Fermat's Library | q-2019-12-02-206 annotated/explained version.

Time-delocalized quantum subsystems and opera-

tions: on the existence of processes with indeﬁnite

causal structure in quantum mechanics

Ognyan Oreshkov

QuIC, Ecole p olytechnique de Bruxelles, C.P. 165, Université libre de Bruxelles, 1050 Brussels, Belgium

It has been shown that it is theoretically possible for there to exist higher-order

quantum processes in which the operations performed by separate parties cannot be

ascribed a deﬁnite causal order. Some of these processes are believed to have a phys-

ical realization in standard quantum mechanics via coherent control of the times of

the operations. A prominent example is the quantum SWITCH, which was recently

demonstrated experimentally. However, the interpretation of such experiments as re-

alizations of a process with indeﬁnite causal structure as opposed to some form of

simulation of such a process has remained controversial. Where exactly are the local

operations of the parties in such an experiment? On what spaces do they act given

that their times are indeﬁnite? Can we probe them directly rather than assume what

they ought to be based on heuristic considerations? How can we reconcile the claim

that these operations really take place, each once as required, with the fact that the

structure of the presumed process implies that they cannot be part of any acyclic cir-

cuit? Here, I oﬀer a precise answer to these questions: the input and output systems

of the operations in such a process are generally nontrivial subsystems of Hilbert

spaces that are tensor products of Hilbert spaces associated with systems at diﬀerent

times—a fact that is directly experimentally veriﬁable. With respect to these time-

delocalized subsystems, the structure of the process is one of a circuit with a causal

cycle. This provides a rigorous sense in which processes with indeﬁnite causal struc-

ture can be said to exist within the known quantum mechanics. I also identify a whole

class of isometric processes, of which the quantum SWITCH is a special case, that

admit a physical realization on time-delocalized subsystems. These results unveil a

novel structure within quantum mechanics, which may have important implications

for physics and information processing.

1 Introduction

According to quantum mechanics, physical quantities in general do not have deﬁnite values unless

measured. Yet, the classical idea that events occur in a well deﬁned causal order persists, since

quantum operations are assumed to always take place in acyclic compositions that respect the

causal structure of spacetime. A natural question is whether this deﬁniteness in the causal order

of operations is a fundamental physical restriction or an artifact of our formulation of quantum

theory. Is it possible that, in suitable circumstances, the order of operations would be indeﬁnite

similarly to other physical variables? How would this be described formally, and what testable

consequences would it entail? These questions may be particularly relevant for understanding

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arXiv:1801.07594v4 [quant-ph] 15 Nov 2019

physics in the regimes of quantum gravity, where the causal structure of spacetime is expected to

be subject to quantum indeﬁniteness [1, 2].

A theoretical model that can be interpreted as describing a situation in which separate opera-

tions occur in a ‘superposition’ of diﬀerent orders was proposed by Chiribella et al. in Ref. [3].

This model, called quantum SWITCH, was conceived as a hypothetical computer program that

takes as an input two black-box quantum gates and outputs a new gate that can be thought of as

the result of applying sequentially the two gates in an order that depends coherently on the logical

value of a qubit that could be prepared in a superposition. Mathematically, such a program is an ex-

ample of a higher-order quantum transformation, or supermap [4, 5]. The authors pointed out that

this transformation cannot be realized by using each of the input gates once in an acyclic circuit

[6], just as the classical version of the program cannot, but nevertheless one can conceive imple-

mentations based on physical circuits with movable wires that simulate the eﬀect of the program,

thereby making a case for the need of a more general theoretical model of quantum computation

than the circuit model. Although at face value the program involved nonclassicality in the order of

operations, the theoretical tools for making this statement precise were not developed at that time.

A framework for investigating the possibility of indeﬁnite causal order in quantum theory by

means of correlations was developed in Ref. [9]. This so-called process framework describes

separate local experiments, each deﬁned by a pair of input and output quantum systems on which

an agent can apply arbitrary quantum operations, without presuming the existence of global causal

order between the experiments. Under a set of natural assumptions (see Sec. 2), the most general

correlations between such experiments can be shown to be given by a generalization of Born’s

rule that involves an extension of the density matrix called the process matrix. Mathematically,

the process matrix can also be understood as a higher-order transformation, but one that maps the

local operations to probabilities. The formalism provides a uniﬁed description of all nonsignaling

and signaling quantum correlations between separate experiments that can be arranged in a causal

conﬁguration [5], as well as probabilistic mixtures of diﬀerent such causal scenarios. Remarkably,

it was found that there are logically consistent bipartite process matrices that are incompatible

with the existence of deﬁnite causal order between the local experiments and hence cannot be

realized in this way. Such processes were called causally nonseparable [9]. The concept of causal

nonseparability was subsequently generalized to more that two parties [10] (see also [11, 12]),

which in particular provided a rigorous framework in which the indeﬁniteness of the causal order

involved in the quantum SWITCH can be deﬁned.

Causally nonseparable processes allow accomplishing certain tasks that cannot be achieved

with operations for which a deﬁnite causal order exists. A striking possibility allowed by some

causally nonseparable processes is the generation of correlations that violate causal inequalities

[9, 13, 14, 15, 10, 16, 17, 20, 18, 19]. These correlations imply incompatibility with deﬁnite causal

order under theory independent assumptions, similarly to the way a violation of a Bell inequal-

ity implies incompatibility with local hidden variables [21]. It is not known at present whether

processes violating causal inequalities have a physical realization, except through post-selection

[22, 23, 24, 25]. However, it is widely believed that a speciﬁc class of causally nonseparable pro-

cesses, which includes the quantum SWITCH, has a physical realization without post-selection

via coherent control of the times at which the local operations occur, as in the original proposal

[3]. The known processes of this kind cannot violate causal inequalities [10, 11], but they can

be proven incompatible with deﬁnite causal order in a device-dependent fashion [26, 11, 27]. In

particular, the quantum SWITCH and its generalizations have been shown to oﬀer advantages over

processes in which the order of operations is deﬁnite for a variety of information-processing tasks

[26, 28, 29, 30, 31]. Concrete implementations of the quantum SWITCH via coherent control of

the times of the operations have been proposed for trapped-ion systems [32], photonic systems

[33], and systems of superconducting qubits [34], and demonstrated with photonic systems in a

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series of increasingly sophisticated experiments [33, 35, 36, 37].

Despite this experimental progress, however, the question of whether implementations of this

kind can be rightfully interpreted as realizations of causally nonseparable processes as opposed to

some form of simulations of such processes has remained a subject of debate. The reason is that,

even though by construction the produced data agrees with what we expect from the corresponding

process, it is unclear whether the circumstances in which the data is produced correspond to a

process. In a quantum process, by deﬁnition, the local operations of the parties occur each once

on a speciﬁc pair of input and output Hilbert spaces. In an implementation based on controlled

operations, it is not obvious on what spaces the local operations occur, if at all. For example, in the

implementations of the quantum SWITCH [33, 35, 36, 37], the experiment can be seen to involve

two applications of controlled unitary operations at two diﬀerent times in the laboratory of each

party (see Sec. 3), which are such that under the particular arrangement, when the control qubit is

prepared in a classically deﬁnite logical state, exactly one of the two would result in a nontrivial

transformation on the target system. However, in the actual implementation, the control qubit is

prepared in a quantum superposition of the two logical states, in which case the belief that the

same operation is applied once on the target system at some indeﬁnite time is merely based on

heuristics. This heuristics can be artiﬁcially strengthened by extending each controlled operation

to act on a system that works as a ‘counter’ that is coherently raised each ‘time’ the nontrivial

controlled operation is applied on the target system, such that the reading of the counter at the end

could be interpreted as evidence that the desired operation has been applied once [29]. However,

since the faithfulness of the counter as evidence for the applied operation is established only in

the case when the target operation occurs at a deﬁnite time, regarding it as evidence in the case

of superpositions requires the same conceptual leap. While such a leap appears appealing, it is

nothing but begging for a rigorous theory that positions the supposed operations in relation to the

standard temporal description of the experiment.

Here, I show that the supposed operations of the parties in such an implementation really take

place on speciﬁc input and output systems—a fact that can be directly veriﬁed experimentally.

These input and output systems are generally time-delocalized subsystems, i.e., nontrivial subsys-

tems of the tensor products of Hilbert spaces associated with diﬀerent times. The fact that we

can think, both mathematically and operationally, of Hilbert spaces that are tensor products of

Hilbert spaces at diﬀerent times is well established [5]—a generic fragment of a standard quantum

circuit is an example of a quantum operation whose input and output Hilbert spaces are of this

kind. However, given two Hilbert spaces of this kind, it is generally not possible to apply arbitrary

quantum operations from one to the other due to the constraints imposed by the causal structure

of spacetime [5]. At the same time, it is well known that the most general faithful realization of

a quantum system inside a given Hilbert space is in the form of a subsystem—a tensor factor of

a subspace of the full Hilbert space [38, 39, 40]. In fact, any experimentally accessible quantum

system is such a subsystem from the perspective of a larger Hilbert space [41, 42]. Thus, it is

also natural to consider operations whose input and output systems are nontrivial subsystems of

the tensor products of Hilbert spaces associated with diﬀerent times. As it turns out, there exist

pairs of such time-delocalized input and output subsystems on which it is possible to apply any

standard quantum operation without post-selection, despite the fact that the input system cannot

be associated with a region of spacetime that is in the causal past of the output system. It is

on such time-delocalized quantum subsystems that causally nonseparable processes are realized.

With respect to these input and output systems, a causally nonseparable process has the structure

of a circuit with a cycle that does not admit a decomposition into a probabilistic mixture of acyclic

circuits or dynamical generalizations of such mixtures [10]. The existence of these irreducible

cyclic structures within quantum mechanics is the main ﬁnding of this paper.

The rest of the paper is organized as follows. In Sec. 2, I review the basics of the process matrix

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framework. In Sec. 3, I describe explicitly the main result in the case of the quantum SWITCH.

In Sec. 4, I discuss the sense in which the result is regarded as justifying the interpretation of the

discussed implementations as ‘realizations’ of the quantum SWITCH, as well as the question of the

resources used in these implementations. In Sec. 5, I outline how the result generalizes to arbitrary

processes in which the operations are delocalized in time through controlled operations. In Sec.

6, I show that all bipartite processes that obey a recently proposed unitary extension postulate

[43], together with their unitary extensions, of which the quantum SWITCH is an example, have a

realization on suitably deﬁned time-delocalized subsystems. In Sec. 7, I deﬁne a class of isometric

extensions of bipartite processes, which is strictly larger than the class of unitary extensions, and

show that these processes also admit a realization on suitable time-delocalized subsystems. It

remains an open question whether the class of bipartite processes that admit such extensions is

larger than those admitting unitary extensions. In Sec. 8, I discuss the results.

2 The process matrix framework

The quantum process framework [9] describes separate local experiments, X = A, B, C, ..., each

deﬁned by an input quantum system X

with Hilbert space H

and an output quantum system

with Hilbert space H

, where an agent can perform an arbitrary quantum operation from

to X

. A quantum operation is most generally described by a collection of completely posi-

tive (CP) and trace-nonincreasing maps {M

→X

}

∈O

, M

→X

: L(H

) → L(H

where L(H

) denotes the space of linear operators over the Hilbert space H

with dimension d

(here we assume ﬁnite dimensions), and i

∈ O

labels the possible outcomes of the operation

with which the diﬀerent CP maps are associated. The sum of the CP maps corresponding to the

complete set of the outcomes of an operation,

→X

∈O

→X

, must be a CP

and trace-preserving (TP) map.

Consider the joint probabilities p(i

, j

, ···|{M

→A

}

∈O

, {M

→B

}

∈O

, ···) for

the outcomes of the local experiments, conditional on the parties choosing to perform speciﬁc

quantum operations. Under the following assumptions [9]—(i) these probabilities are functions

only of the CP maps corresponding to the local outcomes, (ii) they are consistent with the lo-

cal quantum description of coarse-graining and randomization, (iii) the local operations can be

extended to act on arbitrary auxiliary input systems prepared in any joint quantum state—the

probabilities can be written in the form

p(i

, j

, ···|{M

→A

}

∈O

, {N

→B

}

∈O

, ···)

= Tr

···



⊗ N

⊗ ···

i

. (1)

Here, M

∈ L(H

⊗ H

) ≥ 0 is the Choi-Jamiołkowski (CJ) operator [44, 45] of the

CP map M

→A

, and similarly for N

, etc. [Here, the CJ operator M

∈ L(H

⊗

) of a linear map M

→A

: L(H

) → L(H

) is deﬁned as M



→A

⊗

→A



|φ

ihφ



, where |φ

j=1

|ji

∈ H

⊗ H

is a (not

normalized) maximally entangled state on two copies of H

(one of them denoted by H

), the

states

|ji

j=1

form an orthonormal basis of H

, and

|ji

j=1

are their copies in H

→A

is the identity superoperator on L(H

), and T denotes matrix transposition in the above

basis of A

and some basis of A

(see Ref. [22] for a physical interpretation of this isomorphism

and the choice of basis, which is based on the symmetry transformation of time reversal [46]).]

The operator W

··· is an operator on the tensor product of all input and output systems,

called the process matrix. The only constraints that a process matrix must satisfy come from the

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Alice Bob

} {N

} =

p(i, j|{M

}, {N

}, W)

Figure 1: A process as a cyclic circuit. A process matrix W is equivalent to the transpose of the CJ

operator of a channel

W from the outputs of the local operations to their inputs, while the probability

rule (1) is equivalent to composing this channel with the local operations, as illustrated here.

requirement that probabilities must be nonnegative and sum up to 1 (the latter is equivalent to

requiring normalization on all deterministic local operations, or CPTP maps):

···

≥ 0, (2)

···



⊗ N

⊗ ···

i

= 1,

∀M

, N

··· ≥ 0,

= 11

, Tr

= 11

, ··· . (3)

Condition (3) can be equivalently formulated as a simple constraint on the types of nonzero

terms permitted in the expansion of a process matrix in a Hilbert-Schmidt basis, which is highly

useful for constructing process matrices or verifying if a given operator is a valid process matrix

[9]. More speciﬁcally, any Hermitian matrix W

···

can be written in the form

···

i,j,k,l,m,n···

ijklmn···

⊗ σ

⊗ ··· ,

ijklmn···

∈ R, ∀i, j, k, l, m, n, ··· , (4)

where the Hermitian operators {σ

}

−1

µ=0

, with σ

= 11

, Trσ

= d

µν

, and Trσ

= 0 for

j = 1, ..., d

− 1, form a Hilbert-Schmidt basis. An operator of the form (4) satisﬁes condition

(3), if and only if, in addition to the term proportional to 11

···

which comes with

weight w

000000···

···

, it contains only nonzero terms in which there is a nontrivial σ

(diﬀerent from the identity) operator on X

and a trivial one (the identity) on X

for some party

X ∈ {A, B, C, ···} [10, 11].

The process matrix can be understood as a higher-order transformation from the tensor product

of a set of local quantum operations to conditional probability distributions, which is completely

positive and normalized on deterministic local operations [47, 48]. Since a process matrix is

mathematically equivalent to the transpose of the CJ operator of a channel

···→A

···

from the outputs of the local parties to their inputs (which can be seen from the terms permitted

in its Hilbert-Schmidt basis expansion [9]), and the probability formula (Eq. (1)) is equivalent

to the composition of that channel with the local operations, such a higher-order transformation

can be thought of as a circuit with a cycle [9], as illustrated in Fig. 1 for the case of two parties.

Hereafter, when it is clear from the context, we will often refer to the channel

···→A

···

corresponding to a process matrix simply as the process.

In the case of two parties, the most general process matrix compatible with a ﬁxed causal

relation between the operations of Alice and Bob has the form W

= W

⊗

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(the case where Bob is not in the causal past of Alice and hence cannot signal to Alice) or

= W

⊗11

(the case where Alice is not in the causal past of Bob and hence

cannot signal to Bob). Each of these has an implementation via embedding the operations of Alice

and Bob in an acyclic quantum circuit [4]. More generally, we may conceive of situations where

these two possibilities are realized at random with some probabilities, in which case the process

matrix can be written in the form

= qW

⊗ 11

+ (1 − q)W

⊗ 11

q ∈ [0, 1]. (5)

Such process matrices are called causally separable [9]. They correspond to the most general

situation in which a ﬁxed (though possibly unknown) casual conﬁguration between the parties

exist in each run of the experiment, and where the correlations arising in each conﬁguration come

from a standard quantum circuit containing the operations of Alice and Bob. (In the case of three

or more parties, the concept of causal separability is more complicated due to the possibility of

dynamical causal conﬁgurations [10], where the causal order between a subset of the parties can

depend on the operations performed by other parties in their past.)

3 The quantum SWITCH and its physical realization

The quantum SWITCH [3] is a higher-order transformation, or supermap, which takes as an input

two black-box operations, {M

→A

}

∈O

and {N

→B

}

∈O

, ﬁguratively associated with

Alice and Bob, where d

= d

= d, and gives as an output an operation

G→G

}

k∈O

×O

, where H

= H

⊗ H

and H

= H

⊗ H

with d

= d

= 2 and

= d

= d. Its action can be described intuitively as follows. If we think of H

and H

the Hilbert spaces of a control qubit at some initial and some ﬁnal time, respectively, and of H

and H

as the Hilbert spaces of some target system at the same two times, then the eﬀect of the

resultant operation can be thought of as transforming the target system from the initial to the ﬁnal

time by the sequential application of the operations {M

}

∈O

, {N

}

∈O

, where the order

in which the two operations are applied depends coherently on the logical value of the control

qubit. To describe this coherent conditioning precisely, we use the fact that supermaps are deﬁned

on all extensions of the original input operations onto additional systems [3, 4, 5, 47, 48]. Since

any quantum operation can be realized by applying a joint unitary channel on the original input

system plus a suitably initialized auxiliary input system followed by a destructive measurement

on a subsystem of the output system of the channel, the eﬀect of the quantum SWITCH on the

operations of Alice and Bob can be inferred from its eﬀect in the case when the two operations

are extended unitary channels U

→a

and U

→b

, each acting on the original input

and output systems plus separate auxiliary input and output systems, as described in Fig. 2. Let

→a

and U

→b

denote the unitary matrices describing the action of these unitary

channels at the Hilbert-space level. Then, the result of the quantum SWITCH is a unitary channel

G→a

SWITCH

, U

) whose unitary matrix at the Hilbert-space level is

G→a

SWITCH

, U

) ≡ U

QS→a

SWITCH

, U

) =

|0i

h0|

⊗ U

X→b

◦ U

S→a

+|1i

h1|

⊗ U

X→a

◦ U

S→b

, (6)

where X is a dummy system of dimension d over which the transformations are composed (we

have dropped the superscripts indicating the input and output systems of Alice’s and Bob’s opera-

tions in the argument of the supermap to simplify the notation).

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SWITCH

, U

)

S’Q’

S’

Q’

Figure 2: Extending local operations. The quantum SWITCH, just like any higher-order process, is

deﬁned on all extensions of the local operations onto auxiliary systems. Hence, it is fully determined

by its action on extended local unitary channels.

(a) (b)

Figure 3: The quantum SWITCH in the extreme classical cases. When the control qubit is

initialized in either the state |0ih0| [case (a)] or in the state |1ih1| [case (b)], the result of the quantum

SWITCH is eﬀectively that of applying the operations of Alice and Bob on the target system in a

particular order.

In the special cases when the control qubit is prepared in the state |0ih0|

, the quantum

SWITCH eﬀectively applies ﬁrst the operation of Alice and then the operation of Bob on the

target system. When the control qubit is prepared in the state |1ih1|

, it eﬀectively applies ﬁrst

the operation of Bob and then the operation of Alice on the target system (see Fig. 3). When the

control qubit is prepared in a superposition of these basis states, such as |+ih+|

, |+i ≡

|0i+|1i

√

the intuitive understanding is that these two scenarios are somehow realized ‘in superposition’,

as sketched in Fig. 4. Note, however, that the two extreme scenarios are not simultaneously

compatible with a common causal structure [3].

To see the quantum SWITCH as a process matrix, we consider two more parties—David, who

is allowed to prepare diﬀerent input states into the input system G of the resultant channel (i.e., his

possible operations have a trivial input system and output system D

≡ G = QS), and Charlie,

who is allowed to perform measurements on the output system G

of the resultant channel (i.e.,

his possible operations have the input system C

≡ G

= Q

and a trivial output system). It can

be veriﬁed that the process matrix describing the correlations between the four parties is

SWITCH

= |W ihW |

, (7)

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S’Q’

Figure 4: Intuitive description of the quantum SWITCH in the case of superpositions. When

the control qubit is initialized in a state that is a superposition of the logical basis states, such as

|+ih+|, one may intuitively think of the quantum SWITCH as realizing the extreme classical cases ‘in

superposition’.

where

|W i

= |0i

|0i

|Φ

+ |1i

|1i

|Φ

, (8)

with |Φ

i =

|ii|ii, where {|ii}is the basis for the Choi isomorphism for the respective system.

The fact that this process matrix is causally nonseparable follows from the fact that there

are certain preparations that David can make for which the operations of Alice and Bob cannot

be said to take place in a deﬁnite order. The example most commonly considered and the one

implemented in the experiments discussed below corresponds to the case where David prepares

the state |+ih+|

on the control qubit together with some pure state |ψihψ|

on the target system.

In that case, the correlations between Alice, Bob, and Charlie are given by the tripartite process

matrix

= |W ihW |

, (9)

where

|W i

√

(|0i

|ψi

|Φ

+|1i

|ψi

|Φ

). (10)

The causal nonseparability of this process matrix [10, 11] follows from the fact that it is propor-

tional to a rank-one projector and hence cannot be written as a probabilistic mixture of diﬀerent

process matrices, but at the same time it permits some communication from Alice to Bob as well as

from Bob to Alice—conditions that cannot be met simultaneously by a tripartite causally separable

process matrix [10].

In the experimental implementations that we will discuss, the target system is represented

by the internal degrees of freedom of a particle (e.g., the polarization of a photon as in Refs.

[33, 35, 36]) and the control qubit by its path degree of freedom—an idea naturally inspired by

Fig. 3 and Fig. 4. (We stress, however, that our argument does not depend on the exact choice

of physical encoding of the control qubit and the target system, and hence holds also for other

realizations of this encoding, such as the one in Ref. [37], where the control qubit is given by

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photon polarization and the target system by transverse spatial modes.) The setup is such that it

allows the particle to follow two possible paths—along one path (corresponding to the initial state

|0ih0|

) it ﬁrst passes through Alice’s laboratory and then through Bob’s laboratory, and along

the other path (corresponding to the initial state |1ih1|

) it passes through the laboratories in the

opposite order. If inside each laboratory there is a device that applies the correct unitary on the

internal degrees of freedom of the particle plus auxiliary systems when the particle passes through,

it can be easily veriﬁed that the resulting transformation on all systems would be given by Eq. (6)

(see Ref. [35] for a concrete experimental implementation).

The fact that the result of such an experiment is the transformation (6) does not by itself justify

interpreting the experiment as a realization of the quantum SWITCH. The latter requires that this

transformation be produced by applying each of the operations of Alice and Bob once on speciﬁc

input and output systems. Whether this can be said to be the case in such an implementation is one

of the central questions of this paper. The standard argument in favor of this being a realization

of the process is merely based on the intuition that since the local devices would apply the correct

operations on the internal degrees of freedom of the particle whenever the particles passes through

at a given time, and since the particle may be heuristically argued to pass exactly once through

each device (an idea that can be supported by supplying each device with a coherent counter that

counts how many particles go through it), even if the time of passage in the two diﬀerent classical

cases is diﬀerent and in the general case not even deﬁned, we should be able to say that the correct

operation has been applied once. While this line of reasoning may appear intuitive, the problem

with it is that we only have a clear deﬁnition of what it means to apply the correct operations and

possess evidence for this in the extreme cases of deﬁnite times. Simply declaring that this holds

in the more general case without specifying where precisely the operations take place nor how we

could probe them can hardly be considered convincing. (There are many properties that hold in the

extreme case where a qubit is prepared in one of the two logical basis states but do not hold in the

case of nontrivial superpositions—the causal separability of the presumed process resulting from

such an implementation would be one example.) In fact, the above heuristic reasoning suggests

that even in the case of indeﬁnite times, the operations of Alice and Bob are still operations on the

internal degrees of freedom of the particle. As we will see below, this is not correct—Alice and

Bob can indeed be said to apply the correct operations, but the precise input and output systems are

nontrivial subsystems of Hilbert spaces composed of both the control qubit and the target system

at diﬀerent times.

To simplify our analysis, we will restrict our attention to an implementation in which the

operation of Bob is applied at a ﬁxed time, while the operation of Alice may be applied before

Bob’s operation or after it, depending on the logical value of the control qubit. (The experiments

reported in Refs. [33, 35, 36] are based on a symmetric setup with respect to the possible times of

the operations of Alice and Bob, and below we will comment on how our result looks in that case

as well.) From a temporal perspective, the experiment has the circuit structure depicted in Fig. 5,

where the unitaries are given at the Hilbert-space level, the operations of David and Charlie are

left unspeciﬁed, and we use a standard graphical notation for controlled unitaries [49], where a

black dot represents conditioning on the state |1ih1| and a white dot on the state |0ih0|.

Remark. The circuit in Fig. 5 is drawn with respect to the control qubit and the target system,

which in the conceived realization are the path and internal degrees of freedom of a particle. Nei-

ther of these systems is a spatially localized system. One may ask how this description relates to a

system decomposition that reﬂects the spatiotemporal conﬁguration. Assuming that Alice and Bob

reside at separate spatial locations and that either vacuum or one particle can enter their laboratory

at any given time, a natural choice of a spatially local system to associate with each laboratory

would be the d + 1-dimensional Hilbert space that is a direct sum of the local vacuum and the d-

dimensional Hilbert space of the internal degrees of freedom of the particle. The full Hilbert space

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CharlieTime

S’

Q’

David

Bob

Figure 5: Temporal description of the implementation of the quantum SWITCH with Bob’s

operation at a ﬁxed time. From a temporal perspective, the experiment can be described by a

sequence of unitaries acting on the control qubit and the target system plus the local auxiliary systems

of Alice and Bob. Here, Bob’s operation is implemented at a ﬁxed time, while Alice’s supposed operation

is implemented by means of two controlled unitaries.

of the joint system of Alice and Bob at a given time is then the tensor product of these two local

Hilbert spaces. However, in the described experimental setup only a subspace of this Hilbert space

is ever populated—the one corresponding to the presence of exactly one particle, which could be

in either Alice’s or Bob’s laboratory or some superposition of the two locations. This subspace

decomposes into a tensor product of a two-dimensional subsystem encoding the location of the

particle (or the path degree of freedom) and the d-dimensional subsystem corresponding to the

internal degrees of freedom. Thus, with respect to a spatially local choice of systems, the control

qubit and the target system are factors of a proper subspace of the full Hilbert space. As noted

in the introduction, this is the most general realization of a system within a given Hilbert space,

and it will be essential in the generalization discussed in Sec. 7. Note, however, that the present

argument is independent of how precisely the circuit in Fig. 5 is realized—one may even think

that the depicted systems are spatially local and that the realization consists of directly applying

the controlled operations displayed on the ﬁgure by turning on and oﬀ suitable ﬁelds.

By assumption, the input and output systems of David, Bob, and Charlie are associated with

ﬁxed times. We know what it means operationally to say that David, Bob, and Charlie perform

speciﬁc operations with these input and output systems: we can test this by feeding suitable states

in their input systems and performing suitable measurements on their output systems. Alice’s

supposed operation U

→a

, on the other hand, is some still unidentiﬁed part of the circuit

fragment in Fig. 6 (the description in Fig. 6 is given in terms of the transformations at the Hilbert-

space level). It is clear that such a fragment is itself a quantum operation from the composite input

system a

QSB

to the composite output system a

(which is described by the theory of

quantum combs [5]; in particular, it is a 1-comb). In this case, the operation is a unitary channel

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Time

S’

Q’

Figure 6: Circuit fragment containing Alice’s operation. Removing the operations of David, Bob,

and Charlie from Fig. 5, we are left with this circuit fragment in which Alice’s operation is supposedly

taking place in some sense.

whose unitary matrix at the Hilbert-space level is

QSB

→a

SWITCH

) =

|0i

h0|

⊗ U

S→a

⊗ 11

→S

+|1i

h1|

⊗ U

→a

⊗ 11

S→B

, (11)

which can be obtained by multiplying the unitary gates that make up the circuit fragment. This

statement can also be veriﬁed through tomography by feeding suitable states in the joint input and

performing suitable measurements on the joint output, as depicted in Fig. 7.

Our ﬁrst main observation is that the unitary in Eq. (11) can be written in the form

QSB

→a

SWITCH

) = U

→a

⊗ 11

→A

, (12)

or, equivalently, the unitary channel of the quantum comb can be written in the form

QSB

→a

SWITCH

) = U

→a

⊗ I

→A

, (13)

where I denotes the identity channel. Here, H

is a tensor factor of H

QSB

that has dimension d

and H

is its cofactor (H

QSB

= H

⊗H

), and similarly, H

is a tensor factor of H

that has the dimension of d and H

is its co-factor (H

= H

⊗ H

). The factor H

is deﬁned by the algebra of operators of the form

≡ |0ih0|

⊗ O

⊗ 11

+ |1ih1|

⊗ 11

⊗ O

, (14)

and the factor H

by the algebra of operators of the form

≡ |0ih0|

⊗ 11

⊗ O

+ |1ih1|

⊗ O

⊗ 11

. (15)

These are the input and output systems of Alice’s operation.

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Time

S’Q’ a

Q a

measurement

preparation

Figure 7: Tomography of circuit fragments and time-delocalized operations: an example. The

operation implemented by a circuit fragment can be tested via tomography similarly to any standard

operation, by ‘pulling’ all input wires to the past and all output wires to the future via identity trans-

formations, as depicted here for the fragment from Fig. 6. Via the same general scheme, one can also

perform tomography of time-delocalized operations, such as U

→a

in Eq. (12). This simply

requires preparing suitable states in the input subsystem a

, and applying suitable measurements on

the output subsystem a

To see this, notice that [50], for any U

QSB

→a

SWITCH

)

= C-SWAP

S→a

⊗ 11

→Q

)C-SWAP

QSB

, (16)

where C-SWAP

XY Z

denotes the controlled-SWAP unitary operator (which is also Hermitian)

with control qubit X and target systems Y and Z [52]. This can be veriﬁed from the expression

for U

QSB

→a

SWITCH

) in Eq. (11). The operator sandwiched between the two C-SWAP

operators on the right-hand side of Eq. (16) has the same form as the right-hand side of Eq. (12),

except that A

is replaced by S, and A

is replaced by B

. Since the ﬁrst C-SWAP maps some

subsystem of QSB

(call it A

) to the subsystem S, and the second C-SWAP maps the subsystem

to some subsystem of Q

(call it A

), the action of U

QSB

→a

SWITCH

) is precisely

the one given in Eq. (12) with respect to A

and A

. The subsystems A

and A

are related to S

and B

via the above C-SWAP transformations, which is equivalent to the relations (14) and (15)

for the operators on these subsystems.

Having identiﬁed the subsystems A

and A

, the operation performed from A

to A

becomes

a directly testable fact—we can verify it by preparing suitable states on A

and performing suitable

measurements on A

, which ﬁts within the general scheme depicted in Fig. 7.

Note that the subsystem A

has a nontrivial ‘spread’ over B

in the sense that it is not a

subsystem of the complement QS. Similarly, A

has a nontrivial spread over B

. But B

is in the

causal past of B

. In spite of this, it is possible to perform an arbitrary standard operation from

to A

. While this fact may appear counterintuitive, it is important to stress that, at least in this

case, it is not speciﬁc to quantum mechanics—it holds irrespectively of whether we initialize Q in

a quantum superposition or a classical probabilistic mixture of the logical states, and even has an

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SWITCH

Alice Bob

S’Q’

Charlie

David

Figure 8: The quantum SWITCH as an irreducible cyclic circuit. With respect to the time-

delocalized subsystems A

and A

, the discussed implementation has the structure of a circuit with a

cycle. This is trivially true for any experiment involving a set of ﬁrst-order operations since any process

matrix is equivalent to a channel, as discussed earlier. However, when the local operations are part of

a standard acyclic circuit, the box of the process can be further decomposed into a ﬁner-grained circuit

fragment such that the loop is explicitly broken. But when the process is causally nonseparable, as is

the case here, the circuit cannot be decomposed into a ﬁner-grained acyclic circuit or a probabilistic

mixture of acyclic circuits.

analogue in the case where all systems are classical. In those classical cases, however, it is possible

to think that the logical observable on Q has a deﬁnite value and hence the ‘true’ input system of

Alice’s operation is not a ﬁxed one but either S or B

, depending on the control bit (formally

speaking, this is equivalent to associating A

with a tensor factor of two diﬀerent subspaces of

QSB

), and similarly the output system is either B

or S

. However, when Q is initialized in a

quantum superposition and Charlie is allowed to perform arbitrary operations on Q

, such an

interpretation is not possible anymore. In general, with respect to the input and output systems

and A

that we have identiﬁed, the experiment has the cyclic circuit structure depicted in Fig.

8, where the channel connected to the operations of the four parties cannot be decomposed into a

mixture of channels where the cycle might break. (Note that for simplicity we have deﬁned A

as directly connected to QSB

without any temporal separation, and A

as directly connected to

, i.e., the central box in Fig. 8 is not strictly ‘bulky’ from a temporal perspective, despite

what is suggested by the graphical representation. However, we could also have realizations where

the central box is ‘bulky’ if we insert identity channels with nonzero time spans at the places of

the systems Q, S, B

, B

, Q

, S

in Fig. 5.) As ﬁrst pointed out by Chiribella et al. [3], such a

cyclic structure is a necessary property of the quantum SWITCH as a higher-order transformation

on the operations of Alice and Bob [51]. By identifying precisely these operations, our analysis

makes the cyclic structure explicit, vindicating the interpretation of the described implementation

as a realization of the quantum SWITCH. (For more on this point, see Sec. 4.)

Let us now comment on the input and output systems of Alice and Bob in the case of a sym-

metric implementation, such as the one in the experiments [33, 35, 36]. Again, identifying the

control and target systems, we may draw the overall circuit as in Fig. 9, where the controlled

unitaries are such that either Alice’s or Bob’s operation takes place at a given time, depending

on the logical value of the control qubit (the controlled unitaries are displayed with a slight shift

rather than simultaneous for graphical clarity). Following analogous analysis to the one presented

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Charlie

Time

S’

Q’

David

Figure 9: Temporal description of a symmetric implementations of the quantum SWITCH.

The experimental implementations of the quantum SWITCH in Refs. [33, 35, 36] are symmetric with

respect to the possible times of the operations of Alice and Bob. In this case, the temporal description

of the experiment in terms of the control qubit and the target system has the form displayed here.

earlier, one sees that Alice’s input and output systems are given by

≡ |0ih0|

⊗ O

⊗ 11

+ |1ih1|

⊗ 11

⊗ O

, (17)

≡ |0ih0|

⊗ O

⊗ 11

+ |1ih1|

⊗ 11

⊗ O

, (18)

and Bob’s input and output systems are given by

≡ |0ih0|

⊗ 11

⊗ O

+ |1ih1|

⊗ O

⊗ 11

, (19)

≡ |0ih0|

⊗ 11

⊗ O

+ |1ih1|

⊗ O

⊗ 11

. (20)

4 On the notion of ‘realization’, and a comment on resources

It is instructive to discuss in more detail the sense in which we regard the previous results as jus-

tifying the interpretation of those experiments as ‘realizations’ of the higher-order process. The

perspective taken in this paper is that a higher-order process, such as the quantum SWITCH, is a

mathematical concept deﬁned in terms of CP maps over Hilbert spaces. To say that such a math-

ematical concept has a physical realization means that there can be an experiment in which every

element of the description of the concept has a physical counterpart that itself can be considered

a realization of that element. What we have shown is that, in the discussed types of experiments,

every element of the description of the process – Hilbert spaces, and operations on them – has a

physical counterpart. What may still be debated is whether we are willing to regard that counter-

part as a ‘realization’ of the element.

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When it comes to experiments of the kind discussed in the previous section, their interpreta-

tion as realizations of standard circuits composed of time-localized operations is well established.

However, if one proposes an alternative interpretation of the same experiment based on a new

notion of realization of a CP map, it is natural to question whether we are willing to accept this

new notion. But in this case, the proposed notion is not really new. A time-delocalized CP map is

a trivial combination of two already accepted notions: (1) quantum combs as CP maps on Hilbert

spaces that are tensor products of Hilbert spaces associated with deﬁnite times, and (2) subsystems

as realizations of Hilbert spaces. We therefore regard the proposed interpretation in full agreement

with and equally sound as the standard one.

Of course, one may be interested in realizations of quantum operations that obey further con-

ditions. For instance, one may ask that a quantum operation is realized by the action of a spatially

local physical box on the internal degrees of freedom of a particle passing through it. However,

from our perspective this is just a particular kind of realization, and there is no reason to assume

that all realizations of a quantum operation must be of this kind. For example, operations acting

on the Hilbert spaces of ﬁelds, which include the vacuum state, are just as meaningful. The main

goal of the present work is to point out that, in agreement with the established theory, we can have

more general types of systems and operations than those usually considered, which oﬀers a more

general way in which we can contemplate realizations of processes.

A common objection to the assertion that an implementation of the kind described in the pre-

vious section can be regarded as a realization of the quantum SWITCH concerns the fact that the

temporal circuit description of the experiment involves multiple controlled versions of the input

operations (or, more generally, operations on a larger Hilbert space that includes the vacuum).

Indeed, one of the celebrated diﬀerences between the quantum SWITCH and higher-order pro-

cesses in which the input operations are compatible with deﬁnite causal order is that the quantum

SWITCH produces a speciﬁc operation that is a function of U

and U

with a single use of each

and U

, whereas processes with a deﬁnite causal order require at least two uses of U

to produce the same operation. An example of an implementation that uses two copies of

one input operation and a single copy of the other one was described in Eq. (18) of [3]. Such

implementations are commonly understood as ‘simulations’ of the quantum SWITCH, as opposed

to realizations. Although we have seen that with respect to the relevant systems the implemen-

tations in the previous section are exactly equivalent to the quantum SWITCH applied on single

copies of U

and U

, one may still be concerned by the fact that the temporal circuit contains two

controlled-U

operations, where a single use of controlled-U

is a more powerful resource than

a single use of U

(because we can obtain the latter from the former by preparing a suitable state

on the control system). Are we not using too powerful resources for this implementation to be

considered a realization?

To understand this issue, it is crucial to clarify the context in which implementations with

multiple copies are regarded as simulations [3]. The context is that we are given access to time-

localized versions of two unknown unitaries, f and g (we use the notation corresponding to Fig.

(18) of [3]), and we want to produce the unitary U

SWITCH

(f, g). Using standard circuits, we

would need to call at least one of the unknown unitaries, say g, twice. However, if we could apply

the higher-order operation quantum SWITCH on such unitaries, we would achieve the result with

a single use of each f and g. Note that since by assumption these are time-localized versions

of f and g, the ability to apply a quantum SWITCH on them would imply the ability to send

information back in time (this can be inferred from the steps in the proof of Proposition 1 in Ref.

[3]). The sense in which an implementation such as the one in Fig. (18) of [3] can be regarded

as a simulation is that it simulates the eﬀect of having access to a quantum SWITCH of this kind

plus a single use of each f and g, for the purpose of producing U

SWITCH

(f, g).

In the present paper, we are not concerned with the task of producing U

SWITCH

(f, g) by us-

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ing time-localized versions of f and g. We are asking whether the higher-order transformation

quantum SWITCH can be said to have a physical realization. We have seen that the answer is pos-

itive, but the ‘slots’ in which the input operations must be ‘plugged’ are given by time-delocalized

subsystems. Such a quantum SWITCH cannot be applied on time-localized operations (and in par-

ticular does not allow signaling back in time). It is nevertheless a realization of the higher-order

transformation in the sense we have described. Furthermore, it can be regarded as a resource in

comparison to causally separable processes in an appropriately redeﬁned context. For example,

instead of being given access to time-localized versions of the input unitaries f and g, we could be

given access to these operations applied on time-delocalized subsystems (or systems of our choice,

as long as this is physically permissible). In such a scenario, the process connecting the opera-

tions can still be causally separable. Indeed, one can see that we can create any causally separable

process between Alice, Bob, Charlie, and David, when Alice’s operation is time-delocalized as in

the scheme in Fig. 5, by inserting in that circuit suitable operations after David, before and after

Bob, and before Charlie. (Essentially, this is because we can ﬁx the order between Alice and Bob

by ﬁxing the state of the control qubit right before the quantum 1-comb that contains the opera-

tion of Alice.) Therefore, the comparison between the quantum SWITCH and causally separable

processes is meaningful in this scenario, and the proven advantages hold.

It is clear that a time-delocalized version of a given operation is a diﬀerent physical resource

than a time-localized version. One could ask how these resources relate to each other, e.g., what

resources of one kind are needed to obtain a resource of the other kind. (A rigorous formulation

of this question would require deﬁning the resources to which we have access for free, and the

way we can use all resources.) For instance, we have seen that having access to two uses of

time-localized controlled-g, where g is unknown, allows us to create one time-delocalized copy

of g of the kind on which a quantum SWITCH can be applied. Obviously, the former resource

is strictly more powerful than the latter since we cannot use a single time-delocalized versions of

g to obtain two copies of controlled-g that we could use in a circuit independently. In practice,

we could be given access to either of these two resources. There is a common tendency to think

that the gates that make up the temporal circuit in implementations of the quantum SWITCH

(e.g., Fig. 5) represent ‘the’ resource of interest. This is likely because these gates are associated

with single uses of physical devices (e.g., optical elements) that in the laboratory context could

naturally be applied independently. But the latter is not mandatory—we can imagine that we only

have access to time-delocalized versions of g rather than time-localized versions of controlled-g

if, for example, the operations of interest are provided by another party. Regarding the intuition

that operations are associated with single uses of physical devices, this idea is compatible with

time-delocalized operations too: consider a machine, which upon pressing of a button applies one

Hamiltonian pulse, followed, after some time, by another Hamiltonian pulse, so as to implement

the sequence of gates in Fig. 6 [53]. Such a device could itself be used multiple times in principle,

and one could ask how many times it has been used. In this concrete example, it is used exactly

once by construction.

Finally, it is instructive to look at what the time-delocalized subsystem perspective can tell us

about simulations of the quantum SWITCH (in the sense described earlier) via multiple copies

of some input operation, such as the one in Eq. (18) of Ref. [3]. For that example, it is easy

to see that there are suitable time-delocalized subsystems with respect to which the circuit takes

the form of a quantum SWITCH on the two input operations f and g, plus a second copy of g

applied in parallel on some additional system. One can also see that it is physically possible to

implement only the quantum SWITCH part without the residual copy of g, which from a temporal

perspective would look like applying a controlled-g twice, rather than g twice in the circuit in Eq.

(18) of Ref. [3], similarly to Fig. 6. In other words, the implementation with two copies of g

contains a realization of the quantum SWITCH on time-delocalized subsystems, plus a residual

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Time

Q’

2N-1

N-1

Figure 10: A time-delocalized operation realized through a sequence of controlled gates. Here,

a controlled unitary depicted with “i” in the circle on the control system represents a controlled unitary

that is ‘triggered’ if and only if the logical value of the control observable is i. We can permit arbitrary

gates Z

to act on the control system at any time (here shown in between the controlled gates for

clarity) as long as they commute with the control observable. We can also permit implementations in

which the gates that make up the circuit fragment are modiﬁed from the ones in this ﬁgure in a way

that preserves the transformation realized by the fragment. A fragment of this type can be seen to give

rise to a unitary that contains the factor U

→a

, where A

and A

are deﬁned in Eqs.(21) and

(22).

time-delocalized version of g. In particular, we see that two time-localized copies of g can be used

to produce a time-delocalized copy of g of the type required for the quantum SWITCH, plus a

residue.

5 General processes based on coherent control of the times of the oper-

ations

The result of Sec. 3 generalizes straightforwardly to multipartite process implementations where

the operation of each party may be delocalized over a number of possible times by means of

controlled unitaries conditioned on a logical observable (most generally deﬁned by a complete

set of orthogonal projectors {P

}

i=1

over some control system) that is preserved during a period

containing these times, as described in Fig. 10. [Note that we can permit gates Z

to act on

the control system during the period of the controlled gates, as shown in the ﬁgure, as long as

they commute with the logical observable. We can further modify the gates that make up the

circuit fragment in any way that leaves the transformation realized by the fragment invariant (e.g.,

appending local unitaries on the control system to the controlled operations, which are undone

by their inverses in the past or in the future).] The input system of such an operation would be a

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Figure 11: Unitary extendibility of a bipartite process. A bipartite process obeys the unitary

extension postulate proposed in Ref. [43] if and only if there exists a four-partite process of the kind

on the right-hand side of this ﬁgure that is a unitary channel, such that the original process is obtained

from this four-partite process conditionally on David preparing his output system in some pure state

|0ih0|, and Charlie tracing out his input system. The unitary extendibility of multipartite processes is

deﬁned analogously [43].

subsystem of QS

···S

2N−1

deﬁned by the algebra of operators

≡

i=1

⊗ O

2i−1

⊗ 11

2i−1

, (21)

where S

2i−1

denotes the complementary subsystem of S

2i−1

in S

···S

2N−1

, and the output

system would be a subsystem of Q

···S

deﬁned by the algebra of operators

≡

i=1

⊗ O

⊗ 11

, (22)

where S

denotes the complementary subsystem of S

in S

···S

In Refs. [10, 11] it was shown that a class of multipartite processes containing operations

realized through this procedure cannot violate causal inequalities. It seems likely that no process

based on such operations would be able to violate causal inequalities, for reasons similar to those

used in the arguments in Refs. [10, 11]. This conjecture is left open for future investigation.

6 Unitarily extendible bipartite processes

In Ref. [43], Araújo et al. proposed a postulate that might distinguish processes that have a phys-

ical realization from those that do not, showing that certain processes violating causal inequalities

fail to respect it. The postulate says that any physically admissible process should be possible to

obtain from an extended process involving two additional parties, one having only a nontrivial out-

put system and the other one only a nontrivial input system, in a way analogous to the one in Fig.

11 for the case of two parties, where the extended process is equivalent to a unitary channel from

to A

. (Note that it is crucial in this deﬁnition that the unitary channel U is not

just any unitary dilation of the channel

W, but is itself a valid process, i.e., the transpose of its CJ

matrix is a valid process matrix.) As we have seen in the previous section, the quantum SWITCH

is a four-partite process of the kind on the right-hand side of Fig. 11, and hence it provides a

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Alice Bob

Charlie

David

Figure 12: A class of unitary processes that have a realization on time-delocalized subsystems.

We consider the class of four-partite processes of the kind depicted in this ﬁgure, which are equivalent to

unitary channels from D

to A

. We show that they admit realizations on time-delocalized

subsystems (see Theorem 1).

unitary extension for a class of bipartite processes (obtained for diﬀerent choices of the state on

). These bipartite processes, however, are causally separable [10, 11], even though the quan-

tum SWITCH is not. At the time of the initial submission of this paper, it was not know whether

there are causally nonseparable bipartite processes that obey the above postulate [54]. But by the

same argument sketched earlier for the case of the quantum SWITCH, the unitary extension of any

bipartite process would in general be causally nonseparable except in the extreme case where the

bipartite process is compatible with a ﬁxed causal order (i.e., it is nonsignaling from Alice to Bob

or from Bob to Alice). Here, we show that all bipartite processes that obey the postulate, together

with their unitary extensions, have a realization on suitable time-delocalized quantum subsystems.

Theorem 1 (Realizability of unitary extensions of bipartite processes). Consider a unitary

four-partite process of the kind depicted in Fig. 12. In agreement with the known quantum me-

chanics, there exist time-local systems D

, B

, C

, ordered in time in this order, a subsystem

of D

, and a subsystem A

of B

, such that the circuit-like process depicted on the

left-hand side of Fig. 13, for arbitrary operations of David, Bob, and Charlie (left unspeciﬁed in

the ﬁgure for simplicity), and an arbitrary unitary operation U

→a

of Alice (which can

be used to obtain arbitrary operations in Alice’s slot), where a

and a

are ancillary input and

output systems, admits an experimental implementation on these systems without post-selection.

The implementation has a temporal description as on the right-hand side of Fig. 13, where the

unitaries U

and U

are functions of U

Remark on graphical notation. In Fig. 13, as well as in all of the following ﬁgures in this and

the next section, the boxes with (red) solid boundaries and the solid wires connecting them denote

the objects about which we are stating equivalences. The dashed fragments inside the solid boxes

describe compositions of operations that can yield the respective total transformations in the solid

boxes.

Proof. In agreement with the known quantum mechanics, one can ﬁnd time-local systems

, B

, C

, ordered in time as on the right-hand side of Fig. 13, such that it is possible

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)

Time

Bob

David

Charlie

Bob

Charlie

≡

David

Figure 13: Equivalence of speciﬁc circuits and unitary higher-order processes. According to

Theorem 1, there is a subsystem A

of the time-local systems D

depicted on the right-hand

side, and a subsystem A

of the time-local systems B

depicted on the right-hand side, such that

for appropriately chosen unitary channels U

) and U

), the circuit on the right-hand side is

equivalent to the process on the left-hand side (both understood veriﬁable box-wise).

to implement arbitrary quantum operations between these systems (the time intervals between the

systems may need to be suﬃciently large so as to allow universal transformations to be realized in

agreement with the Hamiltonian that governs the fundamental ﬁelds). This ensures the realizability

of the operations of David, Bob, and Charlie, as depicted on the right-hand side of Fig. 13, as well

as of arbitrary operations between them. Therefore, the theorem would follow if we can show the

existence of a subsystem A

of D

and a subsystem A

of B

, and unitary transformations

) and U

) dependent on U

, such that the equivalence in Fig. 14 holds for every U

Note that if the equivalence in Fig. 14 holds, then it must in particular hold that the resultant

transformation from a

to a

on the left-hand side is equal to the corresponding

resultant transformation on the right-hand side, as depicted in Fig. 15. But for any process of

the type considered here (not necessarily unitary), and for any choice of channel for Alice (not

necessarily unitary), the transformation resulting from the composition of Alice’s channel with the

process is equivalent to a supermap [4] on Bob’s operation. This can be easily seen from the fact

that a process, by deﬁnition, is normalized on all deterministic operations performed by the parties.

As shown in Ref. [5], any deterministic supermap on a single operation is equivalent to a quantum

1-comb and can be realized by a circuit fragment composed of isometries, where some subsystem

is discarded at the end. This means that in general we have the equivalence in Fig. 16 for suitable

channels W

) and W

) that are functions of M

. In the case when both

W and M

are unitary (and hence their composition is unitary [43]), these channels can themselves be taken

unitary. Indeed, as shown in Ref. [5], the construction of the circuit fragment that implements

a quantum comb can be taken such that it achieves the minimal Stinespring dilation [55] for the

full comb, which in the case of a unitary comb yields a circuit fragment composed of unitary

channels as in Fig. 15. Moreover, these unitary channels are unique up to a joint transformation

that is formally equivalent to a change of basis for the system E. (For how to construct such a

circuit fragment, see Ref. [5].) Therefore, if there exists a circuit fragment as on the right-hand

side of Fig. 14 such that the equivalence in Fig. 14 holds, that circuit fragment must belong

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≡

)

Figure 14: The main equivalence behind Theorem 1. The proof of Theorem 1 is based on proving

the equivalence between the process fragment on the right-hand side and the quantum 1-comb on the

left-hand side, for suitable subsystems A

and A

of the boundary systems on the left-hand side.

)

≡

Figure 15: The composition of Alice’s unitary channel with the unitary process yields a unitary

quantum 1-comb. For any choice of extended unitary channel on Alice’s side, the process reduces to

a supermap from Bob’s operation to a new operation, which can be realized physically by a unitary

circuit fragment connected to Bob’s operation [4]. This is a special case of Fig. 16.

≡

)

Figure 16: The composition of Alice’s channel with a process of the considered kind yields a

quantum 1-comb. For any choice of operation on Alice’s side, a process of the type considered here

reduces to a supermap from Bob’s operation to a new operation, which can be realized physically by a

circuit fragment connected to Bob’s operation [4].

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≡

Figure 17: Alternative realization of the unitary channel of the quantum comb in Fig.15.

Deforming the left-hand side of Fig. 15 by ‘pulling’ up and down the ‘wires’ corresponding to Bob’s

input and output systems is formally equivalent to plugging the SWAP operation in Bob’s slot, which

must yield the result of a supermap applied on Alice’s operation, implying the alternative realization

inside the red box on the right-hand side of this ﬁgure.

to the equivalence class arising from the construction in Ref. [5]. We will show that there exist

subsystems A

of D

and A

of B

, such that for any unitary fragment obtained through

this construction, the equivalence in Fig. 14 holds, i.e., any such fragment provides a realization

of the process fragment on the left-hand side of Fig. 14.

The proof that such subsystems exist uses crucially the fact that the channel of a process does

not allow signaling from the output of one party to the input of the same party, which means

that the unitary channel U

→A

maps A

isomorphically [56] to some subsystem

of B

, and similarly maps some subsystem

of D

isomorphically to A

. To formally

identify these subsystems, it is convenient to deform the fragment contained in the red box on

the left-hand side of Fig. 15 by ‘pulling’ the input and output ‘wires’ of Bob up and down,

respectively, so as to put the red box in a more standard input-to-output layout, as on the left-hand

side of Fig. 17. The overall transformation in the red box does not change under this graphical

deformation. But we may also interpret the deformation as describing the plugging of a SWAP

transformation into Bob’s slot in the process fragment inside the red box, which by the same

argument as above must yield a supermap on Alice’s operation that has a realization as a unitary

circuit fragment. In other words, we have the equivalence displayed in Fig. 17 for speciﬁc unitary

channels U

and U

, which depend on the process U, and which, similarly to above, are uniquely

deﬁned up to a joint transformation formally equivalent to a change of basis on the intermediary

system X (see below). Looking at the circuit inside the red box on the right-hand side of Fig. 17,

we see that there must be a subsystem

of D

which is mapped isomorphically to A

by the

channel U

. It is deﬁned by the algebra of operators of the form

≡ U

†

⊗ 11

, (23)

where U

is the unitary associated with the channel U

at the Hilbert-space level. Similarly, the

system A

is mapped isomorphically to a subsystem

of B

, which is deﬁned by the algebra

of operators of the form

≡ U

⊗ 11

†

, (24)

where U

is the unitary associated with the channel U

at the Hilbert-space level. As noted already,

the unitaries U

and U

are not unique: the quantum comb that they create when composed is

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≡

Figure 18: Equivalent representation of the unitary channel in Fig. 17. Using that the channel

in Fig. 17 maps isomorphically some subsystem

of D

= H

⊗ H

) onto A

and that the channel U

in Fig. 17 maps isomorphically the subsystem A

onto a subsystem

= H

⊗ H

), we conclude the equivalent representation depicted here, where each

of the channels U

and U

on the right-hand side describes the link between two diﬀerent factorizations

of the same Hilbert space.

invariant under transformations of the form U

→ U

, U

→ U

)

†

, where U

is an

arbitrary unitary on X. However, the operators O

and O

deﬁned above are also invariant

under such transformations and hence can be obtained starting from any valid choice of U

and

, i.e., the subsystems

and

are uniquely deﬁned.

Let

denote the co-subsystem of

in D

= H

⊗H

), and let

denote

the co-subsystem of

in B

= H

⊗ H

). It is obvious from Fig. 17 and the

deﬁnition of

and

that the circuit fragment inside the box on the right-hand side transforms

to X via a unitary, and then X to

via another unitary, i.e., it maps unitarily

. We

can always choose an isomorphism between the initial and ﬁnal subsystems (i.e., identify a basis

for one with a basis for the other) such that the unitary connecting them is the identity. The unitary

of the four-partite process matrix can then be written

→A

= 11

→A

⊗ 11

→

⊗ 11

→

. (25)

This means that the unitary inside the box on the right-hand side of Fig. 17 has the form

→a

) = U

→a

⊗ 11

→

. (26)

We can therefore rewrite the transformation on the right-hand side of Fig. 17 as on the right-

hand side of Fig. 18, where the unitary channel U

from D

is nothing but a trivial

transformation (the identity) on the same system, which describes the relation between one sub-

system decomposition of that system and another, and similarly for U

. Such channels are always

present at the respective systems on the boundary of the box, which is why the circuit on the

right-hand side of Fig. 18 can be claimed to be equivalent to the single transformation on the left-

hand side. (The boxes containing U

and U

in this construction are not ‘bulky’ from a temporal

perspective as each has a trivial ‘time span’ from input to output, but this is merely due to the sim-

plifying choices we have made. They could be ‘bulky’ if instead of a single transformation on the

left-hand side we consider the same transformation composed with identity channels of nontrivial

time spans along the boundary systems. One may think that identity channels are also present

along the boundary systems on the left-hand side of Fig. 18, but they have been suppressed due

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≡

Figure 19: Extracting Alice’s operation. Using that the middle box on the right-hand side of Fig. 18

equals the tensor product of two operations, we may split it into two parallel boxes as on the left-hand

here. Since the equivalence in Fig. 18 must hold for all choices of Alice’s unitary as and all operations

that could be applied in the slots of the rest of the parties, whose correlations completely characterize

the process connecting the slots of all parties, we conclude the equivalence displayed here.

to being of zero time span and hence entirely trivial.) Using that the middle box on the right-hand

side of Fig. 18 can be further decomposed into two parallel boxes because it contains a tensor

product of two operations, U

→

⊗ I

→

, and the fact that the equivalence in Fig. 18 must

hold for every choice of Alice’s unitary and all operations that might be applied by David, Bob,

and Charlie (whose correlations fully characterize the process matrix connecting the slots in which

these operations are plugged), we conclude the equivalence in Fig. 19. From Figs. 15, 18, and 19,

it then follows that the the left-hand side of Fig. 19 is equivalent to the right-hand side of Fig. 15,

i.e., the equivalence in Fig. 14 holds for A

≡

and A

≡

as deﬁned above. Therefore, if

we deﬁne the system A

to be the subsystem

and A

to be the subsystem

in the circuit on

the right-hand side of Fig. 13, we obtain the equivalence in Fig. 13. This equivalence is testable

by tomography in a way analogous to that explained in Fig. 7. This completes the proof.

7 A generalization

Consider the class of four-partite processes of the kind depicted in Fig. 20, where the process

is equivalent to an isometric channel V for which the isometry at the Hilbert-space level has the

following property. With respect to at least one of the parties Alice and Bob, say Alice, the

isometry can be written in the form

→A

= 11

→A

⊗ V

→B

, (27)

where H

= H

⊗ H

, and V

→B

is an isometry. Note that V

→B

maps

⊗ H

isomorphically to a subspace of H

, and in particular each of the subsystems

and H

to tensor factors of that subspace. Calling these tensor factors H

and H

respectively, we can equivalently write condition (27) as

→A

= 11

→A

⊗ 11

→

, (28)

where H

⊗ H

is a subspace of H

, i.e.,

= (H

⊗ H

) ⊕ H

. (29)

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Alice Bob

Charlie

David

Figure 20: A class of isometric processes that have a realization on time-delocalized subsys-

tems. We consider the class of four-partite processes of the kind depicted in this ﬁgure, which are

equivalent to isometric channels that map a subsystem of D

isomorphically to A

, or a subsystem

of D

isomorphically to B

. We show that they admit realizations on time-delocalized subsystems

(see Theorem 2).

Note that there exists an isometric extension of any process, and in particular any bipartite pro-

cess, which can be obtained by purifying the channel of the process onto Charlie’s input (in this

case David’s output system is trivial), but this puriﬁcation in general would not have the form (27).

Theorem 2 (Realizability of a class of isometric extensions of bipartite processes). Con-

sider an isometric four-partite process of the kind depicted in Fig. 21, where the isometry satisﬁes

Eq. (27) (or, equivalently, Eqs. (28) and (29)). In agreement with the known quantum mechanics,

there exist time-local systems D

, B

, C

, ordered in time in this order, a subsystem (tensor

factor) A

of D

, and a subsystem (tensor factor of a subspace) A

of B

, such that the

circuit-like process depicted on the left-hand side of Fig. 21, for arbitrary operations of David,

Bob, and Charlie (left unspeciﬁed in the ﬁgure for simplicity), and an arbitrary unitary operation

→a

of Alice (which allows obtaining arbitrary operations in Alice’s slot), where a

and

are ancillary input and output systems, admits an experimental implementation on these sys-

tems without post-selection. The implementation has a temporal description as on the right-hand

side of Fig. 21, where the isometries V

and V

are functions of U

The proof is similar to the one for the unitary case, so we will not repeat all steps but will

only highlight the diﬀerences. Here, the channels U

) and U

) from the unitary case have

been replaced by isometric channels V

) and V

), respectively, which is because the com-

position of Alice’s unitary with the isometric process yields a quantum 1-comb that is isometric.

(A simple way of seeing this is to note that since the CJ operators of all operations involved are

rank-one, so is the operator of the supermap resulting from connecting these operations, and since

the latter is equivalent to a quantum comb, which is a channel, the channel can only be isometric.)

As before, such isometric channels that realize the minimal Stinespring dilation of the quantum

comb (without the need for any auxiliary systems traced out at the end) indeed exist as shown in

Ref. [5], and they can be seen to be unique up to a joint transformation that is formally equivalent

to a change of basis of the intermediary system E.

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)

Time

Bob

David

Charlie

Bob

Charlie

≡

David

Figure 21: Equivalence of speciﬁc circuits and isometric higher-order processes. According to

Theorem 2, there is a subsystem (tensor factor) A

of the time-local systems D

depicted on the

right-hand side, and a subsystem (tensor factor of a subspace) A

of the time-local systems B

depicted on the right-hand side, such that for appropriately chosen isometric channels V

) and

), the circuit on the right-hand side is equivalent to the process on the left-hand side (both

understood veriﬁable box-wise).

From Eq. (28), one can directly see that the analogue of Fig. 17 is now Fig. 22, where U

is the unitary channel describing the relation between the two diﬀerent factorizations of the input

Hilbert space, and the isometry V

describes the embedding of the subspace H

of H

into H

. Again, these transformations are always present at the respective boundary systems,

which is why the single transformation on the left-hand side of Fig. 22 can be claimed to have

the nontrivial compositional structure depicted on the right-hand side. (As before, we remark that

here we have taken these boundary transformations to be ‘non-bulky’ from a temporal perspective

for simplicity, but they could also be ‘bulky’ as explained previously.)

Finally, we note that the isometry of the 1-comb on the right-hand side of Fig. 21, which by

construction equals the transformation on the left-hand side of Fig. 22, has a form analogous to

Eq. (26),

→a

) = U

→a

⊗ 11

→

, (30)

but with

and

being subsystems in the sense of Eq. (29). As before, we deﬁne the input and

output systems of Alice to be the subsystems

and

within the circuit on the right-hand side

of Fig. 21. The operation of Alice can be veriﬁed experimentally by tomographic operations on

the respective subsystems, similarly to all elements of the process. A minor subtlety in comparison

to the previous case is the notion of measurement in a tensor factor of a subspace (the subsystem

in Eq. (29)), which can be naturally deﬁned as performing a measurement in the respective

factor conditionally on ﬁrst projecting the full system into the respective subspace (which in this

case would succeed with unit probability since the 1-comb on the right-hand side of Fig. 21 always

outputs a state in that subspace).

How big is the class of processes deﬁned here? The four-partite processes are obviously

strictly more general than the unitary class discussed in the previous section because they do not

have to be unitary and the unitary case is included. However, without additional considerations,

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≡

Figure 22: The analogue of Fig. 18. From Eq. (28), one can see that the analogue of Fig. 17 is now

Fig. 22. Here, U

is the unitary channel describing the relation between the two diﬀerent factorizations

of the input Hilbert space, and the isometry V

describes the embedding of the subspace H

into H

it is in principle conceivable that all processes in this class admit unitary extensions and hence

they do not describe anything that cannot be achieved with the unitary class. If the latter is not

the case, our demonstration that these processes have a realization on suitable time-delocalized

subsystems would be a counterexample to the postulate conjectured in Ref. [43]. An even more

radical possibility is that any bipartite process has an extension to a process of this kind, which

would imply the realizability of all bipartite processes on suitable time-delocalized subsystems.

These questions are left for future investigation.

8 Discussion

We have shown that a class of causally nonseparable quantum processes has a physical realiza-

tion within standard quantum mechanics in terms of operations whose input and output systems

are time-delocalized subsystems—a concept that is both mathematically well deﬁned and directly

experimentally testable. This result puts on solid grounds the interpretation of recent experimen-

tal demonstrations of the quantum SWITCH [33, 35, 36, 37] as realizations of this higher-order

transformation. A natural addition to this line of demonstrations suggested by the present work is

to probe directly the time-delocalized operations of Alice and Bob, e.g., through tomography, as

sketched in Fig. 7.

The concept of time-delocalized quantum subsystems and operations introduced here substan-

tially expands the landscape of information-processing structures known to be available within

quantum mechanics [5], which opens up a new realm of practical possibilities. The irreducible

cyclic circuits unveiled in this work are just a glimpse of this landscape, the full scope of which

has to be explored through the hierarchy of higher-order supermaps [47].

Apart from understanding the set of physically realizable supermaps, a big open problem is

to understand their information processing power. Since those of them incompatible with deﬁnite

causal order are still part of standard quantum mechanics, the advantages that they can oﬀer over

standard causal processes depend on the precise deﬁnition of the resources used and their costs.

The fact that causally nonseparable processes are more powerful than causally separable ones has

been shown by examples of speciﬁc computation and communication tasks for which the quantum

SWITCH oﬀers advantage [26, 28, 29, 30, 31]. Identifying natural applications for which causal

nonseparability would be useful is an important direction for research.

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A speciﬁc problem, left unanswered by this work, is whether quantum processes violating

causal inequalities have a physical realization. From the circuit perspective emphasized in this

paper, such processes are cyclic circuits that give rise to correlations that cannot be simulated by

probabilistic mixtures of acyclic circuits or dynamical variants thereof [10], even if the systems

and operations involved are described by theories more general than quantum theory. In analogy

with device-independent applications based on Bell-inequality violations [57], the existence of

such processes could open the road to a new kind of device-independent information-processing

applications. The time-delocalized subsystem concept provides a concrete framework in which

the question of whether this resource is available in quantum mechanics can be systematically

investigated. The present proof that all unitary extensions of bipartite processes, as well as a class

of isometric extensions, have a realization on time-delocalized subsystems is a promising step in

this direction. As pointed out in Ref. [43], there are tripartite processes, such as a deterministic

classical process discovered in Ref. [15], which violate causal inequalities and admit a unitary

extension. An obvious question is whether and how the proof presented here could be generalized

to the case of extensions of multipartite processes [58].

One of the results of our analysis of the quantum SWITCH and its implementations is that

the operations of Alice and/or Bob are not operations on the target system alone, as sometimes

assumed on intuitive grounds, but they spread onto the control qubit as well. While this is con-

sistent with the deﬁnition of the quantum SWITCH as a supermap, it may have implications for

arguments that treat these operations as operations on the target system [60].

Finally, a natural question is whether time-delocalized operations such as those identiﬁed here

could be seen in a way analogous to standard operations, with the input system being in the past of

the output system, with respect to a suitable notion of time, e.g., based on quantum time reference

frames. Recent arguments about background-independence in a probabilistic setting [22] as well

as gedanken proposals for realizations of the quantum SWITCH at the interface of quantum me-

chanics and gravity [61] or with Rindler observers [62] oﬀer a promising route for investigating

this subject, as well as the implications that the subsystem picture uncovered in this work may

have for quantum gravity.

Acknowledgements

O. O. thanks J. Barrett,

C. Brukner, G. Chiribella, F. Costa, and P. Perinotti for related discus-

sions. The author is grateful to the anonymous reviewers whose comments and questions helped

improve the manuscript. Their contributions include spotting a few errors in a previous version of

the manuscript, raising the question about the link between time-delocalized subsystems and sim-

ulations of the quantum SWITCH with multiple uses of the input operations, and pointing to the

need for a better explanation of the argument in Section 6, as well as the assumed meaning of ‘re-

alization’. This work was supported by the Wiener-Anspach Foundation, the UK NQIT Quantum

Hub, and the Program of Concerted Research Actions (ARC) of the Université libre de Bruxelles.

O. O. is a Research Associate of the Fonds de la Recherche Scientiﬁque (F.R.S.–FNRS).

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transmission of information back in time. In the realization discussed here, this condition is

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[52] More precisely, C-SWAP

XY Z

= |0ih0|

⊗11

Y Z

+ |1ih1|

⊗SWAP

Y Z

, where SWAP

Y Z

is the SWAP operator on Y and Z, which can be deﬁned as follows. Consider two systems Y

and Z with Hilbert spaces of the same dimension and a linear isomorphism between the states

in these Hilbert spaces. An arbitrary vector in the joint system Y Z can be written in the form

|ψi

Y Z

i,j

|ii

|ji

, where {|ii

} are orthonormal bases for Y and Z, respectively.

The action of the operator SWAP

Y Z

on the vector |ψi

Y Z

is then given by SWAP

Y Z

|ψi

Y Z

i,j

|ji

|ii

Accepted in Quantum 2019-11-08, click title to verify. Published under CC-BY 4.0. 32

[53] Of course, if during the working of the device, an adversary turns on unwanted interactions,

such as a Hamiltonian on the control qubit that is not diagonal in the logical basis, this could

prevent the device from implementing the correct operation on the systems of interest. But

this is the case for any physical device implementing an operation, irrespectively of whether

the operation is localized or delocalized in time.

[54] Very recently, after the submission of this paper, the author and colleagues J. Barrett and

R. Lorenz showed via diﬀerent methods that all bipartite processes that are unitarily ex-

tendible are causally separable, and hence their unitary extensions are variations of the quan-

tum SWITCH (in preparation). Nevertheless, we believe that the proof of realizability pre-

sented here has a particular value since it is based on a diﬀerent idea that could have wider

applications. In particular, it provides the basis for the generalization in Sec. 7, and might be

useful in the search for realizations of more complicated unitary processes.

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admit a representation on time-delocalized subsystems. However, this claim is based on a

misunderstanding of the concept of time-delocalized subsystems. The proof claimed in [59]

amounts to the observation (discussed in this paper) that if we have a unitary process, the

unitary maps isomorphically the output system of any one party, say Alice, onto a subsystem

of the input systems of the rest of the parties, and similarly maps a subsystem of the output

systems of the rest of the parties onto the input system of Alice. This by itself does not imply

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c, M. Milivojevi

c, D. Go

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Caslav Brukner, Simulating spacetime with

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(2017).

Accepted in Quantum 2019-11-08, click title to verify. Published under CC-BY 4.0. 33

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