Operational locality in global theories

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by  Lea Kraemer

Operational locality in global theories
Lea Krämer1, and Lídia del Rio12
1InstituteforTheoreticalPhysics,ETHZurich,Switzerland
2SchoolofPhysics,UniversityofBristol,UnitedKingdom
7 Within a global physical theory, a no- waysgohandinhandimpliesthatinordertomodel
1 tion of locality allows us to find and jus- agents we have to specify both (Section 1). This
0 tify information-processing primitives, like naturally leads us to search for minimal operational
2
non-signalling between distant agents. Here constraints needed to ensure that agents are truly

n we propose exploring the opposite direction: local.
a to take agents as the basic building blocks
J Here we motivate a notion of secrecy between
through which we test a physical theory,
9 agents, which captures whether actions performed
and recover operational notions of locality
1 by an agent (like writing a message, choosing a
from signalling conditions. First we intro-
bit or preparing a quantum state) can be perceived
] duce an operational model for the effective
h by another (Section 2); traditional notions of non-
state spaces of individual agents, as well as
p signalling correspond to an extended secrecy be-
the range of their actions. We then for-
- tweenspace-likeseparatedregions(Section4). This
t mulate natural secrecy conditions between
n work brings together and clarifies concepts of local-
a agents and identify the aspects of locality rel-
ity used in quantum theory, generalized probabilis-
u evant for signalling. We discuss the possibil-
tic theories and field theories. It highlights that the
q ity of taking commutation of transformations
[ statespaceandtransformationsofatheoryarebuta
as a primitive of physical theories, as well as
subjectivechoiceofrepresentationoftheunderlying
2 applications to quantum theory and general-
physical theory from a viewpoint that is convenient
v ized probability frameworks. This “it from
0 toagivenagent,asarguedbySpekkens[2]. Here,we
bit” approach establishes an operational con-
8 tentatively suggest commutation of transformations
nection between local action and local obser-
2 as a primitive of physical theories. In particular,
3 vations, and gives a global interpretation to
we show how to derive local agents (and effective
0 concepts like discarding a subsystem or com-
descriptionsoflocalsubsystems)fromcommutation
. posing local functions.
1 relations on global transformations (Section 3).
0 In modelling local agents acting within a global
7 theory, the intuitive assumption is that both their This work draws from our “Resource theories
1 actions and their knowledge are restricted to a of knowledge” [3], and has natural applications in
:
v bounded region. The canonical example is a scien- multi-player settings, like cryptographic scenarios,
i tist who has full control of her lab and can perform games or resource theories. There is yet a more
X
local tomography. In reality though, the breadth of exciting possible application: to recover the space-
r
a knowledge and the range of action of agents may time structure of a physical theory from the prim-
be decoupled. For example, prisoners can acquire itive notion of test agents, in the spirit of Hardy’s
global knowledge by reading the news, but their ac- operational GR [4] and to the task of localization
tions are limited to small subsystems. Conversely, in wireless sensor networks [1]. The idea is to send
someone locked in a control room may only have out agents (or probes) to unknown positions, see if
local knowledge of the shapes of different buttons, they can communicate with each other, and use the
butpressingonemayhaveglobalconsequences. The signalling graph to define distances between agents,
observationthattheknowledgeandactiondonotal- reconstruct their relative positions, and infer prop-
erties of space-time (Figure 1). For this we must
firstfindappropriate,theory-independentnotionsof
LídiadelRio: [email protected] agents and signalling.
1
a.
b.
signal range
spatial location signalling graph
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.
1 Modelling agents
representation from the point of view of a global
agent; 2) different pictures, like Ω and Ω0, may be
We start with a top-down approach, where we first related and mapped to one another [2, 3]; and 3) Ω
describe a global theory (as seenby a global agent), need not have any special structure a priori besides
and then model restricted agents acting within that beingaset—indeed,theapproachlaidoutherewill
theory. allow us to find an operational subsystem structure
in the set of states.
The transformations in T represent all actions
1.1 Global theory
that the theory allows the global agent to imple-
From the point of view of a given global agent, a ment. We can think of them as the ways in which
global theory may be represented via a state space the agent may test a theory, by applying actions
Ω and a set of transformations T that are available that change state parameters. For example, an ex-
to the agent [5–8]. We can think of the state space plicit theory of a quantum universe may allow only
as the “language” chosen by this global observer to for unitary operations, while a more generous the-
describe nature. For example, Ω could be the set of ory could equip the agent with implicit large ancil-
coordinates and momenta of all celestial bodies; in las, and allow her to implement general quantum
quantum theory, it could be the set of valid density channels, state preparations and even tomography.
matrices over a global Hilbert space. It need not be Again the two views can be related: the latter is an
a static picture: in astronomy, an alternative state effective theory derived from the unitary quantum
space Ω0 could be the set of possible trajectories of theory, by internalizing part of the global space as
celestial bodies, and in quantum theory it could in- belongingtotheagentandherinstruments,andnot
cludeallglobalHamiltoniansthatdeterminethefree to the object of study (the rest of the universe) [3].
evolutionofdensitymatrices. Notethat1)Ωisnot In the context of field theories, this is discussed as
theultimatedescriptionofreality, justaconvenient emergingagency [4]. Inasuperdeterministictheory,
2
thereisonlyonepossiblecourseofevolutionforthe may have nothing to do with spatial locality. Bob
universe, and T consists only of functions that ap- might only have access to crude measurement in-
ply it (for example T ∼= {e−iHt} where the global struments unable to distinguish microscopic details
t
agent is given some choice of time). Formally, T of states, or he may not be able to distinguish a
is a monoid of functions f : Ω → Ω: it contains globalphaseorgauge[4]. Ingeneralizedprobability
theidentitytransformationandisclosedundercon- frameworks, Bob’s perspective can correspond to a
catenation (an associative binary operation), such grouping of individual global outcomes into events
that performing two actions subsequently, f ◦g, is (Appendix E). In algebraic quantum field theory,
still an allowed operation. We discuss the monoidal theseequivalenceclassescouldemergefromalgebras
assumption and possible relaxations in Section 4. of local observables (see e.g. Ref. [10] for a review).
The other ingredient needed to define an agent,
1.2 Local agents aswesawintheintroduction,isadescriptionofthe
actions available to him. As his actions may have a
Localagentsarecharacterizedbylimitedknowledge:
global impact, a minimal approach is to take them
their inability to distinguish global states that ap-
to be a submonoid T ⊆ T of the globally allowed
pearidenticalintheireyes. Wecanformalizethisby B
transformations. Wediscussrelaxationsofthisdefi-
building equivalence classes of states that are indis-
nitioninSection4. Generalizationsofthisapproach
tinguishable from the perspective of an agent. For
can be found in Ref. [3]. There, we also study ex-
example,inquantumtheory,wecouldhaveanagent
plicit ways to move between global and local views
Bob who only has access to a Hilbert space H ;
B (technically, related by Galois insertions), effective
two global states are indistinguishable (or equiva-
theories and other properties of local agents.
lent) from Bob’s perspective if they have the same
marginal in H . This defines an equivalence rela-
B
tion σ ∼ ρ:Tr σ =Tr ρ, where Tr denotes a
B B B B
partial trace over all systems except B. The corre-
sponding equivalence classes are
Definition 1(Globaltheoryandrestrictedagents).
Aglobal theory of agents isdefinedbyapair(Ω,T),
[ρ] :={σ ∈Ω:Tr σ =ρ }.
B B B whereΩ(thestatespace)isaset,andT isamonoid
of transformations f : Ω → Ω, with the concatena-
Taking the quotient over this equivalence relation
tion operation ◦.
gives us a new space state Ω/∼ , which is in one-
B
A restricted agent B acting within the theory is
to-one correspondence with the set of all reduced
defined by a pair (∼ ,T ) , where ∼ is an equiv-
densitymatricesinH . ThisisBob’seffective state B B B
B
alence relation in Ω and T is a submonoid of T
space, sufficient to encode all the information that B
called the set of local operations of the agent. The
he can observe about any global state (Figure 2).
quotient space Ω := Ω/∼ is called the effective
In this case, the map from the global to the local B B
space of agent B. The reduction to the effective
spaces (the canonical map) is given by the partial
space is given by the canonical map
trace:
h :Ω→Ω
h :Ω→Ω/∼ B B
B B
ρ7→[ρ] ∼=Tr ρ=ρ . ρ7→[ρ]B.
B B B
More generally, we can always build the effective
state space of an agent in this way, even if we do
notknowanythingaboutthestructureoftheglobal We can always further coarse-grain the effective
space(forinstancewhetheritcanbesplitintoacon- state space of a given agent B in order to obtain a
venienttensorformH ⊗H ). Theconstructionof more restricted agent C. For example, in renormal-
A B
anagent’seffectivestatespaceΩ :=Ω/∼ isinthe ization group flow, lowering the cutoff corresponds
B B
spiritofLeibnizprincipleofidentityofindiscernibles to coarse-graining over more and more observables
[9]. Yet this operational procedure emphasizes that [11, 12]. The following proposition formalizes this
both discernibility and identity are subjective con- idea [3, Prop. III.5]. All proofs can be found in Ap-
cepts (Figure 2). Limitations on Bob’s perspective pendix A.
3
Ω
A
0A 1A ΩM
0 0 0 1 0 0 Ω Ω
B
[000]
B 0 0 1 1 0 1
Ω 0
B
0 1 0 1 1 0
[010]
B 0 1 1 1 1 1
1
B
a. [000] [100] b.
A A
Figure 2: Building an agent’s effective state space. The different states of a global space Ω 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 Ω 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] and [100] ). Her effective state space Ω =Ω/∼ has only two
A A A A
states, which can be relabelled as 0 and 1 for convenience. Another agent Bob identifies the equivalent
A A
classes [000] and [010] , which leads us to conclude that he can only see the second bit. Note that for
B B
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 Ω 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 Ω .
M
Proposition 1 (Nested agents). Let (Ω,T) be a them, even after post-processing.1
globaltheory,andB,C tworestrictedagents. Then
the following are equivalent: Definition 2 (Secrecy). We say that an agent A
has access to secret operations TS ⊆ T towards
1. C has more restricted knowledge than B, that A A
another agent B if
is [ρ] ⊆[ρ] , ∀ ρ∈Ω,
B C
f ◦g (ρ) ∼ f (ρ)
2. There exists an equivalence relation ∼ in B A B B
B→C
B’s effective state space Ω such that Ω ∼=
B C for all ρ ∈ Ω, g ∈ TS, f ∈ T . If all actions in
Ω /∼ . A A B B
B B→C T are secret towards B and T are secret towards
A B
A we say that the two agents are mutually secret.
We may ask if this definition is robust enough,
2 Secrecy between agents that is, whether further pre- or post-processing by
Alice and Bob could destroy the secrecy of a choice
ofactiong ∈TS. Thenextpropositionshowsthat
A A
2.1 Secrecy no matter how many ‘secret’ transformations in TS
A
Alice implements, or how Bob acts in between to
Having defined agents, we may study conditions for
secrecy and non-signalling between them. Consider
a setup of two agents Alice and Bob, represented 1Bob’s effective space may include his local processing
(“states that I can distinguish after applying all my acces-
by A = (∼ ,T ) and B = (∼ ,T ). Imagine that
A A B B sible operations") or not (“states that I distinguish immedi-
Alice wants to keep her actions (like writing a mes-
ately,beforefurtherprocessing"). Forthesakeofgenerality,
sage or preparing a state) secret from Bob. This we leave the freedom in this decision up to the agent, and
is achieved if Bob cannot tell whether she applied accountforpost-processinginthedefinitionofsecrecy.
4
try and recover information, he will not detect any Proposition 3 (Robustness of extended secrecy).
of the effects of Alice’s actions. In addition, it is If an agent A only uses secret operations g ∈ TS
A A
easy to see that pre-processing with a global func- withrespecttotheagentBinthepresenceoff ∈T,
tion(suchasdistributingentanglementbetweenthe thenfurtherpre-andpost-processingcannotliftthe
two parties) cannot lift secrecy, since Definition 2 secrecy, that is
requires it to hold for all initial states.
(cid:0)(cid:13)N fi ◦gi (cid:1)◦f ◦(cid:16)(cid:13)N f0i ◦g0i (cid:17)◦g(ρ)
i=1 B A i=1 B A
Proposition 2 (Robustness of secrecy). If A has
secretoperationsTS withrespecttoB(accordingto ∼ (cid:0)(cid:13)N fi(cid:1)◦f ◦(cid:16)(cid:13)N f0i (cid:17)◦g(ρ)
A B i=1 B i=1 B
Definition 2), then pre- and post-processing cannot
lift the secrecy, that is forallstatesρ∈Ω,localoperations{gi } ⊆T and
A i A
{fi} ⊆T , global operations g ∈T and N ∈N.
fN ◦gN ◦···◦f2 ◦g2 ◦f1 ◦g1 ◦f(ρ) B i B
B A B A B A
∼B fBN ◦···◦fB2 ◦fB1 ◦f(ρ), In particular, for the case in which Bob only im-
plements post-processing at the very end, Proposi-
for all states ρ ∈ Ω, secret operations {gAi }i ⊆ TAS tion 3 implies that TAS forms a monoid.
and {fi} ⊆ T , global operations f ∈ T and N ∈
B i B
N.
Corollary 4 (Secret monoid). The set TS of secret
A
operationsinthepresenceofaglobalfunctionf ∈T
2.2 Extended secrecy
forms a monoid, i.e. id∈TS and
A
We may also ask whether Alice’s actions stay se-
f ,g ∈TS =⇒ f ◦g ∈TS.
cret to Bob in the presence of an additional global A A A A A A
transformation f ∈ T. Transformations such as a
subsystem swap or a communication channel may Naturally, if we further restrict the actions and
break secrecy; others, like the use of a PR box, do knowledgeofoneoftheagents(asinProposition1),
not.2 For this situation, we define an extended no- secrecy is maintained.
tion of secrecy in the spirit of Definition 2, which
reduces to Definition 2 in the case f = id. Here, Corollary 5 (Restricted agents and secrecy). Let
Bobmaytrytopost-processinformationbeforeand A, B and C be three agents, such that C is more
after the global transformation. restricted than B, that is T ⊆T and [ρ] ⊆[ρ] ,
C B B C
for all ρ∈Ω.
Definition 3 (Extended secrecy). Let A be an If T was secret towards A (in the presence of
B
agent with access to secret operations towards an f ∈ T), the same is true of T . If T was secret
C A
agent B, TAS ⊆ TA . We say that TAS is in addi- towards B (in the presence of f), it is still secret
tion secret (towards B) in the presence of a global towards C (idem).
transformation f ∈T if
f ◦f ◦f0 ◦g (ρ) ∼ f ◦f ◦f0 (ρ), 3 Commuting agents
B B A B B B
for all ρ ∈ Ω, g ∈ TS, f ,f0 ∈ T . We say that
A A B B B Nowweexplorehowsecrecyisaffectedwhentheac-
theagentsaremutuallysecretinthepresenceof f if
tions of two agents A and B commute. This is par-
allactionsinT aresecrettowardsBinthepresence
A ticularlyrelevantinthecontextofthenon-signalling
of f and vice-versa.
principle, since actions at space-like separation nat-
urally commute.
We can now show that, analogously to Proposi-
tion2,furtherpre-andpost-processingbyAliceand
Bob cannot lift the secrecy. Definition4(Commutingagents). Wesaythattwo
agents A and B commute if
2Ingeneralizedprobabilitytheories,PRboxescanbeseen fB ◦gA(ρ)=gA◦fB(ρ),
astransformationsthattakeclassicalinputsandreturnout-
puts(AppendixE). for all ρ∈Ω, g ∈T , f ∈T .
A A B B
5
id 000 100 000 100 000 100
NOT
id 001 C 101 001 101 001 101
010 110 010 110 010 110
011 111 011 111 011 111
a. b. c.
Figure 3: Three-bit example. Consider again the theory described by the state space Ω of 3 bits, and
all transformations on those bits. a. All operations T that change the third bit (of which id and not
C C
are labeled). b. Equivalence classes [x](cid:26)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 ∼(cid:26)C, which coarse-grains over
the functions applied to the third bit, gives us the largest effective state space relative to which functions in
T are secret. c. More coarse-grained equivalence classes [x] (vertical, yellow) and [x] (horizontal, blue),
A B
corresponding to an agent A who can only distinguish the first bit and an agent B who only sees the second
bit,respectively. OperationsinT arestillsecretrelativetothesetwoagents. Inaddition,operationsonthe
C
first bit are secret towards B and vice-versa. These smaller effective state spaces correspond to equivalence
relations on the effective state space Ω(cid:26)C (as in the nested agents of Proposition 1). The two-bit space Ω(cid:26)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](cid:26)C =[x]A∩[x]B.
For example, in field theory commutativity holds Proposition 6 (Secrecy for commuting agents). If
for measurements or field interactions at space-like A and B commute, then if there exists a subset of
separation,andthisisingeneralhowcausalityisre- actions TS ⊆T such that, ∀ρ∈Ω, g ∈TS,f ∈
A A A A B
coveredthere[13]3. Motivatedbythis, weheretake T ,
B
the commutation of actions in space-like separated
regions as a fundamental building block in deriving fB ◦f ◦gA(ρ) ∼B fB ◦f(ρ),
agents that are secret relative to each other. Note
that in particular, finding commuting sets of trans- then TAS is secret towards B in the presence of f.
formationsinT issomethingthatcanbedoneprior In particular, gA(ρ) ∼B ρ for all gA ∈ TA,ρ ∈ Ω
todefinitionsoflocalagents;thisisshownexplicitly implies secrecy of A towards B.
in Ref. [3]. Commutation relations result in a nice
algebraic structure — a lattice — in the space of
transformations[3]. Thisisalsothecaseforthevon 3.1 Secrecy from commutation
Neumann bicommutant in operator algebras [15].
Commutation is also an operational property of the Starting only from commutation relations on the
theory: for example, commutation is independent global transformations, we can construct descrip-
of the choice of reference frames in relativity and tions of local agents that have secret actions with
quantum field theory [13, 14]. If two agents com- respect to each other. More specifically, given any
mute, secrecy follows from simpler conditions. two commuting submonoids T ,T ⊆ T, we can
A B
construct equivalence relations ∼ ,∼ so that two
A B
agentsAlice(∼ ,T )andBob(∼ ,T )havesecret
A A B B
actions with respect to each other.
The first step is to start with transformations T
A
(“Alice’s transformations"), and look for the most
generous effective state space Ω(cid:26)A that is insensitive
to transformations in T . This will model the per-
A
3The simplest illustration of this is the commutation of
spectiveofanagent,Bob,whocannotdetectAlice’s
theKlein-Gordonfieldoperatorsφ(x)andφ(y)atspace-like
actions. Essentially, this perspective identifies sets
separated x and y, [φ(x),φ(y)] = 0. Such a commmutation
condition is also referred to in field theory as the locality of global states that Alice can locally make “con-
postulate [14]. verge” to the same state.
6
Definition 5 (Perspective insensitive to transfor- From a global viewpoint, their actions do not com-
mations). Let T ⊆ T be a submonoid of transfor- mute. However,forsomeonethatonlyhadaccessto
A
mations. First we define a binary relation ∼0 in Ω the combined knowledge of Alice and Bob (the first
(cid:26)A
called convergence through T as two bits), their actions would appear to commute.
A
For such an agent, only local time ordering of Al-
ρ∼(cid:26)0A σ ⇐⇒ ∃ fA,gA ∈TA s.t. fA(ρ)=gA(σ), ice and Bob’s actions matters, as the two processes
f ◦f and f ◦f are indistinguishable. This is
A B B A
We take the transitive closure ∼(cid:26)A of ∼(cid:26)0A to define yetanotherexampleofhowsubsystemsandlocalde-
the perspective insensitive to transformations T ,
A scriptions represent simplified pictures of the global
theory,reducingthedegreesoffreedomofthetheory
ρ∼(cid:26)A σ ⇐⇒ ∃ n∈N, {τi}ni=1 ⊆Ω: to an operational minimum for a given agent, who
ρ∼(cid:26)0A τ1 ∼(cid:26)0A τ2 ∼(cid:26)0A ···∼(cid:26)0A τn ∼(cid:26)0A σ. in this case would not need to model global time
ordering.
The above construction gives us minimal restric-
tionsforindependentagents. Thefollowingtheorem
4 Applications
is adapted from [3].
Theorem 7 (Deriving secret agents). Commuting In the previous sections, we have shown how to de-
submonoids T ,T ⊆ T give rise to descriptions of riveanotionoflocalitywithinaglobaltheorystart-
A B
mutually secret agents ingfromaprimitivenotionofindividualagents,and
their observed secrecy and commutation relations.
A=(∼(cid:26)B,TA), B =(∼(cid:26)A,TB). The operational approach laid out here has the
advantage of carrying very little assumptions about
Indeed,allagentswhoseactionscommutewithT the underlying physical theory. For example, it
A
andforwhomtransformationsinT aresecretmust goes to a higher level of abstraction than general-
A
be described by a coarse-graining of ∼(cid:26)A (Figure 3). ized probability theories by not taking for granted
This and related minor results can be found in Ap- that all agents express their knowledge in terms of
pendix D. In Appendix F we generalize Theorem 7 reliable (classical) statistics about the outcomes of
to extended secrecy in the presence of global func- measurements.
tions. There, wealso extendtheconstructionofthe Our notion of effective state spaces captures the
effective spaces of two agents to the case where the concept of beables of a theory: aspects (or classes)
two monoids of transformations do not commute: ofstatesthatcaninfactbephysicallyobservedand
without commutation, this construction is not as distinguished [4, 18]. Our approach highlights that
simple. beables are observer-dependent: for example, what
appearstobeagaugemayturnouttobeonlyalocal
gauge[4],andthesameappliesto“global”phasesof
3.2 Perceived commutation from secrecy
quantum states or yet-to-be-discovered microscopic
We can now ask if the actions T ,T ⊆ T of two detailsofsomestructure. Wecanneverruleoutthe
A B
mutually secret agents must always commute. The existence of a more refined underlying theory, but
answer is no, not at a global level: unbeknownst to witheffectivestatespaceswecantailorthedescrip-
the two agents, their actions could affect other de- tions used in a theory to the level of detail needed
grees of freedom of the global theory. This can be- for a particular application. This goes in the direc-
comerelevantwhentheactionsoftwoagentsaffecta tion of the work of Colbeck and Renner [19], where
common environment that is not directly accessible it is shown that quantum theory is complete for the
to them but could be recovered by a third party. taskofguessingmeasurementoutcomes,andfurther
For example, consider again the state-space of refinements would be irrelevant.
threebits, whereAlice can onlyseethefirstbitand As presented here, our framework simplifies the
Bob the second. Now imagine that Alice has access modelling of agents for the pedagogical purpose of
to all the transformations that change the first bit highlightingtheadvantagesofthisgeneraldirection.
and, as a side effect, reset the third bit to 0, while In Appendix C we show how one could relax some
Bob has access to all the actions that act on the ofourassumptionstomodelagentsthatarelimited
second bit and, as a side effect, flip the third bit. in time or who can only approximately distinguish
7
A’’
g
A’’ A’ B’ A’ B’
A’
= g o f f g = f g
f A A B A B A B
a. A b. c.
Figure4: Processtheories. Processestheoriesaremodular,bottom-upconstructionsthatcanbefaithfully
represented by diagrams [5–8, 16]. 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
imposedonprocesstheories,whereactionsarenotseenasaffectingtheglobalspacebutexplicitlyassociated
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
therelevantdegreesoffreedom(thatisgoingtoasmallereffectivespace). Otherconditionscanbeimposed:
e.g. in [7], causal loops are forbidden, and outputs are always connected to inputs.
states. In the following, we discuss further applica- beinterpretedasthespeedofsignalpropagation,for
tions and relation to other work. example the speed of light.
The challenge to obtaining a meaningful distance
is two-fold: 1) choosing a “natural” family of trans-
4.1 Non-signalling
formations {u } to represent time evolutions, and
t t
2) choosing a family of agents that do not conflate
One natural application of our extended notion of
different types of coarse-grainings. For example, lo-
secrecy is the traditional non-signalling condition.
cality and macroscopicity each give rise to a natu-
To see this, imagine that the two agents are coop-
ralnotionofdistance, relatingto thespace between
erating, so that Alice is trying to communicate in-
agents and to precision of observation, respectively;
formation to Bob by means of some action g ∈T
A A the latter could be used to quantify chaos given a
onherside. Bobcannoweitherdirectlyapplypost-
family of time evolutions.
processingf ∈T ,orhecanwaitforsometimeto
B B More generally, we can try to use signalling be-
pass, as represented by a function u ∈ T that im-
t tween agents to infer properties of space-time of a
plements global time evolution over time t. If Alice
given theory, as illustrated in Figure 1. Some steps
and Bob are mutually secret in the presence of u ,
t in this direction have been given for example in
for all t ≤ T, we conclude that they cannot signal
Refs. [4, 22, 23]. This would be of particular in-
to each other in this time window.
terest in the context of field theories [10, 24, 25].
In Appendix E we show explicitly how our no-
We leave the generalization of the operational ap-
tion of extended secrecy implies traditional non-
proach depicted in Figure 1 to reconstruct position
signalling in the framework of generalized proba-
as future work.
bilistic theories [20, 21], where the state space con-
sists of probability distributions over outcomes of
possible measurements on physical systems. 4.3 Relation to modular approaches
Our global approach complements modular,
4.2 Reconstructing space-time bottom-up constructions [26], like process theories
based on symmetric monoidal categories [5–8, 16].
Building up on the example above, if two agents For the purpose of comparison with our work,
cannot signal in the presence of u for t≤T and in modular theories can be understood as theories of
t
addition can signal in the presence of u for t > T, individual systems (or “objects”) and local actions
t
thiscanbeusedtodefineadistancebetweenthetwo (“processes”) on those systems, which allow for
agents,viad∝T. Theproportionalityconstantcan parallel and sequential composition of processes
8
H * H*
B C
B M C M
B C
H H
Y Z B C
P EB C | A
Y Z | X
H *
A
X λ
Z A M
A
H
a. b. A
Figure 5: Causal structures. a. In classical causal models, nodes are associated with random variables
correspondingtoevents,whilearrowscarrycausalinfluence, asspecifiedbyconditionalprobabilitydistribu-
tions like P . These models and distributions may be extended if one later learns of additional causes,
YZ|X
like λ . b. A model for quantum causal structures proposed in [17], where each node is associated with the
Z
intervention of an agent in a local space. A node i is represented by input and output Hilbert spaces, H
i
and H∗, and by a quantum instrument M that links the two and corresponds to the agent’s intervention
i i
(for example, M could be a local measurement followed by a local preparation dependent on the outcome).
i
Causal influence is explicitly carried by quantum maps like E which acts like a channel from A to BC.
BC|A
on different systems. Typically, they assume that: complex global theories, a strong subsystem struc-
1) Processes with matching output and input ture may not be clear cut from the start. The cau-
systems can be composed sequentially. That is, a tious researcher can first use our approach to test
process f :A→A0 can be composed with a process differentreduceddescriptionsforindependencecon-
g :A0 →A00, to form a new process g◦f :A→A00 ditions. If she succeeds in finding independent ef-
satisfying fectivespaces—whichisnotalwayspossible—she
g◦f(A)=g(f(A)) may then frame them as subsystems and attempt a
modular construction.
(Figure 4.a). 2) Any two systems A and B can
Ataconceptuallevel,ourapproachgivesaglobal
be combined in parallel to form a composite system
denotedbyA⊗B. 3)Anytwoprocessesf :A→A0 interpretationtoaspectsofprocesstheoriesthatare
and g :B →B0 can be composed to yield a process more epistemic than physical. For example, if we
f ⊗g :A⊗B →A0⊗B0 satisfying think of subsystems as building blocks of a global
space, it appears natural to see “discarding a sub-
(f ⊗g)(A⊗B)=f(A)⊗g(B) system” as a physical action, like throwing away a
piece of lego (Figure 4.c). However, if we start from
(Figure 4.b). This last assumption implies that
theglobalspaceandseesubsystemsasarbitraryre-
processes act locally without disturbing other sys-
stricteddescriptions,then“discardingasubsystem”
tems, and that actions on independent systems al-
corresponds to a coarse-graining over the relevant
ways commute. This allows us to represent process
degrees of freedom (for example going from Ω
AB
theoriesintermsofdiagramsthatcanbeeasilycom-
to an effective space Ω ), a change of perspective
A
posed (Figure 4).
rather than a physical transformation.
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 strength- 4.4 Relation to causal structures
ens Coecke’s argument that non-signalling can be
derived from a simpler condition [7] (Appendix B). Our notion of secrecy between agents is analogous
In general, our top-down view can be taken as a to causal independence between events in graphs
precursor and sanity check for process theories. In used to study causality in physics. Causal struc-
9
tures [2, 27–31] try to capture the causal relations causal graph or independent agents.
between events within a larger context (Figure 5). Even with a clever choice of initial variables, it
Bothcausality(asexpressedbyReichenbach’sprin- could be that the guiding principle is not powerful
ciple)andsecrecyareguidingprinciplesofacertain enoughtoprovidemeaningfulrepresentationsforall
way of representing a theory (causal structures and physical situations. This is likely the case in both
restricted agents respectively) that help us under- approaches, which are still rooted in classical intu-
stand a complex situation — they are not neces- itions — resulting in concepts like agents, Reichen-
sarily fundamental features of the laws of nature. bach’s principle, and time order. In trying to ex-
How useful the representation is depends both on plain a physical scenario in terms of these classical
the guiding principle and on the choice of variables notions, we risk running into paradoxes such as the
of interest (like events or agents). inconsistencies between quantum agents in [32]. It
Let us illustrate this. In classical causal graphs, remains to explore whether both our approach and
eventsarerepresentedbyrandomvariables,inprin- causal models can handle this kind of physical chal-
ciple subject to intervention (Figure 5.a). As we lenges,andwhetherextensionstocoverthemwould
move from purely classical scenarios to more phys- stillbeintuitiveenoughtohelpusmakesenseofthe
ical situations, like those involving quantum mea- world.
surements, the formalism of causal structures is
evolving to focus on agents and on explicit phys-
Acknowledgements
ical transformations as carriers of causal influence,
similarlytoourapproach. Forexample,inthequan-
tumcausalstructuresof[17], eventscancorrespond We thank Bob Coecke and Barry Sanders for dis-
to quantum systems where agents can act locally cussions on discarding and non-signalling, Lucien
(Figure 5.b). Generally speaking, “events” embody Hardy, Ryszard Paweł Kostecki and Renato Ren-
a particular coarse-graining of a global picture into nerfordiscussionsonlocality,John-MarkAllenand
variables or subsystems of interest. As such, a sin- Mirjam Weilenmann for discussions on causality,
gle causal graph cannot reveal all the features of a David Jennings and Markus Müller for bringing up
complex theory — a different decomposition may approximate distinguishability, Roger Colbeck for
explore new causal relations.4 The choice of rele- feedback on this manuscript, Miyuko for sanctuary,
vant nodes can be guided by operationalism: 1) we and Sandu Popescu for the mantra “one idea, one
startbypicking“variables”thatwecareabout(like paper”. This one goes out to Matt and Rob — let’s
the outcomes of an experiment, or a subsystem cor- disagree again soon.
respondingtotheperspectiveandrangeofinterven- LKissupportedbytheEuropeanResearchCoun-
tionofanagent);2)wethenuseReichenbach’sprin- cilvia grantNo. 258932, theSwiss National Science
ciple and independence conditions to complete the Foundation through the National Centre of Com-
causal graph, by identifying further nodes and con- petencein Research Quantum Science and Technol-
straining the channels between them.5 This proce- ogy (QSIT), and the European Commission via the
dureissimilarinspirittohowinthepresentworkwe project RAQUEL. LdR acknowledges support from
could start with the description of a few agents and ERC AdG NLST and EPSRC grant DIQIP, from
usesecrecyandcommutationconstraintstoidentify the FQXi grant Physics of the observer, and from
other subspaces and transformations of interest, or thePerimeterInstituteforTheoreticalPhysics. Re-
build a notion of locality. How successful we are in search at Perimeter Institute is supported by the
this endeavour depends largely on the (subjective) government of Canada through Industry Canada
starting point — a poor initial choice of events or and by the Province of Ontario through the Min-
agentscouldmakeitimpossibletofindameaningful istry of Economic Development & Innovation.
4Also, we can never know if we only have access to an
effectivestatespace,andthereisadeepertheorythatchanges
allthecausalrelations,e.g.byprovidingnewcommoncauses.
5 E.g. in studying the process of coherent copy α|0iA+
β|1iA→α|0iB|0iC+β|1iB|1iC in[17],westartwith3nodes
ofinterestA,BandC,andareforcedbyReichenbach’sprin-
cipletocompletethegraphwitha4thnode.
10