to the case where the two
monoids of transformations do not commute: without commutation, this
construction is not as simple.

###  3.2 Perceived commutation from secrecy

We can now ask if the actions
$\mathcal{T}_{A},\mathcal{T}_{B}\subseteq\mathcal{T}$ of two mutually secret
agents must always commute. The answer is no, not at a global level:
unbeknownst to the two agents, their actions could affect other degrees of
freedom of the global theory. This can become relevant when the actions of two
agents affect a common environment that is not directly accessible to them but
could be recovered by a third party.

For example, consider again the state-space of three bits, where Alice can
only see the first bit and Bob the second. Now imagine that Alice has access
to all the transformations that change the first bit and, as a side effect,
reset the third bit to $0$, while Bob has access to all the actions that act
on the second bit and, as a side effect, flip the third bit. From a global
viewpoint, their actions do not commute. However, for someone that only had
access to the combined knowledge of Alice and Bob (the first two bits), their
actions would appear to commute. For such an agent, only local time ordering
of Alice and Bob’s actions matters, as the two processes $f_{A}\circ f_{B}$
and $f_{B}\circ f_{A}$ are indistinguishable. This is yet another example of
how subsystems and local descriptions represent simplified pictures of the
global theory, reducing the degrees of freedom of the theory to an operational
minimum _for a given agent_ , who in this case would not need to model global
time ordering.

##  4 Applications

Figure 4:  Process theories. Processes theories are modular, bottom-up
constructions that can be faithfully represented by diagrams [16, 8, 5, 7, 6].
Lines represent _systems_ (or “objects”) and boxes _processes_ on systems:
wires fed in from below a box can be understood as inputs to the process,
while wires coming out on top represent the outputs of the process. Diagrams
can be composed due to the strong subsystem structure imposed on process
theories, where actions are not seen as affecting the global space but
explicitly associated with local systems. a. Processes can be composed in
sequence when their output and input systems match. b. Processes can be
composed in parallel on combined systems. c. Discarding a subsystem (e.g.
taking the partial trace) is indicated by three horizontal lines; in our
approach this corresponds to coarse-graining over the relevant degrees of
freedom (that is going to a smaller effective space). Other conditions can be
imposed: e.g. in [7], causal loops are forbidden, and outputs are always
connected to inputs.  Figure 5:  Causal structures. a. In classical causal
models, nodes are associated with random variables corresponding to events,
while arrows carry causal influence, as specified by conditional probability
distributions like $P_{YZ|X}$. These models and distributions may be extended
if one later learns of additional causes, like $\lambda_{Z}$. b. A model for
quantum causal structures proposed in [17], where each node is associated with
the intervention of an agent in a local space. A node $i$ is represented by
input and output Hilbert spaces, $H_{i}$ and $H_{i}^{*}$, and by a quantum
instrument $\mathcal{M}_{i}$ that links the two and corresponds to the agent’s
intervention (for example, $\mathcal{M}_{i}$ could be a local measurement
followed by a local preparation dependent on the outcome). Causal influence is
explicitly carried by quantum maps like $\mathcal{E}_{BC|A}$ which acts like a
channel from $A$ to $BC$.

In the previous sections, we have shown how to derive a notion of locality
within a global theory starting from a primitive notion of individual agents,
and their observed secrecy and commutation relations.

The operational approach laid out here has the advantage of carrying very
little assumptions about the underlying physical theory. For example, it goes
to a higher level of abstraction than generalized probability theories by not
taking for granted that all agents express their knowledge in terms of
reliable (classical) statistics about the outcomes of measurements.

Our notion of effective state spaces captures the concept of _beables_ of a
theory: aspects (or classes) of states that can in fact be physically observed
and distinguished [18, 4]. Our approach highlights that beables are observer-
dependent: for example, what appears to be a gauge may turn out to be only a
local gauge [4], and the same applies to “global” phases of quantum states or
yet-to-be-discovered microscopic details of some structure. We can never rule
out the existence of a more refined underlying theory, but with effective
state spaces we can tailor the descriptions used in a theory to the level of
detail needed for a particular application. This goes in the direction of the
work of Colbeck and Renner [19], where it is shown that quantum theory is
_complete_ for the task of guessing measurement outcomes, and further
refinements would be irrelevant.

As presented here, our framework simplifies the modelling of agents for the
pedagogical purpose of highlighting the advantages of this general direction.
In Appendix C we show how one could relax some of our assumptions to model
agents that are limited in time or who can only approximately distinguish
states. In the following, we discuss further applications and relation to
other work.

###  4.1 Non-signalling

One natural application of our extended notion of secrecy is the traditional
non-signalling condition. To see this, imagine that the two agents are
cooperating, so that Alice is trying to communicate information to Bob by
means of some action $g_{A}\in\mathcal{T}_{A}$ on her side. Bob can now either
directly apply post-processing $f_{B}\in\mathcal{T}_{B}$, or he can wait for
some time to pass, as represented by a function $u_{t}\in\mathcal{T}$ that
implements global time evolution over time $t$. If Alice and Bob are mutually
secret in the presence of $u_{t}$, for all $t\leq T$, we conclude that they
cannot signal to each other in this time window.

In Appendix E we show explicitly how our notion of extended secrecy implies
traditional non-signalling in the framework of generalized probabilistic
theories [20, 21], where the state space consists of probability distributions
over outcomes of possible measurements on physical systems.

###  4.2 Reconstructing space-time

Building up on the example above, if two agents cannot signal in the presence
of $u_{t}$ for $t\leq T$ and in addition can signal in the presence of $u_{t}$
for $t>T$, this can be used to define a _distance_ between the two agents, via
$d\propto T$. The proportionality constant can be interpreted as the speed of
signal propagation, for example the speed of light.

The challenge to obtaining a meaningful distance is two-fold: 1) choosing a
“natural” family of transformations $\\{u_{t}\\}_{t}$ to represent time
evolutions, and 2) choosing a family of agents that do not conflate different
types of coarse-grainings. For example, locality and macroscopicity each give
rise to a natural notion of distance, relating to the space between agents and
to precision of observation, respectively; the latter could be used to
quantify chaos given a family of time evolutions.

More generally, we can try to use signalling between agents to infer
properties of space-time of a given theory, as illustrated in Figure 1. Some
steps in this direction have been given for example in Refs. [4, 22, 23]. This
would be of particular interest in the context of field theories [24, 25, 10].
We leave the generalization of the operational approach depicted in Figure 1
to reconstruct position as future work.

###  4.3 Relation to modular approaches

Our global approach complements modular, bottom-up constructions [26], like
process theories based on symmetric monoidal categories [16, 8, 5, 7, 6]. For
the purpose of comparison with our work, modular theories can be understood as
theories of individual systems (or “objects”) and local actions (“processes”)
on those systems, which allow for parallel and sequential composition of
processes on different systems. Typically, they assume that: 1) Processes with
matching output and input systems can be composed sequentially. That is, a
process $f:A\to A^{\prime}$ can be composed with a process $g:A^{\prime}\to
A^{\prime\prime}$, to form a new process $g\circ f:A\to A^{\prime\prime}$
satisfying

$g\circ f(A)=g(f(A))$

(Figure 4.a). 2) Any two systems $A$ and $B$ can be combined in parallel to
form a composite system denoted by $A\otimes B$. 3) Any two processes $f:A\to
A^{\prime}$ and $g:B\to B^{\prime}$ can be composed to yield a process
$f\otimes g:A\otimes B\to A^{\prime}\otimes B^{\prime}$ satisfying

$(f\otimes g)(A\otimes B)=f(A)\otimes g(B)$

(Figure 4.b). This last assumption implies that processes act locally without
disturbing other systems, and that actions on independent systems always
commute. This allows us to represent process theories in terms of diagrams
that can be easily composed (Figure 4).

Our approach is more general in that we do not assume the strong subsystem
structure imposed by conditions 2) and 3). As such, our work strengthens
Coecke’s argument that non-signalling can be derived from a simpler condition
[7] (Appendix B). In general, our top-down view can be taken as a precursor
and sanity check for process theories. In complex global theories, a strong
subsystem structure may not be clear cut from the start. The cautious
researcher can first use our approach to test different reduced descriptions
for independence conditions. If she succeeds in finding independent effective
spaces — which is not always possible — she may then frame them as subsystems
and attempt a modular construction.

At a conceptual level, our approach gives a global interpretation to aspects
of process theories that are more epistemic than physical. For example, if we
think of subsystems as building blocks of a global space, it appears natural
to see “discarding a subsystem” as a physical action, like throwing away a
piece of lego (Figure 4.c). However, if we start from the global space and see
subsystems as arbitrary restricted descriptions, then “discarding a subsystem”
corresponds to a coarse-graining over the relevant degrees of freedom (for
example going from $\Omega_{AB}$ to an effective space $\Omega_{A}$), a change
of perspective rather than a physical transformation.

###  4.4 Relation to causal structures

Our notion of secrecy between agents is analogous to _causal independence_
between events in graphs used to study causality in physics. Causal structures
[27, 2, 28, 29, 30, 31] try to capture the causal relations between events
within a larger context (Figure 5). Both causality (as expressed by
Reichenbach’s principle) and secrecy are guiding principles of a certain way
of representing a theory (causal structures and restricted agents
respectively) that help us understand a complex situation — they are not
necessarily fundamental features of the laws of nature. How useful the
representation is depends both on the guiding principle and on the choice of
variables of interest (like events or agents).

Let us illustrate this. In classical causal graphs, events are represented by
random variables, in principle subject to intervention (Figure 5.a). As we
move from purely classical scenarios to more physical situations, like those
involving quantum measurements, the formalism of causal structures is evolving
to focus on agents and on explicit physical transformations as carriers of
causal influence, similarly to our approach. For example, in the quantum
causal structures of [17], events can correspond to quantum systems where
agents can act locally (Figure 5.b). Generally speaking, “events” embody a
particular coarse-graining of a global picture into variables or subsystems of
interest. As such, a single causal graph cannot reveal all the features of a
complex theory — a different decomposition may explore new causal
relations.444Also, we can never know if we only have access to an effective
state space, and there is a deeper theory that changes all the causal
relations, e.g. by providing new common causes. The choice of relevant nodes
can be guided by operationalism: 1) we start by picking “variables” that we
care about (like the outcomes of an experiment, or a subsystem corresponding
to the perspective and range of intervention of an agent); 2) we then use
Reichenbach’s principle and independence conditions to complete the causal
graph, by identifying further nodes and constraining the channels between
them.555 E.g. in studying the process of coherent copy
$\alpha|0\rangle_{A}+\beta|1\rangle_{A}\to\alpha|0\rangle_{B}|0\rangle_{C}+\beta|1\rangle_{B}|1\rangle_{C}$
in [17], we start with 3 nodes of interest $A$, $B$ and $C$, and are forced by
Reichenbach’s principle to complete the graph with a 4th node. This procedure
is similar in spirit to how in the present work we could start with the
description of a few agents and use secrecy and commutation constraints to
identify other subspaces and transformations of interest, or build a notion of
locality. How successful we are in this endeavour depends largely on the
(subjective) starting point — a poor initial choice of events or agents could
make it impossible to find a meaningful causal graph or independent agents.

Even with a clever choice of initial variables, it could be that the guiding
principle is not powerful enough to provide meaningful representations for all
physical situations. This is likely the case in both approaches, which are
still rooted in classical intuitions — resulting in concepts like agents,
Reichenbach’s principle, and time order. In trying to explain a physical
scenario in terms of these classical notions, we risk running into paradoxes
such as the inconsistencies between quantum agents in [32]. It remains to
explore whether both our approach and causal models can handle this kind of
physical challenges, and whether extensions to cover them would still be
intuitive enough to help us make sense of the world.

## Acknowledgements

We thank Bob Coecke and Barry Sanders for discussions on discarding and non-
signalling, Lucien Hardy, Ryszard Paweł Kostecki and Renato Renner for
discussions on locality, John-Mark Allen and Mirjam Weilenmann for discussions
on causality, David Jennings and Markus Müller for bringing up approximate
distinguishability, Roger Colbeck for feedback on this manuscript, Miyuko for
sanctuary, and Sandu Popescu for the mantra “one idea, one paper”. This one
goes out to Matt and Rob — let’s disagree again soon.

LK is supported by the European Research Council via grant No. 258932, the
Swiss National Science Foundation through the National Centre of Competence in
Research _Quantum Science and Technology_ (QSIT), and the European Commission
via the project _RAQUEL_. LdR acknowledges support from ERC AdG NLST and EPSRC
grant _DIQIP_ , from the FQXi grant _Physics of the observer_ , and from the
Perimeter Institute for Theoretical Physics. Research at Perimeter Institute
is supported by the government of Canada through Industry Canada and by the
Province of Ontario through the Ministry of Economic Development & Innovation.

## Appendix

Appendix A:

Proofs of all statements in the manuscript.

Appendix B:

Relation between our secrecy conditions and Coecke’s “non-signalling from
terminality” argument in modular theories [7].

Appendix C:

Relaxing some of the assumptions of the present approach.

Appendix D:

Additional minor results on commuting agents and minimal constructions.

Appendix E:

Application to generalized probability theories.

Appendix F:

Generalizing the derivation of secret agents to extended secrecy in the
presence of global functions, and to the case where the two monoids of
transformations do not commute.

##  Appendix A Proofs

See 1

###### Proof.

For each direction:

  * $1\to 2$.

We build the equivalence relation in $\Omega_{B}$ as

$\displaystyle[\rho]_{B}\sim_{B\to C}[\sigma]_{B}\iff$
$\displaystyle[\rho^{\prime}]_{C}=[\sigma^{\prime}]_{C},$
$\displaystyle\forall\
\rho^{\prime}\in[\rho]_{B},\sigma^{\prime}\in[\sigma]_{B}.$

Since $[\rho]_{B}\subseteq[\rho]_{C}$ for all $\rho\in\Omega$, $\sim_{B\to C}$
is a well-defined equivalence relation, and the reduced space
$\Omega_{B}/\sim_{B\to C}$ is in one-to-one correspondence with the space of
the equivalence classes $[\rho]_{C}$.

  * $2\to 1$.

By assumption, the reduction ${\bf h}_{C}$ is isomorphic to ${\bf h}_{B\to
C}\circ{\bf h}_{B}.$ Therefore $[\rho]_{C}\cong[[\rho]_{B}]_{B\to C}$, and
$[\rho]_{B}\subseteq[\rho]_{C}$.

∎

See 2

###### Proof.

We apply the non-signalling condition multiple times. Define
$\rho^{(j)}:=\left(\bigcirc_{i=1}^{j}f_{B}^{i}\circ g_{A}^{i}\right)\circ
f(\rho)$. Starting from the left-hand side, we have

$\displaystyle\left(\bigcirc_{i=1}^{N}f_{B}^{i}\circ g_{A}^{i}\right)\circ
f(\rho)$ $\displaystyle=\ f_{B}^{N}\circ g_{A}^{N}(\rho^{(N-1)})$
$\displaystyle\sim_{B}f_{B}^{N}(\rho^{(N-1)})$ $\displaystyle=\ f_{B}^{N}\circ
f_{B}^{N-1}\circ g_{A}^{N-1}(\rho^{(N-2)})$
$\displaystyle\sim_{B}f_{B}^{N}\circ f_{B}^{N-1}(\rho^{(N-3)})$
$\displaystyle\vdots$ $\displaystyle\sim_{B}f_{B}^{N}\circ\dots\circ
f_{B}^{1}\circ f(\rho).$

∎

See 3

###### Proof.

The proof is analogous to the proof of Proposition 2 and uses Definition 3; we
also define

$\tilde{\rho}:=f\circ\left(\bigcirc_{i=1}^{N}{f^{\prime}}_{B}^{i}\circ{g^{\prime}}_{A}^{i}\right)\circ
g(\rho)$

and

$\rho^{(j)}:=\left(\bigcirc_{i=1}^{j}{f^{\prime}}_{B}^{i}\circ{g^{\prime}}_{A}^{i}\right)\circ
g(\rho).$

Then

$\displaystyle\left(\bigcirc_{i=1}^{N}f_{B}^{i}\circ g_{A}^{i}\right)\circ
f\circ\left(\bigcirc_{i=1}^{N}{f^{\prime}}_{B}^{i}\circ{g^{\prime}}_{A}^{i}\right)\circ
g(\rho)$ $\displaystyle\sim_{B}f_{B}^{N}\circ\dots\circ
f_{B}^{1}(\tilde{\rho})\quad{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}[\text{secrecy}]}$
$\displaystyle\sim_{B}f_{B}^{N}\circ\dots\circ f_{B}^{1}\circ
f\circ{f^{\prime}}_{B}^{N}(\rho^{(N-1)})\quad{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}[\text{Def.\leavevmode\nobreak\
\ref{def:extended_secrecy}}]}$ $\displaystyle\vdots$
$\displaystyle\sim_{B}f_{B}^{N}\circ\dots\circ f_{B}^{1}\circ
f\circ{f^{\prime}}_{B}^{N}\circ\dots\circ{f^{\prime}}_{B}^{1}\circ g(\rho).$

∎

See 5

###### Proof.

Since $\mathcal{T}_{C}\subseteq\mathcal{T}_{B}$, it is secret towards $A$.
This also restricts the post-processing that $C$ can do, and since
$\rho\sim_{B}\sigma\implies\rho\sim_{C}\sigma$, we have that $\mathcal{T}_{A}$
is secret towards $C$. ∎

See 6

###### Proof.

To show secrecy of $\mathcal{T}_{A}$ towards $B$, we have

$\displaystyle f_{B}\circ g_{A}(\rho)$
$\displaystyle=g_{A}\circ\underbrace{f_{B}(\rho)}_{\in\ \Omega}$
$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}[\text{commutativity}]}$
$\displaystyle\sim_{B}\ f_{B}(\rho),\ $
$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}[\text{assumption}]}$

for all $\rho\in\Omega,g_{A}\in\mathcal{T}_{A},f_{B}\in\mathcal{T}_{B}$. To
show secrecy in the presence of $f$, we use

$\displaystyle f_{B}\circ f\circ f^{\prime}_{B}\circ g_{A}(\rho)$
$\displaystyle=f_{B}\circ f\circ
g_{A}\circ\underbrace{f^{\prime}_{B}(\rho)}_{\in\ \Omega}$
$\displaystyle\sim_{B}f_{B}\circ f\circ f^{\prime}_{B}(\rho),$

for all
$\rho\in\Omega,g_{A}\in\mathcal{T}_{A},f_{B},f^{\prime}_{B}\in\mathcal{T}_{B}$.
∎

######  Lemma 8.

The perspective $\sim_{\cancel{A}}$ induced by a submonoid
$\mathcal{T}_{A}\subseteq\mathcal{T}$ is an equivalence relation in $\Omega$.

###### Proof.

By construction $\sim_{\cancel{A}}$ is transitive, reflexive and symmetric. ∎

See 7

###### Proof.

We must show that

$\displaystyle f_{B}\circ g_{A}(\rho)\ $ $\displaystyle\sim_{\cancel{A}}\
f_{B}(\rho),$

for all $\rho\in\Omega,g_{A}\in\mathcal{T}_{A}$ and $f_{B}\in\mathcal{T}_{B}$.
By Proposition 6, we only need to show $g_{A}(\rho)\sim_{\cancel{A}}\rho$, for
all $\rho\in\Omega,\ g_{a}\in\mathcal{T}_{A}$. This holds since
$\text{id}\in\mathcal{T}_{A}$ (as $\mathcal{T}_{A}$ is a monoid), and so
$g_{A}(\rho)\sim_{\cancel{A}}\text{id}(\rho)$. We proceed analogously to find
the effective state space of $A$. ∎

##  Appendix B Relation to terminality

Figure 6:  Terminality and non-signalling in process [7]. a. Terminality:
discarding a system $A$ after applying a function $f_{A}$ is the same as
discarding the system directly. b. Non-signalling: in this setup, a global
preparation process $f_{\perp}$ distributes two systems to Alice and Bob, who
perform local operations $f_{A}$ and $f_{B}$ respectively on these systems and
additional inputs $A$ and $B$. Given terminality, if system $A$ is discarded
after this, no information about the original input on $A$ can travel to
system $B$. This holds even in the presence of an effect $f_{\top}$ acting on
joint outputs of $f_{A}$ and $f_{B}$.

In Ref. [7], Coecke argues that a process theory [16, 8] is non-signalling if
it satisfies a simpler condition dubbed _terminality_. Terminality states that
local processes on a system right before “discarding” it cannot have any
observable effect (Figure 6.a). Discarding subsystems is a concept that
corresponds to tracing out or coarse-graining over local information. For
example, in quantum theory it is implemented by the partial trace: terminality
is naturally satisfied for completely positive trace-preserving operations on
the discarded systems, but does not hold for non-deterministic effects such as
projections onto particular outcomes of measurements [7].

In our language, the condition of terminality corresponds to the independence
condition $f_{A}(\rho)\sim_{B}\rho,$ where $f_{A}$ are local functions on a
system $A$ and $\sim_{B}$ corresponds to the local picture of other systems
$B$ outside $A$. Recall that the assumptions behind process theories like [7]
impose some structure on transformations and agents, in particular commutation
between agents’ local actions. As we saw in Proposition 6, this independence
condition together with commutation already implies secrecy.

It remains to see if our secrecy condition is equivalent to the non-signalling
of [7], depicted in Figure 6.b. This non-signalling corresponds roughly to
secrecy under pre- and post-processing, as implemented by an initial state
preparation $f_{\perp}$ and a deterministic effect $f_{\top}$. In our picture,
robustness of secrecy under pre- and post-processing is ensured by Proposition
2. In this case, pre-processing with a function $f_{\perp}$ can be included
without loss of generality in the initial state, and post-processing with
$f_{\top}$ is eliminated by the choice of local perspective $\sim_{B}$.

Hence, the result of [7] that terminality implies non-signalling follows from
our propositions 2 and 6 together. Our premise that actions by different local
agents commute is weaker than the assumptions employed by [7]. In conclusion,
our approach strengthens the argument in [7] for the significance of a
condition like terminality. At the same time, we take a more general approach
to subsystems than the bottom-up model of process theories in [7], thus
highlighting the role of commutation in the context of defining local agents
and non-signalling.

##  Appendix C Relaxing some assumptions

Let us now give some guidelines on how to relax two of the assumptions of our
framework, in order to cover more realistic representations of agents.

###  C.1 Approximate distinguishability

Often agents may not have clear-cut distinguishability criteria. For example,
an agent may categorize light frequencies into basic colours such as green and
blue — there may be some frequencies that the agent could file as both green
and blue. In the language of PBR [33], the reduced states “blue” and “green”
would be epistemic and not ontic (with respect to the underlying state space
of frequencies $\Omega$). Agents could also have a notion of approximate
distinguishability, for example of the sort “I can distinguish these two
states with probability $1-p$.”

We propose a simple approach to address these cases. 1) Build a generous
effective state space $\Omega_{B}$ by assigning different reduced states to
every two states in $\Omega$ that can be distinguished _in principle_ by the
agent. 2) Build an _approximation structures_ in the effective state space
[3]. An approximation structure comprehends all neighbourhoods
$\\{\mathcal{B}^{\varepsilon}(\rho)\\}_{\rho\in\Omega_{B},\varepsilon\in\mathcal{E}}$
parameterized by whatever measure $\mathcal{E}$ is operational for the agent.
For example, one valid approximation structure for quantum states corresponds
to the $\varepsilon$-balls induced by the trace distance; another could be
just the _cover_ $\\{$blue, green, …$\\}$ of the possible colours assigned to
each frequency. 3) Build notions of _approximate secrecy_ , where we can
demand for example

${\bf h}_{B}\circ f_{B}\circ g_{A}(\rho)\ \in\mathcal{B}^{\varepsilon}({\bf
h}_{B}\circ f_{B}(\rho)),$

for all $\rho\in\Omega,\ g_{A}\in\mathcal{T}_{A},\ f_{B}\in\mathcal{T}_{B}$,
instead of the stricter condition of secrecy, where we demand that the two
final states are completely indistinguishable from $B$’s perspective. The
properties of approximate secrecy are inherited from the approximation
structure.

###  C.2 Time-limited agents

In this work we model local actions as monoids $\mathcal{T}_{A}$ and
$\mathcal{T}_{B}$. When applying secrecy to find non-signalling conditions
between time-limited agents, the monoidal structure of actions is only a
convenient approximation, which allows us to concatenate post-processing
actions indefinitely. The intuition behind this approximation is that Alice
and Bob’s actions can be implemented essentially instantaneously, compared to
the relevant time scales. One example would be the action of choosing a bit as
an input to a measurement, by pressing a button in Alice’s lab (see Appendix
E). With this interpretation, $\mathcal{T}_{A}$ and $\mathcal{T}_{B}$ can
consistently be modelled as monoids, because it is assumed that the
concatenation of two instantaneous actions can again be implemented
instantaneously. In this model, time evolution is explicitly modelled by
global functions $u_{t}\in\mathcal{T}$; this could include the actual effect
of pressing the button.

When functions in $\mathcal{T}_{A}$ and $\mathcal{T}_{B}$ on Alice’s and Bob’s
sides take some finite time $t>0$ to implement, we may instead of full
monoidal structure only have $f_{A}\circ g_{A}\in\mathcal{T}$ if the functions
$f_{A}$ and $g_{A}$ together take less than a given time $T$ to implement. In
this case, the notion of secrecy or non-signalling and our results that relate
to it can still be recovered for functions and concatenations of functions
that do not exceed this time-frame $T$.

##  Appendix D Commuting agents: additional results

In the main text, we have noted that the perspective $\sim_{\cancel{A}}$
induced by transformations $\mathcal{T}_{A}$ is minimal: any agent
$(\sim_{B},\mathcal{T}_{B})$ whose actions $\mathcal{T}_{B}$ commute with
$\mathcal{T}_{A}$ and towards whom transformations in $\mathcal{T}_{A}$ are
secret must in fact be described by a coarse-graining of $\sim_{\cancel{A}}$.

######  Proposition 9 (Induced perspective is minimal).

Let $B=(\sim_{B},\mathcal{T}_{B})$ be an agent towards whom $\mathcal{T}_{A}$
is secret, and such that $\mathcal{T}_{A}$ and $\mathcal{T}_{B}$ commute. Then

$\displaystyle[\rho]_{B}$
$\displaystyle\supseteq[\rho]_{\cancel{A}},\quad\forall\ \rho\in\Omega,$

with $\sim_{\cancel{A}}$ the equivalence relation induced by
$\mathcal{T}_{A}$.
This implies that there exists an equivalence relation $\sim_{\cancel{A}\to
B}$ in the effective state space $\Omega/\sim_{\cancel{A}}$ such that

$\Omega_{B}\cong(\Omega/\sim_{\cancel{A}})/\sim_{\cancel{A}\to B}.$

###### Proof.

Since $\mathcal{T}_{A}$ is secret towards $B$,

$\rho\sim_{B}g_{A}(\rho)$

and so, due to transitivity of $\sim_{B}$, also

$\exists\ f_{A},g_{A}\in\mathcal{T}_{A}\text{ s.t.\
}f_{A}(\rho)=g_{A}(\sigma)\implies\rho\sim_{B}\sigma.$

Again due to transitivity it directly follows that

$\rho\sim_{\cancel{A}}\sigma\implies\rho\sim_{B}\sigma$

and so

$[\rho]_{B}\supseteq[\rho]_{\cancel{A}}$

for all $\rho\in\Omega$. We may thus employ Proposition 1. ∎

######  Corollary 10.

Let $\mathcal{T}_{A},\mathcal{T}_{B}\subseteq\mathcal{T}$ be monoids such that
$\mathcal{T}_{A}\subseteq\mathcal{T}_{B}$. Then the induced equivalence
relations $\sim_{\cancel{A}}$ and $\sim_{\cancel{B}}$ satisfy

$[\rho]_{\cancel{B}}\supseteq[\rho]_{\cancel{A}},\ \forall\ \rho\in\Omega.$

This again implies that there exists an equivalence relation
$\sim_{\cancel{A}\to\cancel{B}}$ in the effective state space
$\Omega/\sim_{\cancel{A}}$ such that

$\Omega/\sim_{\cancel{B}}\
\cong(\Omega/\sim_{\cancel{A}})/\sim_{\cancel{A}\to\cancel{B}}.$

###### Proof.

This follows from the fact that $\mathcal{T}_{A}$ is secret towards
$\sim_{\cancel{B}}$, together with Proposition 9. The second statement follows
again from Proposition 1. ∎

Finally, the following proposition shows that equivalence classes
$[\rho]_{\cancel{A}}$ are preserved by commuting transformations
$\mathcal{T}_{B}$, providing an operational interpretation to the perspective
of agents $(\sim_{\cancel{A}},\mathcal{T}_{B})$: namely, states that are
indistinguishable from Bob’s point of view remain indistinguishable after he
applies functions $f_{B}\in\mathcal{T}_{B}$,

$\omega\sim_{\cancel{A}}\rho\implies
f_{B}(\omega)\sim_{\cancel{A}}f_{B}(\rho).$

######  Proposition 11 (Induced perspective is operational).

Let $\mathcal{T}_{A},\mathcal{T}_{B}\subseteq\mathcal{T}$ be commuting
transformations. Then the perspective $\sim_{\cancel{A}}$ induced by
$\mathcal{T}_{A}$ satisfies

$\omega\sim_{\cancel{A}}\rho\implies
f_{B}(\omega)\sim_{\cancel{A}}f_{B}(\rho),$

for any $f_{B}\in\mathcal{T}_{B}$ and $\omega,\rho\in\Omega$.

###### Proof.

From the definition of $\sim_{\cancel{A}}$ it follows that

$\displaystyle\omega\sim_{\cancel{A}}\rho\implies$ $\displaystyle\exists\
n\in\mathbb{N},\\{\tau_{i}\\}_{1\leq i\leq n}\text{ with }\tau_{i}\in\Omega:$
$\displaystyle\rho\sim^{\prime}_{\cancel{A}}\tau_{1}\sim^{\prime}_{\cancel{A}}\tau_{2}\sim^{\prime}_{\cancel{A}}\dots\sim_{\cancel{A}}\tau_{n}\sim^{\prime}_{\cancel{A}}\sigma$

with

$\nu\sim^{\prime}_{\cancel{A}}\omega\iff\exists\
f_{A},g_{A}\in\mathcal{T}_{A}\text{ s.t.\ }f_{A}(\nu)=g_{A}(\omega).$

But then, because $\mathcal{T}_{A}$ and $\mathcal{T}_{B}$ commute,

$\displaystyle\nu\sim^{\prime}_{\cancel{A}}\omega$ $\displaystyle\iff$
$\displaystyle\exists\ f_{A},g_{A}\in\mathcal{T}_{A}\ \text{ s.t.\
}f_{A}(\nu)=g_{A}(\omega)$ $\displaystyle\implies$ $\displaystyle\exists\
f_{A},g_{A}\in\mathcal{T}_{A}\ \text{ s.t.\ }f_{B}\circ f_{A}(\nu)=f_{B}\circ
g_{A}(\omega)$ $\displaystyle\iff$ $\displaystyle\exists\
f_{A},g_{A}\in\mathcal{T}_{A}\ \text{ s.t.\ }f_{A}\circ f_{B}(\nu)=g_{A}\circ
f_{B}(\omega)$ $\displaystyle\implies$ $\displaystyle
f_{B}(\nu)\sim^{\prime}_{\cancel{A}}f_{B}(\omega)$

for all $f_{B}\in\mathcal{T}_{B}$. From the definition of $\sim_{\cancel{A}}$
as the transitive closure of $\sim^{\prime}_{\cancel{A}}$, it then also
follows that

$\rho\sim_{\cancel{A}}\sigma\implies
f_{B}(\rho)\sim_{\cancel{A}}f_{B}(\sigma).$

∎

##  Appendix E Application to GPTs

Generalized probability theories (GPTs [21, 20, 34, 35, 36]) are a framework
to infer as much as possible about a physical system without making
assumptions about its inner workings (like the assumption that states can be
represented as vectors in a Hilbert space). Instead, it is assumed that agents
can implement and label a number of physical procedures, like preparations,
transformations and, crucially, measurements. Note that the agents need not
know the actual physical state prepared; in order to label a procedure, they
only need to be confident that they can repeat it. Indeed, the basic
assumption behind GPT frameworks is that agents can extract significant
measurement statistics (for example, by repeating a procedure many times).
Hence, GPTs model outputs of measurements as random variables, and agents’
knowledge of procedures as probability distributions.

The usual approach to build GPTs is bottom-up, starting with local procedures
that can be composed to reach a global theory. Here we are interested in the
opposite direction: given a global GPT, can we find meaningful notions of
local variables? Firstly, we need to model global and local knowledge.

###  E.1 Basic formalism

While we are inspired by known GPT models [21, 20, 34, 35, 36], we take a
slightly different and simplified approach here. The idea is that agents only
have direct access to classical random variables (like input settings and
outputs of a physical measurement). Since they correspond to accessible
information, we denote probability distributions over these random variables
by _states_. _Transformations_ $f$ are naturally modelled by conditional
probability distributions $P^{f}_{Z|X}$ that take input to output states, such
that $f(P_{X})=P^{\prime}_{Z}$, with

$\displaystyle P^{\prime}_{Z}(Z)$ $\displaystyle=\sum_{x\in
X}P^{f}_{Z|X}(z|x)\ P_{X}(x).$

For example, suppose that we want to model an experiment where an agent
performs a quantum measurement by pressing two buttons: button $X$ prepares a
quantum state $\rho^{x}$ and button $Y$ measures it according to the POVM
$\\{E^{y}_{z}\\}_{z}$ with possible outcomes $\\{z\\}_{z\in Z}$. The
distributions $P_{XY}$ over inputs and $P^{\prime}_{Z}$ over outputs
correspond to accessible “states.” We model the transformation as a
conditional distribution $P^{f}_{Z|XY}$ with
$P^{f}_{Z|XY}(z|x,y)=\operatorname{Tr}(E^{y}_{z}\ \rho_{x})$. The final
distribution $P^{\prime}_{Z}$ of outcomes given an input distribution $P_{XY}$
is therefore

$\displaystyle P^{\prime}_{Z}(z)$ $\displaystyle=\sum_{x\in X}\sum_{y\in
Y}P^{f}_{Z|XY}(z|x,y)\ P_{XY}(x,y)$ $\displaystyle=\sum_{x\in X}\sum_{y\in
Y}\operatorname{Tr}(E^{y}_{z}\ \rho_{x})\ P_{XY}(x,y).$

Note that it is the conditional distribution that encodes the “physical”
information about a particular setting (like the quantum state and POVM),
which may be inaccessible to the agents.

To compare with the models of [20, 21], our transformations are analogous to
their states. However, in our agent-driven approach, we restrict the set of
allowed measurements to those accessible to a particular agent in the resource
theory — they can thus be seen as a subset of the _fiducial measurements_ that
define a state in [20, 21].

In our model, _global states_ correspond to distributions over a global random
variable $X$. Restricted agents are those unable to distinguish some of the
outcomes of the global variable. We can model this via arbitrary groupings of
outcomes $x\in X$ into equivalence classes, i.e.  _events_ $\\{B_{b}\\}_{b}$.
The _reduction_ function ${\bf h}_{B}$ to the effective state space of an
agent $B$ simply sums over all the probabilities of the individual outcomes
$x\in B_{b}$ in each event $B_{b}$ and returns the probability associated with
the event,

$\displaystyle P_{B}={\bf h}_{B}(P_{X}),\ \text{ with }\ P_{B}(b)$
$\displaystyle=\sum_{x\in B_{b}}P_{X}(x).$

###  E.2 Secrecy and non-signalling

Figure 7:  Extended secrecy in GPTs. Consider two agents Alice and Bob and a
two-bit random variable $AB$ (the four possible outcomes are represented by
vertical lines). Our global state space consists of distributions over two
bits (whose names we change after each step to make a proof more readable).
Let the reduction to the state space of Bob be a coarse-graining over the
first bit, $P_{Y}(y)=P_{XY}(0,y)+P_{XY}(1,y)$, as shown on top. Secrecy in the
presence of a function $f$ (Eq. 1) corresponds to
$P_{Y^{\prime}}=P^{\prime}_{Y^{\prime}}$ in the diagram. This is satisfied,
for example if $f$ represents to the use of a PR-box, while $g_{A}$ and
$f_{B}$ correspond to choices of inputs and post-processing: Bob cannot guess
Alice’s choice of input after the use of a PR-box. This is equivalent to the
traditional notion of non-signalling [19] applied to the PR-box.

Consider now two agents $A$ and $B$ whose actions commute. In order to
guarantee secrecy of $A$ towards $B$, we only need to satisfy the independence
condition
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}g_{A}}(P_{X})\sim_{B}P_{X}$
(for all global $P_{X}$ and all
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}g_{A}}\in\mathcal{T}_{A}$,
see Proposition 6). In our language, this condition reads

$\displaystyle\sum_{y\in B_{b}}\sum_{x\in
X}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}P^{g_{A}}_{Y|X}(y|x)}\
P_{X}(x)$ $\displaystyle=\sum_{x\in B_{b}}P_{X}(x),\quad\forall\ b.$

For simplicity, we took $Y$ and $X$ to be identical random variables that
represent the global state before and after the transformation, and ${\bf
h}_{B}$ is a particular coarse-graining of outcomes into events. The condition
then states that ${\bf h}_{B}$ is insensitive to the transformation
(conditional probability distribution) $P^{g_{A}}_{Y|X}$ from inputs $x\in X$
to outcomes $y\in Y$.

If $A$ and $B$ commute and are mutually secret, we can ask if an additional
global transformation $f\in\mathcal{T}$ allows for signalling between them
(Definition 3). Our condition for extended secrecy in the presence of $f$,
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}f_{B}}\circ{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}f}\circ{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}g_{A}}(P_{X})\sim_{B}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}f_{B}}\circ{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}f}(P_{X}),$
for all global $P_{X}$,
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}f_{B}}\in\mathcal{T}_{B}$
and
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}g_{A}}\in\mathcal{T}_{A}$,
becomes

$\displaystyle\sum_{v\in
B_{b}}\sum_{x,y,z}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}P^{f_{B}}_{V|Z}(v|z)}\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}P^{f}_{Z|Y}(z|y)}\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}P^{g_{A}}_{Y|X}(y|x)}\
P_{X}(x)=$ $\displaystyle\sum_{v\in
B_{b}}\sum_{x,z}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}P^{f_{B}}_{V|Z}(v|z)}\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}P^{f}_{Z|X}(z|x)}\
P_{X}(x),\quad\forall\ b.$ (1)

Non-signalling functions are then those that do not let information encoded in
$P^{A}_{Y|X}$ propagate to Bob’s perspective $\sim_{B}$, that is, such that
$\mathcal{T}_{A}$ is secret with respect to $(\sim_{B},\mathcal{T}_{B})$ in
the presence of $f$. Here, again for simplicity $X,Y,Z,V$ were chosen as
identical random variables, and $P^{f}_{Z|X}=P^{f}_{Z|Y}$ both represent the
conditional probability distribution corresponding to $f$. For a simple
example in a 2-bit space, see Figure 7.

We can compare our definition of secrecy in the presence of $f$ to traditional
notions of non-signalling. Consider again the simple case of a two-bit input
and output space of Figure 7. The definition of non-signalling found for
example in [19] reads $P_{Y|AB}=P_{Y|B}$, or

$\displaystyle\sum_{x=0,1}P_{XY|AB}(xy|a=0,b)$
$\displaystyle=\sum_{x=0,1}P_{XY|AB}(xy|a=1,b),\quad\forall\ b,y\in\\{0,1\\}$
(2)

This condition formalizes the idea that Bob cannot learn anything about
Alice’s input $a$ by looking solely at his output $y$ and input $b$. In our
framework, Alice’s choice of input is encoded in a local transformation
$g_{A}\in\mathcal{T}_{A}$ (for example, $g_{A}^{0}$ could correspond to
pressing a button to choose input $0$ and $g_{A}^{1}$ to choose $1$), and
therefore “Bob’s ignorance about Alice’s input” translates to “Bob’s ignorance
about Alice’s action $g_{A}$.”

In the following we establish a direct equivalence between these two notions
of non-signalling in this simple case; we expect this equivalence to hold in
more general settings. Let us first flesh out the assumptions behind the
equivalence. A gentle warning: we have labelled all the intermediate bits
differently “to avoid confusion” (Figure 7). Since we assume a priori that
Alice and Bob have mutual secrecy (without $f$), we take that $g_{A}$ only
acts locally on Alice’s bit,

$P^{g_{A}}_{A^{\prime}B^{\prime}|AB}(a^{\prime},b^{\prime}|a,b)=P^{g_{A}}_{A^{\prime}|A}(a^{\prime}|a)\
\delta(b^{\prime},b),$

so that

$g_{A}(Q_{AB}(a,b))=\sum_{a,b}P^{g_{A}}_{A^{\prime}|A}(a^{\prime}|a)\
\delta(b^{\prime},b)\
Q_{AB}(a,b)=\sum_{a}P^{g_{A}}_{A^{\prime}|A}(a^{\prime}|a)\ Q_{AB}(a,b).$

Similarly, Bob’s post-processing is encoded in $f_{B}\in\mathcal{T}_{B}$ which
we also assume to be truly local, that is

$P^{f_{B}}_{X^{\prime}Y^{\prime}|XY}(x^{\prime},y^{\prime}|x,y)=P^{f_{B}}_{Y^{\prime}|Y}(y^{\prime}|y)\
\delta(x^{\prime},x).$

The final distribution $P_{Y^{\prime}}$ for Bob becomes

$\displaystyle P_{Y^{\prime}}(y^{\prime})$
$\displaystyle={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{\bf
h}_{B}}\circ{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}f_{B}}\circ{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}f}\circ{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}g_{A}}(Q_{AB}(a,b))$
$\displaystyle={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\sum_{a}}\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{\bf
h}_{B}}\circ{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}f_{B}}\circ{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}f}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}P^{g_{A}}_{A^{\prime}|A}(a^{\prime}|a)}\
Q_{AB}(a,b))$
$\displaystyle={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\sum_{a}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\sum_{a^{\prime},b}}\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{\bf
h}_{B}}\circ{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}f_{B}}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}P^{f}_{XY|A^{\prime}B^{\prime}}(x,y|a^{\prime},b)}\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}P^{g_{A}}_{A^{\prime}|A}(a^{\prime}|a)}\
Q_{AB}(a,b))$
$\displaystyle={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\sum_{a}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\sum_{a^{\prime},b}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\sum_{y}}\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{\bf
h}_{B}}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}P^{f_{B}}_{Y^{\prime}|Y}(y^{\prime}|y)}\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}P^{f}_{XY|A^{\prime}B^{\prime}}(x,y|a^{\prime},b)}\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}P^{g_{A}}_{A^{\prime}|A}(a^{\prime}|a)}\
Q_{AB}(a,b))$
$\displaystyle={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\sum_{a}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\sum_{a^{\prime},b}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\sum_{y}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\sum_{x}}\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}P^{f_{B}}_{Y^{\prime}|Y}(y^{\prime}|y)}\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}P^{f}_{XY|A^{\prime}B^{\prime}}(x,y|a^{\prime},b)}\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}P^{g_{A}}_{A^{\prime}|A}(a^{\prime}|a)}\
Q_{AB}(a,b).$

The condition for secrecy in the presence of $f$, Eq. 1, is then

$\displaystyle\sum_{a,a^{\prime},b,x,y}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}P^{f_{B}}_{Y^{\prime}|Y}(y^{\prime}|y)}\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}P^{f}_{XY|A^{\prime}B^{\prime}}(x,y|a^{\prime},b)}\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}P^{g_{A}}_{A^{\prime}|A}(a^{\prime}|a)}\
Q_{AB}(a,b)$ $\displaystyle=$
$\displaystyle\sum_{a,b,x,y}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}P^{f_{B}}_{Y^{\prime}|Y}(y^{\prime}|y)}\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}P^{f}_{XY|AB}(x,y|a,b)}\
Q_{AB}(a,b),$ $\displaystyle\qquad\qquad\qquad\qquad\forall\
y^{\prime}\in\\{0,1\\},g_{A}\in\mathcal{T}_{A},f_{B}\in\mathcal{T}_{B},Q_{AB}\in\Omega.$
(3)

######  Proposition 12 (Equivalence to non-signalling in GPTs).

In the setting of Figure 7, “our” condition of non-signalling, Eq. 3, is
equivalent to the “traditional” notion, Eq. 2.

###### Proof.

In our language, the non-signalling condition of Eq. 2 reads

$P^{f}_{XY|A^{\prime}B}(xy|b,a^{\prime}=0)=P^{f}_{XY|A^{\prime}B}(xy|b,a^{\prime}=1)=:P^{f}_{XY|A^{\prime}B}(xy|b).$

To show that Eq. 3 implies the above, we choose the particular local actions

$\displaystyle P^{g^{0}_{A}}_{A^{\prime}|A}$
$\displaystyle=\delta(a^{\prime},0),\qquad
P^{g^{1}_{A}}_{A^{\prime}|A}=\delta(a^{\prime},1),\qquad
P^{f_{B}}_{Y^{\prime}|Y}(y^{\prime}|y)=\delta(y^{\prime},y).$

There, $g^{0}_{A}$ corresponds to Alice’s choice of input $0$, $g^{1}_{A}$ to
her choice of $1$, and $f_{B}$ to no post-processing by Bob. These choices
directly imply for all $Q_{AB}$,

$\displaystyle\sum_{x}P^{f}_{XY|AB}(xy|b,a=0)\ Q_{B}(b)$
$\displaystyle=\sum_{x}P^{f}_{XY|AB}(xy|b,a=1)\ Q_{B}(b),$

and so traditional non-signalling follows. For the other direction, we have
simply

$\displaystyle P_{Y^{\prime}}(y^{\prime})$
$\displaystyle=\sum_{a,a^{\prime},b,x,y\in\\{0,1\\}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}P^{f_{B}}_{Y^{\prime}|Y}(y^{\prime}|y)}\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}P^{f}_{XY|A^{\prime}B}(xy|a^{\prime}b)}\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}P^{g_{A}}_{A^{\prime}|A}(a^{\prime}|a)}\
Q_{AB}(ab)$
$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}[\text{non-
signalling (Eq.\leavevmode\nobreak\ \ref{eq:traditional})}]}\qquad$
$\displaystyle=\sum_{a,b,x,y}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}P^{f_{B}}_{Y^{\prime}|Y}(y^{\prime}|y)}\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}P^{f}_{XY|B}(xy|b)}\
\underbrace{\left[{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}P^{A}_{A^{\prime}|A}(0|a)}+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}P^{A}_{A^{\prime}|A}(1|a)}\right]}_{=1}\
Q_{AB}(a,b)$
$\displaystyle=\sum_{a,b,x,y}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}P^{B}_{Y^{\prime}|Y}(y^{\prime}|y)}\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}P^{f}_{XY|AB}(xy|b)}\
Q_{AB}(a,b).$

∎

This shows that for example PR boxes satisfy our definition of non-signalling
functions. Examples for functions that are signalling are conditional
probability distributions that swap the states on the two systems, or bitwise
addition of the inputs $a$ and $b$ on the two sides into the outputs $x$ and
$y$.

##  Appendix F Deriving secrecy without commutation

In principle, the way we have constructed an induced perspective
$\sim_{\cancel{A}}$ from a monoid $\mathcal{T}_{A}$ can be extended to
construct equivalence relations that yield secret agents
$(\sim_{A},\mathcal{T}_{A})$ and $(\sim_{B},\mathcal{T}_{B})$ in the presence
of a global function $f$, and even in the case where functions
$\mathcal{T}_{A}$ and $\mathcal{T}_{B}$ do not commute. This is done in the
following proposition, which, as is shown in the subsequent corollary, reduces
to the definition of induced perspectives $\sim_{\cancel{A}}$ and
$\sim_{\cancel{B}}$ in the case $f=\text{id}$ and commuting
$\mathcal{T}_{A},\mathcal{T}_{B}$.

######  Proposition 13 (Deriving secret agents).

Let $(\Omega,\mathcal{T})$ be a gobal theory,
$\mathcal{T}_{A}^{S},\mathcal{T}_{B}\subseteq\mathcal{T}$ be two monoids of
transformations, and let $f\in\mathcal{T}$.
Then the smallest equivalence class $\sim_{B}$ on $\Omega$ towards which
$\mathcal{T}_{A}^{S}$ is secret in the presence of $f$,

$f_{B}\circ f\circ f^{\prime}_{B}\circ g_{A}(V)\sim_{B}f_{B}\circ f\circ
f^{\prime}_{B}(V),$

for all $V\in S^{\Omega},g_{A}\in\mathcal{T}_{A}^{S},f_{B}\in\mathcal{T}_{B}$,
is built as follows:
Define the relation $\sim$ on $\Omega$ as

$\rho\sim\sigma\iff\exists\ f_{A},g_{A}\in\mathcal{T}_{A}^{S}\text{ s.t.\
}f_{A}(\rho)=g_{A}(\sigma).$

Then define another relation $\sim^{\prime}$ as

$\displaystyle\rho\sim^{\prime}\sigma\iff\begin{cases}\rho\sim\sigma\text{ or
}\\\ \exists\
\rho^{\prime},\sigma^{\prime}\in\Omega,f_{B}\in\mathcal{T}_{B}\text{ s.t.\
}\rho=f_{B}(\rho^{\prime}),\sigma=f_{B}(\sigma^{\prime}),\rho^{\prime}\sim\sigma^{\prime}\text{
or }\\\ \exists\
\rho^{\prime},\sigma^{\prime}\in\Omega,f_{B},f^{\prime}_{B}\in\mathcal{T}_{B}\text{
s.t.\ }\rho=f_{B}\circ f\circ f^{\prime}_{B}(\rho^{\prime}),\sigma=f_{B}\circ
f\circ
f^{\prime}_{B}(\sigma^{\prime}),\rho^{\prime}\sim\sigma^{\prime}.\end{cases}$

Finally, the relation $\sim_{B}$ on $\Omega$ is the transitive closure of
$\sim^{\prime}$, namely through

$\rho\sim_{B}\sigma\iff\exists\
n\in\mathbb{N},\tau_{1},\dots,\tau_{n}\in\Omega\text{ s.t.\
}\rho\sim^{\prime}\tau_{1},\tau_{1}\sim^{\prime}\tau_{2},\dots,\tau_{n}\sim^{\prime}\sigma.$

###### Proof.

Both the relation $\sim$ and $\sim^{\prime}$ are by construction reflexive and
symmetric. The relation $\sim_{B}$ is then by construction also transitive,
and thus constitutes an equivalence relation. The relation $\sim_{B}$
furthermore gives rise to the smallest perspective towards which
$\mathcal{T}_{B}$ is secret in the presence of $f$:
$\omega=g_{A}(\rho)\implies\rho\sim_{B}\omega.$ By construction then also
$f_{B}\circ f\circ f^{\prime}_{B}\circ g_{A}(\omega)\sim_{B}f_{B}\circ f\circ
f^{\prime}_{B}(\omega)$ and $f_{B}\circ g_{A}(\omega)\sim_{B}f_{B}(\omega)$,
for all
$\omega\in\Omega,f_{B},f^{\prime}_{B}\in\mathcal{T}_{B},g_{A}\in\mathcal{T}_{A}^{S}$.
∎

######  Corollary 14.

In the case of commuting
$\mathcal{T}_{A}^{S},\mathcal{T}_{B}\subseteq\mathcal{T}$, the equivalence
relation $\sim_{B}$ constructed in Proposition 13 that gives rise to secrecy
in the presence of $f$ simplifies accordingly and can be constructed as
follows.
Define the relation $\sim$ on $\Omega$ as

$\rho\sim\sigma\iff\exists\ f_{A},g_{A}\in\mathcal{T}_{A}^{S}\text{ s.t.\
}f_{A}(\rho)=g_{A}(\sigma).$

Then define another relation $\sim^{\prime}$ as

$\displaystyle\rho\sim^{\prime}\sigma\iff\begin{cases}\rho\sim\sigma\text{ or
}\\\ \exists\
\rho^{\prime},\sigma^{\prime}\in\Omega,f_{B}\in\mathcal{T}_{B}\text{ s.t.\
}\rho=f_{B}\circ f(\rho^{\prime}),\sigma=f_{B}\circ
f(\sigma^{\prime}),\rho^{\prime}\sim\sigma^{\prime}.\end{cases}$

Then, the relation $\sim_{B}$ on $\Omega$ is the transitive closure of
$\sim^{\prime}$, namely through

$\rho\sim_{B}\sigma\iff\exists\
n\in\mathbb{N},\tau_{1},\dots,\tau_{n}\in\Omega\text{ s.t.\
}\rho\sim^{\prime}\tau_{1},\tau_{1}\sim^{\prime}\tau_{2},\dots,\tau_{n}\sim^{\prime}\sigma.$

If in addition $f=\mathbb{I}$, the relation $\sim_{B}$ simplifies to

$\rho\sim_{B}\sigma\iff\exists\
n\in\mathbb{N},\tau_{1},\dots,\tau_{n}\in\Omega\text{ s.t.\
}\rho\sim\tau_{1},\tau_{1}\sim\tau_{2},\dots,\tau_{n}\sim\sigma$

with $\sim$ as above. This recovers the construction of induced perspectives
$\sim_{\cancel{A}}$ in Definition 5.

###### Proof.

In the case when functions in $\mathcal{T}_{A}^{S}$ and $\mathcal{T}_{B}$
commute, we can see that

$\displaystyle\rho\sim\sigma$ $\displaystyle\iff\exists\
f_{A},g_{A}\in\mathcal{T}_{A}^{S}\text{ s.t.\ }f_{A}(\rho)=g_{A}(\sigma)$
$\displaystyle\implies\exists\ f_{A},g_{A}\in\mathcal{T}_{A}^{S}\text{ s.t.\
}f_{B}\circ f_{A}(\rho)=f_{B}\circ g_{A}(\sigma)$ $\displaystyle\iff\exists\
f_{A},g_{A}\in\mathcal{T}_{A}^{S}\text{ s.t.\ }f_{A}\circ
f_{B}(\rho)=g_{A}\circ f_{B}(\sigma)$ $\displaystyle\iff f_{B}(\rho)\sim
f_{B}(\sigma)$

for all $f_{B}\in\mathcal{T}_{B}$. This implies the respective simplifications
of the relation $\sim_{B}$, and recovers the induced perspective
$\sim_{\cancel{A}}$ of $\mathcal{T}_{A}^{S}$ in the case of $f=\text{id}$. ∎

## References

  * [1] L. Cheng, C. Wu, Y. Zhang, H. Wu, M. Li and C. Maple, “A Survey of Localization in Wireless Sensor Network”, International Journal of Distributed Sensor Networks 2012, 1 (2012).
  * [2] R. W. Spekkens, “The paradigm of kinematics and dynamics must yield to causal structure”, arxiv:1209.0023.
  * [3] L. del Rio, L. Krämer and R. Renner, “Resource theories of knowledge”, arxiv:1511.08818.
  * [4] L. Hardy, “Operational General Relativity: Possibilistic, Probabilistic, and Quantum”, arxiv:1608.06940.
  * [5] B. Coecke, T. Fritz and R. W. Spekkens, “A mathematical theory of resources”, arxiv:1409.5531.
  * [6] T. Fritz, “The mathematical structure of theories of resource convertibility I”, arxiv:1504.03661.
  * [7] B. Coecke, “Terminality implies non-signalling”, Electronic Proceedings in Theoretical Computer Science 172, 27 (2014).
  * [8] B. Coecke, “A universe of processes and some of its guises”, arxiv:1009.3786, in: “Deep Beauty: Understanding the Quantum World through Mathematical Innovation”, pp. 128—-186.
  * [9] G. Leibniz, “Tentamina theodicææ de bonitate dei: libertate hominis et origine mali”, C.H. Bergerus (1739).
  * [10] G. Valente, “Local disentanglement in relativistic quantum field theory”, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 44, 424 (2013).
  * [11] K. G. Wilson and J. Kogut, “The renormalization group and the $\epsilon$ expansion”, Physics Reports 12, 75 (1974).
  * [12] J. Polchinski, “Renormalization and effective lagrangians”, Nuclear Physics B 231, 269 (1984).
  * [13] M. E. Peskin and D. V. Schroeder, “An introduction to quantum field theory”, Addison-Wesley (1995).
  * [14] T. Banks, “Modern quantum field theory : a concise introduction”, Cambridge University Press (2008).
  * [15] J. von Neumann, “Zur Algebra der Funktionaloperationen und Theorie der normalen Operatoren”, Mathematische Annalen 102, 370 (1930).
  * [16] B. Coecke, “Kindergarten Quantum Mechanics: Lecture Notes”, in: “AIP Conference Proceedings”, pp. 81–98, AIP (2006).
  * [17] J.-M. A. Allen, J. Barrett, D. C. Horsman, C. M. Lee and R. W. Spekkens, “Quantum common causes and quantum causal models”, arxiv:1609.09487.
  * [18] J. S. Bell, “Speakable and Unspeakable in Quantum Mechanics — Collected Papers on Quantum Philosophy, 2nd Edition”, Cambridge University Press (2004).
  * [19] R. Colbeck and R. Renner, “The Completeness of Quantum Theory for Predicting Measurement Outcomes”, in: “Quantum Theory: Informational Foundations and Foils”, pp. 497–528, ed.: G. Chiribella and R. W. Spekkens, Springer Netherlands (2016), Dordrecht.
  * [20] L. Hardy, “Quantum Theory From Five Reasonable Axioms”, arxiv:0101012.
  * [21] J. Barrett, “Information processing in generalized probabilistic theories”, Physical Review A 75, 032304 (2007).
  * [22] P. A. Hoehn and M. P. Mueller, “An operational approach to spacetime symmetries: Lorentz transformations from quantum communication”, arxiv:1412.8462.
  * [23] C. Cao, S. M. Carroll and S. Michalakis, “Space from Hilbert Space: Recovering Geometry from Bulk Entanglement”, arxiv:1606.08444.
  * [24] H. Halvorson and M. Mueger, “Algebraic Quantum Field Theory”, arxiv:0602036.
  * [25] S. A. M. Wolters and H. Halvorson, “Independence Conditions for Nets of Local Algebras as Sheaf Conditions”, arxiv:1309.5639.
  * [26] L. Hardy, “The operator tensor formulation of quantum theory”, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 370, L. Hardy (2012).
  * [27] R. Penrose, “Techniques in Differential Topology in Relativity”, Society for Industrial and Applied Mathematics (1972).
  * [28] J. Pearl, “Causality: models, reasoning, and inference”, Cambridge University Press (2000).
  * [29] L. Masanes, M. P. Müller, R. Augusiak and D. Pérez-García, “Existence of an information unit as a postulate of quantum theory.”, Proceedings of the National Academy of Sciences of the United States of America 110, 16373 (2013).
  * [30] K. Ried, M. Agnew, L. Vermeyden, D. Janzing, R. W. Spekkens and K. J. Resch, “A quantum advantage for inferring causal structure”, Nature Physics 11, 414 (2015).
  * [31] G. Chiribella and R. W. Spekkens, “Quantum Theory: Informational Foundations and Foils”, Springer Netherlands (2016), Dordrecht.
  * [32] D. Frauchiger and R. Renner, “Single-world interpretations of quantum theory cannot be self-consistent”, arxiv:1604.07422.
  * [33] M. F. Pusey, J. Barrett and T. Rudolph, “On the reality of the quantum state”, Nature Physics 8, 476 (2012), arxiv:1111.3328.
  * [34] W. K. Wootters, “Quantum mechanics without probability amplitudes”, Foundations of Physics 16, 391 (1986).
  * [35] P. G. L. Mana, “Why can states and measurement outcomes be represented as vectors?”, quant-ph/0305117.
  * [36] P. G. L. Mana, “Probability tables”, quant-ph/0403084.