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 oer 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 dierent
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
Accepted in Quantum 2019-11-08, click title to verify. Published under CC-BY 4.0. 1
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 dierent 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 eect 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 dierent 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 oer 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 dierent 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 dierent times. The fact that we
can think, both mathematically and operationally, of Hilbert spaces that are tensor products of
Hilbert spaces at dierent 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 dierent 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
Accepted in Quantum 2019-11-08, click title to verify. Published under CC-BY 4.0. 3
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
I
with Hilbert space H
X
I
and an output quantum system
X
O
with Hilbert space H
X
O
, where an agent can perform an arbitrary quantum operation from
X
I
to X
O
. A quantum operation is most generally described by a collection of completely posi-
tive (CP) and trace-nonincreasing maps {M
X
I
X
O
i
X
}
i
X
O
X
, M
X
I
X
O
i
X
: L(H
X
I
) L(H
X
O
),
where L(H
X
) denotes the space of linear operators over the Hilbert space H
X
with dimension d
X
(here we assume finite dimensions), and i
X
O
X
labels the possible outcomes of the operation
with which the dierent CP maps are associated. The sum of the CP maps corresponding to the
complete set of the outcomes of an operation,
M
X
I
X
O
=
P
i
X
O
X
M
X
I
X
O
i
X
, must be a CP
and trace-preserving (TP) map.
Consider the joint probabilities p(i
A
, j
B
, ···|{M
A
I
A
O
i
A
}
i
A
O
X
, {M
B
I
B
O
j
B
}
j
B
O
B
, ···) 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
A
, j
B
, ···|{M
A
I
A
O
i
A
}
i
A
O
A
, {N
B
I
B
O
j
B
}
j
B
O
B
, ···)
= Tr
h
W
A
I
A
O
B
I
B
O
···
M
A
I
A
O
i
A
N
B
I
B
O
j
B
···
i
. (1)
Here, M
A
I
A
O
i
A
L(H
A
I
H
A
O
) 0 is the Choi-Jamiołkowski (CJ) operator [44, 45] of the
CP map M
A
I
A
O
i
A
, and similarly for N
B
I
B
O
i
B
, etc. [Here, the CJ operator M
A
I
A
O
L(H
A
I
H
A
O
) of a linear map M
A
I
A
O
: L(H
A
I
) L(H
A
O
) is defined as M
A
I
A
O
:=
I
A
I
A
I
M
A
0
I
A
O
|φ
+
ihφ
+
|
A
I
A
0
I

T
, where |φ
+
i
A
I
A
0
I
=
P
d
A
I
j=1
|ji
A
I
|ji
A
0
I
H
A
I
H
A
0
I
is a (not
normalized) maximally entangled state on two copies of H
A
I
(one of them denoted by H
A
0
I
), the
states
n
|ji
A
I
o
d
A
I
j=1
form an orthonormal basis of H
A
I
, and
n
|ji
A
0
I
o
d
A
I
j=1
are their copies in H
A
0
I
,
I
A
I
A
I
is the identity superoperator on L(H
A
I
), and T denotes matrix transposition in the above
basis of A
I
and some basis of A
O
(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
A
I
A
O
B
I
B
O
··· 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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W
Alice Bob
A
O
A
I
B
I
B
O
{M
i
} {N
j
} =
p(i, j|{M
i
}, {N
j
}, 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
f
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):
W
A
I
A
O
B
I
B
O
···
0, (2)
Tr
h
W
A
I
A
O
B
I
B
O
···
M
A
I
A
O
N
B
I
B
O
···
i
= 1,
M
A
I
A
O
, N
B
I
B
O
··· 0,
Tr
A
O
M
A
I
A
O
= 11
A
I
, Tr
B
O
N
B
I
B
O
= 11
B
I
, ··· . (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
A
I
A
O
B
I
B
O
C
I
C
O
···
can be written in the form
W
A
I
A
O
B
I
B
O
C
I
C
O
···
=
X
i,j,k,l,m,n···
w
ijklmn···
σ
A
I
i
σ
A
O
j
σ
B
I
k
σ
B
O
l
σ
C
I
m
σ
C
O
n
··· ,
w
ijklmn···
R, i, j, k, l, m, n, ··· , (4)
where the Hermitian operators {σ
X
µ
}
d
2
X
1
µ=0
, with σ
X
0
= 11
X
, Trσ
X
µ
σ
X
ν
= d
X
δ
µν
, and Trσ
X
j
= 0 for
j = 1, ..., d
2
X
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
A
I
A
O
B
I
B
O
C
I
C
O
···
which comes with
weight w
000000···
=
1
d
A
I
d
B
I
d
C
I
···
, it contains only nonzero terms in which there is a nontrivial σ
(dierent from the identity) operator on X
I
and a trivial one (the identity) on X
O
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
f
W
A
O
B
O
···→A
I
B
I
···
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
f
W
A
O
B
O
···→A
I
B
I
···
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
A
I
A
O
B
I
B
O
= W
A
I
A
O
B
I
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11
B
O
(the case where Bob is not in the causal past of Alice and hence cannot signal to Alice) or
W
A
I
A
O
B
I
B
O
= W
A
I
B
I
B
O
11
A
O
(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
W
A
I
A
O
B
I
B
O
cs
= qW
A
I
A
O
B
I
11
B
O
+ (1 q)W
A
I
B
I
B
O
11
A
O
,
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
I
A
O
i
A
}
i
A
O
A
and {N
B
I
B
O
j
B
}
j
B
O
B
, figuratively associated with
Alice and Bob, where d
A
I
= d
A
O
= d
B
I
= d
B
O
= d, and gives as an output an operation
{M
GG
0
k
}
kO
A
×O
B
, where H
G
= H
Q
H
S
and H
G
0
= H
Q
0
H
S
0
with d
Q
= d
Q
0
= 2 and
d
S
= d
S
0
= d. Its action can be described intuitively as follows. If we think of H
Q
and H
Q
0
as
the Hilbert spaces of a control qubit at some initial and some final time, respectively, and of H
S
and H
S
0
as the Hilbert spaces of some target system at the same two times, then the eect 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
i
A
}
i
A
O
A
, {N
j
B
}
j
B
O
B
, 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 eect of the quantum SWITCH on the
operations of Alice and Bob can be inferred from its eect in the case when the two operations
are extended unitary channels U
a
I
A
I
a
O
A
O
A
and U
b
I
B
I
b
O
B
O
B
, each acting on the original input
and output systems plus separate auxiliary input and output systems, as described in Fig. 2. Let
U
a
I
A
I
a
O
A
O
A
and U
b
I
B
I
b
O
B
O
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
U
a
I
b
I
Ga
O
b
O
G
0
SWITCH
(U
A
, U
B
) whose unitary matrix at the Hilbert-space level is
U
a
I
b
I
Ga
O
b
O
G
0
SWITCH
(U
A
, U
B
) U
a
I
b
I
QSa
O
b
O
Q
0
S
0
SWITCH
(U
A
, U
B
) =
|0i
Q
0
h0|
Q
U
b
I
Xb
O
S
0
B
U
a
I
Sa
O
X
A
+|1i
Q
0
h1|
Q
U
a
I
Xa
O
S
0
A
U
b
I
Sb
O
X
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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W
SWITCH
=
A
O
A
I
B
I
B
O
a
I
a
O
b
I
b
O
b
O
a
O
a
I
b
I
U
A
U
B
U
SWITCH
(U
A
, U
B
)
S’Q’
SQ
S’
Q’
~
Q
S
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|
Q
, the quantum
SWITCH eectively 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|
Q
, it eectively 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+|
Q
, |+i
|0i+|1i
2
,
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 dierent 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
O
G = QS), and Charlie,
who is allowed to perform measurements on the output system G
0
of the resultant channel (i.e.,
his possible operations have the input system C
I
G
0
= Q
0
S
0
and a trivial output system). It can
be verified that the process matrix describing the correlations between the four parties is
W
D
O
A
I
A
O
B
I
B
O
C
I
SWITCH
= |W ihW |
D
O
A
I
A
O
B
I
B
O
C
I
, (7)
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A
O
A
I
B
I
B
O
a
I
a
O
b
I
b
O
SQ
S’Q’
U
B
U
A
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
D
O
A
I
A
O
B
I
B
O
C
I
= |0i
Q
0
|0i
Q
|Φ
+
i
A
I
S
|Φ
+
i
A
O
B
I
|Φ
+
i
B
O
S
0
+ |1i
Q
0
|1i
Q
|Φ
+
i
B
I
S
|Φ
+
i
B
O
A
I
|Φ
+
i
A
O
S
0
, (8)
with |Φ
+
i =
P
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+|
Q
on the control qubit together with some pure state |ψihψ|
S
on the target system.
In that case, the correlations between Alice, Bob, and Charlie are given by the tripartite process
matrix
W
A
I
A
O
B
I
B
O
C
I
= |W ihW |
A
I
A
O
B
I
B
O
C
I
, (9)
where
|W i
A
I
A
O
B
I
B
O
C
I
=
1
2
(|0i
Q
0
|ψi
A
I
|Φ
+
i
A
O
B
I
|Φ
+
i
B
O
S
0
+|1i
Q
0
|ψi
B
I
|Φ
+
i
B
O
A
I
|Φ
+
i
A
O
S
0
). (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 dierent
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|
Q
) it first passes through Alice’s laboratory and then through Bob’s laboratory, and along
the other path (corresponding to the initial state |1ih1|
Q
) 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 dierent classical
cases is dierent 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 dierent 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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U
A
B
A
CharlieTime
B
I
B
O
S
S’
Q
Q’
U
A
David
U
B
B
I
a
I
b
I
b
O
a
O
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 o 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
I
A
I
a
O
A
O
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
I
QSB
O
to the composite output system a
O
Q
0
S
0
B
I
(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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A
Time
B
I
B
O
S
S’
Q
Q’
B
I
a
I
a
O
U
A
U
A
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
U
a
I
QSB
O
a
O
Q
0
S
0
B
I
SWITCH
(U
A
) =
|0i
Q
0
h0|
Q
U
a
I
Sa
O
B
I
A
11
B
O
S
0
+|1i
Q
0
h1|
Q
U
a
I
B
O
a
O
S
0
A
11
SB
I
, (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
U
a
I
QSB
O
a
O
Q
0
S
0
B
I
SWITCH
(U
A
) = U
a
I
A
I
a
O
A
O
A
11
A
I
A
O
, (12)
or, equivalently, the unitary channel of the quantum comb can be written in the form
U
a
I
QSB
O
a
O
Q
0
S
0
B
I
SWITCH
(U
A
) = U
a
I
A
I
a
O
A
O
A
I
A
I
A
O
, (13)
where I denotes the identity channel. Here, H
A
I
is a tensor factor of H
QSB
O
that has dimension d
and H
A
I
is its cofactor (H
QSB
O
= H
A
I
H
A
I
), and similarly, H
A
O
is a tensor factor of H
Q
0
S
0
B
I
that has the dimension of d and H
A
O
is its co-factor (H
Q
0
S
0
B
I
= H
A
O
H
A
O
). The factor H
A
I
is defined by the algebra of operators of the form
O
A
I
|0ih0|
Q
O
S
11
B
O
+ |1ih1|
Q
11
S
O
B
O
, (14)
and the factor H
A
O
by the algebra of operators of the form
O
A
O
|0ih0|
Q
0
11
S
0
O
B
I
+ |1ih1|
Q
0
O
S
0
11
B
I
. (15)
These are the input and output systems of Alice’s operation.
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A
Time
B
I
B
O
B
I
U
A
U
A
S’Q’ a
O
S
Q a
I
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
I
A
I
a
O
A
O
A
in Eq. (12). This simply
requires preparing suitable states in the input subsystem a
I
A
I
, and applying suitable measurements on
the output subsystem a
O
A
O
.
To see this, notice that [50], for any U
A
,
U
a
I
QSB
O
a
O
Q
0
S
0
B
I
SWITCH
(U
A
)
= C-SWAP
Q
0
S
0
B
I
(U
a
I
Sa
O
B
I
A
11
QB
O
Q
0
S
0
)C-SWAP
QSB
O
, (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
a
I
QSB
O
a
O
Q
0
S
0
B
I
SWITCH
(U
A
) 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
I
is replaced by S, and A
O
is replaced by B
I
. Since the first C-SWAP maps some
subsystem of QSB
O
(call it A
I
) to the subsystem S, and the second C-SWAP maps the subsystem
B
O
to some subsystem of Q
0
S
0
B
I
(call it A
O
), the action of U
a
I
QSB
O
a
O
Q
0
S
0
B
I
SWITCH
(U
A
) is precisely
the one given in Eq. (12) with respect to A
I
and A
O
. The subsystems A
I
and A
O
are related to S
and B
1
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
I
and A
O
, the operation performed from A
I
to A
O
becomes
a directly testable fact—we can verify it by preparing suitable states on A
I
and performing suitable
measurements on A
O
, which fits within the general scheme depicted in Fig. 7.
Note that the subsystem A
I
has a nontrivial ‘spread’ over B
O
in the sense that it is not a
subsystem of the complement QS. Similarly, A
O
has a nontrivial spread over B
I
. But B
I
is in the
causal past of B
O
. In spite of this, it is possible to perform an arbitrary standard operation from
A
I
to A
O
. 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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W
SWITCH
Alice Bob
A
O
A
I
B
I
B
O
SQ
S’Q’
Charlie
David
~
Figure 8: The quantum SWITCH as an irreducible cyclic circuit. With respect to the time-
delocalized subsystems A
I
and A
O
, 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
O
, depending on the control bit (formally
speaking, this is equivalent to associating A
I
with a tensor factor of two dierent subspaces of
QSB
O
), and similarly the output system is either B
I
or S
0
. However, when Q is initialized in a
quantum superposition and Charlie is allowed to perform arbitrary operations on Q
0
S
0
, such an
interpretation is not possible anymore. In general, with respect to the input and output systems
A
I
and A
O
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
I
as directly connected to QSB
O
without any temporal separation, and A
O
as directly connected to
B
I
Q
0
S
0
, 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
I
, B
O
, Q
0
, S
0
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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A
Charlie
Time
B
I
S
S’
Q
Q’
U
A
David
a
I
b
I
b
O
a
O
B
U
B
U
A
B
U
B
F
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
O
A
I
|0ih0|
Q
O
S
11
F
+ |1ih1|
Q
11
S
O
F
, (17)
O
A
O
|0ih0|
Q
0
O
F
11
S
0
+ |1ih1|
Q
0
11
F
O
S
0
, (18)
and Bob’s input and output systems are given by
O
B
I
|0ih0|
Q
11
S
O
F
+ |1ih1|
Q
O
S
11
F
, (19)
O
B
O
|0ih0|
Q
0
11
F
O
S
0
+ |1ih1|
Q
0
O
F
11
S
0
. (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 oers 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 dierences 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
A
and U
B
with a single use of each
U
A
and U
B
, whereas processes with a definite causal order require at least two uses of U
A
or
U
B
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
A
and U
B
, one may still be concerned by the fact that the temporal circuit contains two
controlled-U
A
operations, where a single use of controlled-U
A
is a more powerful resource than
a single use of U
A
(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 eect 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 dierent 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
Accepted in Quantum 2019-11-08, click title to verify. Published under CC-BY 4.0. 16