Admissible Causal Structures and Correlations Eleftherios-Ermis Tselentis1 2and Amin Baumeler3 4 1Institute for Quantum Optics and Quantum Information IQOQI-Vienna

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Admissible Causal Structures and Correlations

Eleftherios-Ermis Tselentis

1, 2

and

Amin Baumeler

3, 4

Institute for Quantum Optics and Quantum Information (IQOQI-Vienna),

Austrian Academy of Sciences, 1090 Vienna, Austria

Faculty of Physics, University of Vienna, 1090 Vienna, Austria

Facolt`a di scienze informatiche, Universit`a della Svizzera italiana, 6900 Lugano, Switzerland

Facolt`a indipendente di Gandria, 6978 Gandria, Switzerland

It is well-known that if one assumes quantum theory to hold locally, then processes with indeﬁnite

causal order and cyclic causal structures become feasible. Here, we study qualitative limitations on

causal structures and correlations imposed by local quantum theory. For one, we ﬁnd a necessary

graph-theoretic criterion—the “siblings-on-cycles” property—for a causal structure to be admissible:

Only such causal structures admit a realization consistent with local quantum theory. We conjecture

that this property is moreover suﬃcient. This conjecture is motivated by an explicit construction of

quantum causal models, and supported by numerical calculations. We show that these causal models,

in a restricted setting, are indeed consistent. For another, we identify two sets of causal structures

that, in the classical-deterministic case, forbid and give rise to non-causal correlations respectively.

INTRODUCTION

At heart of Einstein’s equivalence principle is the im-

possibility to detect the gravitational ﬁeld via local ex-

periments.

For general relativity, this principle dictates

that physics in suﬃciently small, i.e., local, space-time

regions is described by special relativity. This princi-

ple naturally extends to the quantum case: local experi-

ments are described by quantum theory. In this quantum

formulation, however, the gravitational ﬁeld, the refer-

ence frames, and the space-time regions might necessitate

quantum descriptions. While diﬀerent approaches target

these descriptions (see, e.g., Refs. [

–

]), another—the

process-matrix framework [

]—abstracts away the general-

relativistic freight and focuses on the idealized prescription

of local quantum experiments in countably many regions

only (without imposing any global constraints). Similar

to the various formulations of the equivalence principles,

this approach can be used to constrain competing theories

of quantum gravity: If a candidate theory of quantum

gravity exceeds the limits of the latter, then local experi-

ments in that theory must disagree with quantum theory.

This, in turn, gives a prescription to experimentally falsify

that candidate theory.

The process-matrix framework, i.e., the assumption

of local quantum theory, reconciles the inherently prob-

abilistic nature of quantum theory with the dynamical

causal structures of general relativity [

]. For one, it

extends quantum indeﬁniteness of physical degrees of

freedom like position and momentum to causal connec-

tions. Exemplary, while the position of a mass in general

This means that if a non-gravitational experiment is carried out

in a suﬃciently small space-time region

with a gravitational

ﬁeld, then for any space-time region

R′

free of gravitation there

exists a suitable reference frame where the same experimental

procedure yields the identical experimental data. This statement

and its variations are discussed in Ref. [1].

relativity determines the causal order among events in

its future, the quantum-switch process [

] does so co-

herently [

–

]. For another, this framework allows for

violations of causal inequalities [

]. Causal inequalities,

similar to Bell inequalities [

], are device-independent

tests of a global causal order. If the observed correlations

violate such an inequality, then they cannot be causally ex-

plained: Any explanation where only past data inﬂuences

future observations fails. These correlations are called

non-causal, and arise in setups that resemble [

] closed

time-like curves (CTCs) [

]. As notoriously shown

by G¨odel [

], CTCs appear in solutions to Einstein’s

equation of general relativity.

This stipulation of local quantum theory is also of

interest in theoretical computer science. A pillar of com-

puter sciences is that machines (programs) and data are

treated on an equal footing. This paradigm ﬁnds its

climax in Church’s notion of computation—the

calcu-

lus [

]—where any data is a function, and therefore

functions are of higher-order: functions on functions.

The process-matrix framework describes the ﬁrst level of

higher-order quantum computation [

]: Its objects—

the process matrices—map quantum gates to quantum

gates. For instance, the previously mentioned quantum

switch maps two quantum gates

A, B

to the functional-

ity

(α|0⟩+β|1⟩)⊗ |ψ⟩ 7→ α|0⟩ ⊗ BA|ψ⟩+β|1⟩ ⊗ AB|ψ⟩

where the order of gate application is controlled by

the ﬁrst qubit. This is achieved, e.g., through pro-

grammable connections between gates [

]. The quan-

tum switch brings forth a reduction in query complexity

when compared to the standard circuit model of compu-

tation [8,20–22].

The causal relations among local quantum experi-

ments (gates) are conveniently expressed with causal

structures. A causal structure is a directed graph where

the vertices represent laboratories, and where the edges

indicate the possibility of a local laboratory to directly

inﬂuence another (see Figure 1). The causal relations

arXiv:2210.12796v2 [quant-ph] 16 Sep 2023

BCD

(a)

(b)

P F

(c)

Figure 1: (a) A quantum circuit and its acyclic causal

structure. (b) The quantum switch—an instance of the

process-matrix framework—has a cyclic causal structure:

Depending on the prepared state at P, a quantum

system is sent through the H-shaped region from

or from Bto A. (c) If Ais traversed by a closed

time-like curve, then

’s output inﬂuences the input, and

a departure from quantum theory becomes necessary.

among the gates of any quantum circuit form an acyclic

causal structure: Naturally, a gate at depth

of the circuit

has no causal inﬂuence on the input to any other gate at

the same or smaller depth. This is radically contrasted by

processes: The quantum switch, for instance, has a cyclic

causal structure [

]. Still, not every causal structure is

compatible with local quantum theory: For the output of

a laboratory to inﬂuence the same laboratory’s input, we

require a departure from quantum theory by introducing

non-linear dynamics [24–26].

In this work, we study the causal structures that admit

a quantum realization—a question raised in Ref. [

In other words: We study the possible causal relations

among laboratories under the assumptions that within

each laboratory no deviation from quantum theory is

observable. We ﬁnd a necessary graph-theoretic criterion

(the causal structure of every quantum process satisﬁes

this criterion) and conjecture that the criterion is also

suﬃcient. The conjecture is motivated by a construction

of causal models for the causal structures of interest,

and is moreover numerically tested for all directed graph

with up to six nodes. In addition, we provide two graph-

theoretic criteria from which, in the classical-deterministic

case, only causal or also non-causal correlations arise.

Supporting the above conjecture, we show that the causal

structures that satisfy the criterion for non-violation are

admissible.

The presentation is structured in the following way.

First, we provide the mathematical tools necessary for

the present treatment. This is followed by our results on

admissible and inadmissible causal structures. Thereafter,

we relate causal structures with causal inequalities. We

conclude with a series of open questions.

PRELIMINARIES

We brieﬂy comment on the notation used. If

a Hilbert space, then

(

) is the set of linear operators

. We use

for the set

{

, . . . , n −

}

. If a sym-

bol is used with and without a subscript from

, then

the bare symbol denotes the collection under the natu-

ral composition, e.g., we will use

to denote (

)

k∈Zn

If the subscript is a subset of

, then the composi-

tion is taken only over those elements. Moreover, we

use

as shorthand for

Zn\S

, and

for

\{i}

. If

is a completely positive map, then

ρε

is its Choi oper-

ator [

]. For a directed graph

G= (V, E ⊆V×V)

denotes the set of nodes and Ethe set of directed edges.

Adirected path

π= (v0, . . . , vℓ)

is a sequence of distinct

nodes with

{(vi, vi+1)|0≤i<ℓ} ⊆ E

. A directed cy-

cle

C= (v0, . . . , vℓ)

is a directed path with (

vℓ, v0

)

∈E

The induced graph

[

V′

] is the graph

G′

= (

V′, E′

)

where

V′⊆V

and all edges

E′⊆E

have endpoints in

V′

i.e., (

i, j

)

∈E′

if and only if

i, j ∈V′

and (

i, j

)

∈E

. A di-

rected cycle

is called induced directed cycle or chordless

cycle if the induced graph

[

] is a directed cycle graph.

For a node

k∈V

, we use

(

), and

Anc

(

) to

express the set of parents, children, and ancestors of

respectively, and similarly

(

) and

Anc

(

) by

taking the union over a set

. The cardinality of the

set

(

) is the in-degree

degin

(

). A node with zero

in-degree is called source. We use

CPa

(

) for the union

of the common parents of all elements

i̸

j∈S

,i.e.,

CPa

(

) :=

Si̸=j∈SPa

(

)

∩Pa

(

). Two nodes

i, j ∈V

are

called siblings if and only if they have common parents,

i.e.,

CPa

(

{i, j}

)

∅

. A directed path

is said to contain

siblings if and only if CPa(π)̸=∅.

Correlations

Correlations observed among

parties (regions)

are expressed with the conditional probability distribu-

tion

(

a|x

) where for each party

we have

ak∈ Ak

and

xk∈ Xk

. The set

is the set of experimental

settings and Akthe set of experimental observations.

Deﬁnition 1 (Causal correlations [

]).The

-party

correlations

(

a|x

) are causal if and only if they can be

decomposed as

p(a|x) = X

k∈Zn

qkp(ak|xk)pxk

ak(a\k|x\k),(1)

where

∀k∈Zn

qk≥

Pk∈Znqk

= 1, and

pxk

ak(a\k|x\k)

are (

n−

1)-party causal correlations. If this decomposition

is infeasible, the correlations are called non-causal.

The motivation behind this deﬁnition is that each party

can inﬂuence its future only, including the causal order

of the parties in its future. From this follows that there

µa0|x0

0µa1|x1

1µa2|x2

Figure 2: Schematic of three parties and a process. The

gray area represents the process: It takes the systems on

the future boundaries of the parties and maps it to the

past boundaries of the parties. The red connections

indicate an example where the parties are causally

ordered increasingly. A priori, however, the process is

not assumed to respect any ordering of the parties.

exists at least one party

whose observation does not

depend on the data of any other party (the value of

depends solely on

). Moreover, the causal order among

the parties might be subject to randomness, e.g., a coin

ﬂip.

Processes

In the process-matrix framework [

], each party (re-

gion) is deﬁned through a past and future space-like

boundary (see Figure 2). For party

, we denote

the Hilbert space on the past, by

the Hilbert

space on the future boundary, and by

CPk

(

CPTPk

)

the set of all completely positive (trace-preserving)

maps from

(

) to

(

). A quantum experiment

for party

is a quantum channel from

equipped with a classical input (the setting) and a clas-

sical output (the observation). Hence, it is a fam-

ily

{µak|xk

k∈CPk}(ak,xk)∈Ak×Xk

of maps, such that for

all

xk∈ Xk

, we have

Pak∈Akµak|xk

k∈CPTPk

. A pro-

cess interlinks all parties without any assumption on

their causal relations, but with the sole assumption that

no deviation from quantum theory is locally observable,

i.e., the probability distribution over the observations

is a multi-linear function of the quantum experiments

and well-deﬁned (even if the parties share an arbitrary

quantum system).

Deﬁnition 2 (Process [

]).An

-party quantum process

is a positive semi-deﬁnite operator W∈ L(I ⊗ O) with

∀{µk∈CPTPk}k∈Zn: Tr "WO

k∈Zn

ρµk#= 1 .(2)

Note that in this deﬁnition, the experimental set-

tings and observations are absent, or equivalently, the

sets

Ak,Xk

are singletons. Eq.

(2)

states that the total

probability of observing this single outcome under this

single setting is one; this holds for any setting and is inde-

pendent of any resolution of the completely positive trace-

preserving map into maps with classical outputs. Ore-

shkov, Costa, and Brukner [

] already observed that the

quantum process

is the Choi operator of a completely

positive trace-preserving map from all future boundaries

to all past boundaries of the parties (cf. Figure 2). For

a given choice of

-party quantum process and experi-

ments, the correlations are computed with the generalized

Born rule

p(a|x) := Tr "WO

k∈Zn

ρµak|xk

k#.(3)

If one assumes that the parties perform classical-

deterministic experiments, as opposed to quantum experi-

ments, we arrive at the following special case. The spaces

on the past and future boundaries

Ik,Ok

are sets (as op-

posed to Hilbert spaces), and an experiment for party

is a function

µk:Ik× Xk→ Ok× Ak

(as opposed to a

family of maps), where we use

µ0

k:Ik× Xk→ Ok

for the

ﬁrst, and

µ1

k:Ik× Xk→ Ak

for the second component.

Just as in the quantum case, a classical-deterministic

process

turns out to be a function from the future

to the past boundaries, and Eq.

(2)

translates into an

intuitive condition: For any choice of experiment, there

exists a unique consistent assignment of values to the

input spaces; the map ω◦µhas a unique ﬁxed point.

Theorem 1 (Classical-determintic process [

]).

-party classical-deterministic process is a func-

tion ω:O → I with

∀{µk:Ik→ Ok}k∈Zn,∃!(rk)k∈Zn:r=ω(µ(r)) ,(4)

where ∃!is the uniqueness quantiﬁer.

Here, the correlations among the

-parties are com-

puted via

p(a|x) := ω ⋆ µa|x=X

i,o

[ω(o) = i] [(o, a) = µ(x, i)] ,(5)

where we use [

] for the Kronecker delta

δn,m

, and

⋆

for the link product [

]. Note that every

-party classical-

deterministic process

also corresponds to an

-party

quantum process [27]

Wω:= X

o∈O

|o⟩⟨o|O⊗ |ω(o)⟩⟨ω(o)|I,(6)

where

|o⟩

Nk∈Zn|ok⟩

, with

{|o′

k⟩}o′

k∈Ok

being a basis

for all k∈Zn, and similarly for |ω(o)⟩.

We brieﬂy illustrate the above theorem in the single-

party case (

= 1). Assume the function

is the iden-

tity function from

{

}

{

}

. Here,

describes a closed time-like curve: The output of that

single party is identically mapped to the same party’s

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AdmissibleCausalStructuresandCorrelationsEleftherios-ErmisTselentis1,2and¨AminBaumeler3,41InstituteforQuantumOpticsandQuantumInformation(IQOQI-Vienna),AustrianAcademyofSciences,1090Vienna,Austria2FacultyofPhysics,UniversityofVienna,1090Vienna,Austria3Facolt`adiscienzeinformatiche,Universit`adellaSvi...

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Admissible Causal Structures and Correlations Eleftherios-Ermis Tselentis1 2and Amin Baumeler3 4 1Institute for Quantum Optics and Quantum Information IQOQI-Vienna

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