Abstract
It has been shown that it is theoretically possible for there to exist quantum and classical processes in which the operations performed by separate parties do not occur in a welldefined causal order. A central question is whether and how such processes can be realised in practice. In order to provide a rigorous framework for the notion that certain such processes have a realisation in standard quantum theory, the concept of timedelocalised quantum subsystem has been introduced. In this paper, we show that realisations on timedelocalised subsystems exist for all unitary extensions of tripartite processes. This class contains processes that violate causal inequalities, i.e., that can generate correlations that witness the incompatibility with definite causal order in a deviceindependent manner, and whose realisability has been a central open problem. We consider a known example of such a tripartite classical process that has a unitary extension, and study its realisation on timedelocalised subsystems. We then discuss this finding with regard to the assumptions that underlie causal inequalities, and argue that they are indeed a meaningful concept to show the absence of a definite causal order between the variables of interest.
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Introduction
The concept of causality is essential for physics and for our perception of the world in general. Our usual understanding is that events take place in a definite causal order, with past events influencing future events, but not vice versa. One may however wonder whether this notion is really fundamental, or whether scenarios without such an underlying order can exist. In particular, the questions of what quantum theory implies for our understanding of causality, and what new types of causal relations arise in the presence of quantum effects, have recently attracted substantial interest. This investigation is motivated both by foundational and by applied questions. On the one hand, it is expected to lead to new conceptual insights into the tension between general relativity and quantum theory^{1,2,3}. On the other hand, it also opens up new possibilities in quantum information processing^{4}.
A particular model for the study of quantum causal relations is the process matrix framework^{2}, where one considers multiple parties which perform operations that locally abide by the laws of quantum theory, but that are not embedded into any a priori causal order. As it turns out, this framework indeed allows for situations where the causal order between the parties is not welldefined (see e.g. refs. ^{2,5,6,7,8,9}). Moreover, some of these processes, called noncausal, can produce correlations that violate causal inequalities^{2,6,7,8,10,11}, which witnesses the incompatibility with definite causal order in a deviceindependent manner, similarly to the way a violation of a Bell inequalities witnesses the incompatibility with local hidden variables^{12}. A central question is which of these processes with indefinite causal order have a practical realisation, and in what physical situations they can occur. It has been speculated that indefinite causal order could arise in exotic physical regimes, such as at the interface of quantum theory and gravity^{1,2,3}. However, there are also processes with indefinite causal order that have an interpretation in terms of standard quantum theoretical concepts. A paradigmatic example is the quantum switch^{4}, a process in which the order between two operations is controlled coherently by a twodimensional quantum system. This control qubit may be prepared in a superposition state, which leads to a superposition of causal orders. Although the quantum switch cannot violate causal inequalities^{5,6,13,14} (however, see recent results in the presence of additional causal assumptions^{15,16}), it can be proven incompatible with a definite causal order in a devicedependent sense^{5,6}.
In order to demonstrate indefinite causal order in practice, a number of experiments that realise such coherent control of orders have been implemented in the laboratory^{17,18,19,20,21,22,23,24,25,26,27}, however their interpretation as genuine realisations of indefinite causal order has remained controversial^{28,29,30,31}. Indeed, the claim that indefinite causal order can be realised in standard quantum scenarios seems contradictory at first sight—after all, such experiments admit a description in terms of standard quantum theory, where physical systems by assumption evolve with respect to a fixed background time, and it is therefore not manifest how the causal order between operations could possibly be indefinite. A resolution of this apparent contradiction was proposed in ref. ^{29}, where it was shown that certain processes with indefinite causal order can be seen to take place as part of standard quantum mechanical evolutions if the latter are described in terms of suitable systems. The twist is to consider a more general type of system than usually studied, namely socalled timedelocalised subsystems, which are nontrivial subsystems of composite systems whose constituents are associated with different times. This concept provides a rigorous underpinning for the interpretation of previous laboratory experiments as realisations of processes with indefinite causal order—when the experiment is described with respect to such an alternative, operationally equally meaningful factorisation of the Hilbert space, it acquires precisely the form of the process with indefinite causal order. It was then shown in ref. ^{29} that this argument extends to an entire class of quantum processes, namely unitary extensions of bipartite processes, as well as a class of isometric extensions, whose relation to the unitary class is not yet fully understood. The generalisation of these constructions to more parties has however remained an open question. In particular, it has remained an open question whether processes violating causal inequalities can be realised in a similar way. It is in fact generally believed that such processes could not be realised deterministically within the known physics^{14}.
In this paper, we extend the proof of realisability on timedelocalised subsystems to all unitary extensions of tripartite processes. This class contains examples of processes that can violate causal inequalities, showing that they have realisations with the tools of known physics in a welldefined sense.
This work is structured as follows. We set the stage by reviewing the process matrix framework, as well as the notion of timedelocalised subsystems. We present the general tripartite construction, and we study an example of a tripartite noncausal process on timedelocalised subsystems. We then analyse our finding with regard to the assumptions that underlie causal inequalities, and argue that their violation witnesses the absence of a definite causal order in a meaningful way.
Results
Notations
We start by introducing some notations. We denote the Hilbert space of some quantum system Y by \({{{{{{\mathcal{H}}}}}}}^{Y}\), the dimension of \({{{{{{\mathcal{H}}}}}}}^{Y}\) by d_{Y} and the space of linear operators over \({{{{{{\mathcal{H}}}}}}}^{Y}\) by \({{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{Y})\). Each such Hilbert space comes with a preferred, computational basis generally denoted \({\{{\left i\right\rangle }^{Y}\}}_{i}\). The identity operator on \({{{{{{\mathcal{H}}}}}}}^{Y}\) is denoted by \({{\mathbb{1}}}^{Y}\). We also use the notation \({{{{{{\mathcal{H}}}}}}}^{YZ}: = {{{{{{\mathcal{H}}}}}}}^{Y}\otimes {{{{{{\mathcal{H}}}}}}}^{Z}\) for the tensor product of two Hilbert spaces \({{{{{{\mathcal{H}}}}}}}^{Y}\) and \({{{{{{\mathcal{H}}}}}}}^{Z}\) (whose computational basis is built as the tensor product of the two subsystems’ computational bases). For two isomorphic Hilbert spaces \({{{{{{\mathcal{H}}}}}}}^{Y}\) and \({{{{{{\mathcal{H}}}}}}}^{Z}\), we denote the identity operator between these spaces (i.e. the canonical isomorphism, which maps each computational basis state \({\left i\right\rangle }^{Y}\) of \({{{{{{\mathcal{H}}}}}}}^{Y}\) to the corresponding computational basis state \({\left i\right\rangle }^{Z}\) of \({{{{{{\mathcal{H}}}}}}}^{Z}\)) by \({{\mathbb{1}}}^{Y\to Z}: = {\sum }_{i}{\left i\right\rangle }^{Z}{\left\langle i\right }^{Y}\), and its pure Choi representation (see the Methods section “The Choi isomorphism and the link product”) by \({\left.\left {\mathbb{1}}\right\rangle \right\rangle }^{YZ}: = {\sum }_{i}{\left i\right\rangle }^{Y}\otimes {\left i\right\rangle }^{Z}\). (Generally, superscripts on vectors indicate the Hilbert space they belong to, which may be omitted when clear from the context). Moreover, we will often abbreviate X_{I}X_{O} to X_{IO} for the incoming and outgoing systems of the party X (see below).
The process matrix framework
In the following, we briefly outline the process matrix framework, originally introduced in ref. ^{2}. We consider multiple parties X = A, B, C, … performing operations that are locally described by quantum theory. That is, each party has an incoming quantum system X_{I} with Hilbert space \({{{{{{\mathcal{H}}}}}}}^{{X}_{I}}\) and an outgoing quantum system X_{O} with Hilbert space \({{{{{{\mathcal{H}}}}}}}^{{X}_{O}}\), and can perform arbitrary quantum operations from X_{I} to X_{O}. A quantum operation is most generally described by a quantum instrument, that is, a collection of completely positive (CP) maps \({\{{{{{{{\mathcal{M}}}}}}}_{X}^{[{o}_{X}]}\}}_{{o}_{X}}\), with each \({{{{{{\mathcal{M}}}}}}}_{X}^{[{o}_{X}]}:{{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{{X}_{I}})\to {{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{{X}_{O}})\) associated to a classical outcome o_{X}, and with the sum over the classical outcomes yielding a completely positive and tracepreserving (CPTP) map.
The process matrix framework was conceived to study the most general correlations that can arise between such parties, without making any a priori assumption about the way they are connected. In ref. ^{2}, it was shown that these correlations can most generally be expressed as
Here, \({M}_{X}^{[{o}_{X}]}\in {{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{{X}_{IO}})\) are the Choi representations of the local CP maps \({{{{{{\mathcal{M}}}}}}}_{X}^{[{o}_{X}]}\) and “ * ” denotes the link product^{32,33}, a mathematical operation that describes the composition of quantum operations in terms of their Choi representation (see the Methods section “The Choi isomorphism and the link product”). \(W\in {{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{{A}_{IO}{B}_{IO}{C}_{IO}\ldots })\) is a Hermitian operator called the process matrix. The requirement that the probabilities in Eq. (1) should be nonnegative, even when the operations of the parties are extended so as to act on additional, possibly entangled ancillary input systems, is equivalent to W ≥ 0. The requirement that the probabilities should be normalised (i.e., they should sum up to 1 for any choice of local operations) is equivalent to W satisfying certain linear constraints^{2,5,6,9,34}, and having the trace \({{{{{\rm{Tr}}}}}}(W)={d}_{{A}_{O}}{d}_{{B}_{O}}{d}_{{C}_{O}}\ldots \,\).
The process matrix is the central object of the formalism, which describes the physical resource or environment through which the parties are connected. Mathematically, the process matrix defines (i.e., it is the Choi representation of) a quantum channel \({{{{{\mathcal{W}}}}}}:{{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{{A}_{O}{B}_{O}{C}_{O}\ldots })\to {{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{{A}_{I}{B}_{I}{C}_{I}\ldots })\) from all output systems of the parties to their input systems. Equation (1) then describes the composition of that channel with the local operations, which can be interpreted as a circuit with a cycle as represented graphically (for the bipartite case) in Fig. 1a.
Through the topdown approach outlined here, one recovers standard quantum scenarios, such as joint measurements on multipartite quantum states, or, more generally, quantum circuits in which the parties apply their operations in a fixed causal order (and the process matrix corresponds to the acyclic circuit fragment consisting of the operations in between the parties^{33,35}). However, one also finds processes that are incompatible with any definite causal order between the local operations. Such processes are said to be causally nonseparable^{2,5,6,9}. Furthermore, some causally nonseparable processes can generate correlations P(o_{A}, o_{B}, o_{C}, …∣i_{A}, i_{B}, i_{C}, …), where i_{X} are local classical inputs based on which the local operations are chosen, that violate socalled causal inequalities^{2,6,7,8,10,11}, which certifies their incompatibility with a definite causal order in a deviceindependent way. Such processes are referred to as noncausal.
A class of processes that is of particular interest in this paper is that of unitarily extendible processes, which were first discussed in ref. ^{34}. A unitary extension of a process matrix W is a process matrix which involves an additional party P with a trivial, onedimensional input Hilbert space, as well as an additional party F with a trivial, onedimensional outgoing Hilbert space, such that the corresponding channel from P_{O}A_{O}B_{O}C_{O}… to F_{I}A_{I}B_{I}C_{I}… is unitary, and such that the original process matrix W is recovered when P prepares some fixed state and F is traced out. That is, the extended process matrix is a rankone projector \(\left.\left U\right\rangle \right\rangle \left\langle \left\langle U\right \right.\), where \(\left.\left U\right\rangle \right\rangle \) is the pure Choi representation (see Methods) of a unitary \(U:{{{{{{\mathcal{H}}}}}}}^{{P}_{O}{A}_{O}{B}_{O}{C}_{O}\ldots }\to {{{{{{\mathcal{H}}}}}}}^{{F}_{I}{A}_{I}{B}_{I}{C}_{I}\ldots }\), which satisfies
The additional parties P and F can be interpreted as being in the global past, respectively global future, of all other parties, since they do not receive, respectively send out, a quantum system.
Note that the unitary extension also needs to be a valid process matrix, i.e., it needs to satisfy the abovementioned constraints which ensure that it yields valid probabilities when the parties (including P and F) perform arbitrary local operations. In ref. ^{34}, it was found that some process matrices do not admit such a unitary extension, and unitary extendibility was postulated as a necessary condition for a process matrix to describe a physically realisable scenario. It was also shown that unitary extensions are equivalent to processes that preserve the reversibility of quantum operations. That is, when the slots of P and F are left open, and all other parties perform unitary operations \({{{{{{\mathcal{U}}}}}}}_{X}:{{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{{X}_{I}{X}_{I}^{{\prime} }})\to {{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{{X}_{O}{X}_{O}^{{\prime} }})\), which act on X_{I} and X_{O} as well as some (possibly trivial) additional ancillary incoming and outgoing systems \({X}_{I}^{{\prime} }\) and \({X}_{O}^{{\prime} }\), the resulting global operation, which takes the initial systems \({P}_{O}{A}_{I}^{{\prime} }{B}_{I}^{{\prime} }{C}_{I}^{{\prime} }\ldots \) to the final systems \({F}_{I}{A}_{O}^{{\prime} }{B}_{O}^{{\prime} }{C}_{O}^{{\prime} }\ldots \), is again unitary (Fig. 1b).
In this case, the Choi representations of the local operations, as well as the unitarily extended process matrix, are rankone projectors, and we can describe their composition in terms of their pure Choi representations. The global unitary operation \({{{{{{\mathcal{U}}}}}}}_{{{{{{\mathcal{G}}}}}}}({{{{{{\mathcal{U}}}}}}}_{A},{{{{{{\mathcal{U}}}}}}}_{B},{{{{{{\mathcal{U}}}}}}}_{C},\cdots \,):{{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{{P}_{O}{A}_{I}^{{\prime} }{B}_{I}^{{\prime} }{C}_{I}^{{\prime} }\ldots })\to {{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{{F}_{I}{A}_{O}^{{\prime} }{B}_{O}^{{\prime} }{C}_{O}^{{\prime} }\ldots })\), in its pure Choi representation, is given by
where \(\left.\left {U}_{X}\right\rangle \right\rangle \in {{{{{{\mathcal{H}}}}}}}^{{X}_{IO}{X}_{IO}^{{\prime} }}\) are the pure Choi representations of the local unitary operations \({{{{{{\mathcal{U}}}}}}}_{X}\), and “*” denotes here the socalled vector link product^{13} (cf. Methods). In the following, we are going to refer to \(\left.\left U\right\rangle \right\rangle \) as the process vector of the unitary process under consideration.
The process matrices that we are concerned with in this work are unitary extensions of bipartite or tripartite process matrices. Moreover, any local operation can be dilated to a unitary channel acting on the original incoming and outgoing systems together with an additional incoming and outgoing ancilla, followed by a measurement of the outgoing ancilla. Throughout the paper, we will therefore not consider the actions of the global past and global future parties explicitly, but rather work with the description as per Eq. (3) in terms of pure Choi representations, which is convenient. We will also take the incoming and outgoing Hilbert spaces of all parties, except for P and F, to be of equal dimension \({d}_{{X}_{I}}={d}_{{X}_{O}} = :d\). This simplification saves us some technicalities, and it does not entail any loss of generality. Namely, if these dimensions do not match, one can treat the process under consideration as a part of an extended process with process vector \(\left.\left U\right\rangle \right\rangle \otimes {\left.\left {\mathbb{1}}\right\rangle \right\rangle }^{{P}_{A}{\tilde{A}}_{I}}\otimes {\left.\left {\mathbb{1}}\right\rangle \right\rangle }^{{\tilde{A}}_{O}{F}_{A}}\otimes {\left.\left {\mathbb{1}}\right\rangle \right\rangle }^{{P}_{B}{\tilde{B}}_{I}}\otimes {\left.\left {\mathbb{1}}\right\rangle \right\rangle }^{{\tilde{B}}_{O}{F}_{B}}\otimes {\left.\left {\mathbb{1}}\right\rangle \right\rangle }^{{P}_{C}{\tilde{C}}_{I}}\otimes {\left.\left {\mathbb{1}}\right\rangle \right\rangle }^{{\tilde{C}}_{O}{F}_{C}}\otimes \ldots \), which involves additional identity channels between additional outgoing (incoming) Hilbert spaces \({{{{{{\mathcal{H}}}}}}}^{{P}_{A}},{{{{{{\mathcal{H}}}}}}}^{{P}_{B}},{{{{{{\mathcal{H}}}}}}}^{{P}_{C}},\ldots \) (\({{{{{{\mathcal{H}}}}}}}^{{F}_{A}},{{{{{{\mathcal{H}}}}}}}^{{F}_{B}},{{{{{{\mathcal{H}}}}}}}^{{F}_{C}},\ldots \)) of the global past (future) party, and additional incoming (outgoing) Hilbert spaces \({{{{{{\mathcal{H}}}}}}}^{{\tilde{A}}_{I}},{{{{{{\mathcal{H}}}}}}}^{{\tilde{B}}_{I}},{{{{{{\mathcal{H}}}}}}}^{{\tilde{C}}_{I}},\ldots \) (\({{{{{{\mathcal{H}}}}}}}^{{\tilde{A}}_{O}},{{{{{{\mathcal{H}}}}}}}^{{\tilde{B}}_{O}},{{{{{{\mathcal{H}}}}}}}^{{\tilde{C}}_{O}},\ldots \)) of the remaining parties, whose dimensions are chosen such that \({d}_{{X}_{I}{\tilde{X}}_{I}}={d}_{{X}_{O}{\tilde{X}}_{O}}=d\) for all parties (except P and F).
Timedelocalised subsystems and operations
In this section, we discuss the concept of timedelocalised subsystem, first introduced in ref. ^{29}. Briefly summarised, the idea is that a quantum circuit, consisting of operations that act at definite times on specific input and output systems, can be described in terms of a different choice of systems, corresponding to an alternative factorisation of the joint Hilbert spaces of the input and output systems of operations at different times. In general, the new systems may be delocalised relative to the old ones and thus spread over different times. When described in terms of such alternative timedelocalised subsystems, the circuit generally contains cycles as considered in the process matrix framework (Fig. 1). We first discuss the general formalisation of this idea, then we recall how it applies to the case of the quantum switch, as well as general unitary extensions of bipartite processes, for which it was shown in ref. ^{29} that realisations on such timedelocalised subsystems always exist.
The concept of timedelocalised subsystem arises from combining two notions from standard quantum theory, namely the definition of quantum subsystem decompositions in terms of tensor product structures, and the fact that a fragment of a quantum circuit containing multiple operations implements itself a quantum operation from all its incoming to all its outgoing systems.
In quantum theory, the division of a physical system into subsystems is formally described through the choice of a tensor product structure. Equipping a given Hilbert space \({{{{{{\mathcal{H}}}}}}}^{Y}\), corresponding to some quantum system Y, with a tensor product structure means choosing an isomorphism (i.e., a unitary transformation) \(J:{{{{{{\mathcal{H}}}}}}}^{Y}\to {\otimes }_{i=1}^{n}{{{{{{\mathcal{H}}}}}}}^{{Y}_{n}}\), where \({{{{{{\mathcal{H}}}}}}}^{{Y}_{1}},\ldots,{{{{{{\mathcal{H}}}}}}}^{{Y}_{n}}\) are Hilbert spaces of dimensions \({d}_{{Y}_{1}},\ldots,{d}_{{Y}_{n}}\), with \({\Pi }_{i=1}^{n}{d}_{{Y}_{n}}={d}_{Y}\). Such a choice establishes a notion of locality on \({{{{{{\mathcal{H}}}}}}}^{Y}\), and defines a decomposition of the system Y into subsystems Y_{1}, …, Y_{n}. For instance, the operators in \({{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{Y})\) that are local on the subsystem Y_{i} are those of the form \({J}^{{{\dagger}} }({O}^{{Y}_{i}}\otimes {{\mathbb{1}}}^{{Y}_{1},\ldots,{Y}_{i1}{Y}_{i+1}\ldots {Y}_{n}})\,J\) with \({O}^{{Y}_{i}}\in {{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{{Y}_{i}})\). (Equivalently, the tensor product structure can also be defined in terms of the algebras of operators that are local on the different subsystems^{36}). Since the choice of such a tensor product structure is not unique, there are many different ways to view \({{{{{{\mathcal{H}}}}}}}^{Y}\) as the state space of some quantum system with multiple subsystems.
In standard quantum theory, physical systems evolve with respect to a fixed background time. At an abstract level, such standard quantum mechanical time evolution can be described in terms of a quantum circuit, that is, a collection of quantum operations (pictorially represented by boxes) that are composed through quantum systems (pictorially represented by wires) in an acyclic network. The operations in such a quantum circuit thus act on their incoming and outgoing quantum systems (which may consist of several subsystems) at definite times. One may however also consider quantum operations that act on several subsystems associated with different times. In fact, this possibility arises naturally within the quantum circuit framework. Namely, if one considers a generic fragment of a quantum circuit containing many operations, that fragment implements a quantum operation from the joint system of all wires that enter into it, to the joint system of all wires that go out of it^{33}, where the incoming and outgoing wires are generally associated with Hilbert spaces at different times (see Fig. 2a for an example).
One may choose to describe such a quantum operation implemented by a fragment with respect to a different subsystem decomposition. Formally, this is achieved by composing its incoming, respectively outgoing, wires with some isomorphisms that define a different tensor product structure on the corresponding joint Hilbert spaces (Fig. 2b). The resulting subsystems are then in general not associated with a definite time. This is what one understands by timedelocalised subsystems.
To describe the full circuit in terms of these newly chosen timedelocalised subsystems, the operation implemented by the complement of the fragment under consideration needs to be composed with precisely the inverse of the chosen isomorphisms (Fig. 2c). The composition of the two fragments (which, for a circuit with no open wires, corresponds to the joint probability of the measurement outcomes of the different operations in the circuit^{37,38}, see Fig. 1a) then indeed remains the same in the old and new descriptions. This follows from the properties of the link product (see Methods, Eqs. (13) and (14)), which provides a formal tool to connect the different fragments that a quantum circuit is decomposed into^{32,33}.
Importantly, the structure of a given circuit with respect to such a choice of timedelocalised subsystems can also be tested operationally^{29}. In particular, the circuit can be disconnected at the chosen subsystems and each of the timedelocalised operations that occur on these subsystems can be experimentally addressed and verified, similarly to the way one would test the circuit description with respect to the standard timelocal factorisation. In this sense, such an alternative description of the experiment is operationally just as meaningful. This is discussed in more detail in Supplementary Note 1.
Processes with indefinite causal order on timedelocalised subsystems
In the laboratory experiments that have been proposed as implementations of the quantum switch, one considers a target quantum system at two possible times. The operation U_{A} is applied to the target system T_{1} at the earlier time, or to the target system T_{2} at the later time, depending on whether another twodimensional quantum system, the control qubit, is in the computational basis state \(\left 0\right\rangle \) or \(\left 1\right\rangle \), and conversely for the operation U_{B}. There has been much debate (see e.g. refs. ^{28,29,30,31}) about whether experiments of that type can be interpreted as valid realisations of the quantum switch, understood as an abstractly defined scenario in the process matrix formalism^{5}. Indeed, the relation between the above outlined experimental procedure, and the situation considered in the process matrix framework, where one instance of each U_{A} and U_{B} is composed with the process matrix in a circuit with a cycle, is a priori unclear. A heuristic argument that is sometimes invoked to justify that each of the two operations is indeed applied once and only once is that each operation occurs precisely once in each of the two superposed coherent branches, and is therefore used once overall. To further corroborate this, one could introduce a flag or counter system^{14,39} that keeps track of the usage of the operations. To really understand the sense in which the quantum switch is realised in these experiments, it is however desirable to rigorously formalise the link between the standard quantum description of the experiments, and the process matrix scenario. This question was addressed in ref. ^{29}. It was shown that the temporally ordered quantum circuit that describes the experimental situation outlined above indeed takes the structure of a circuit with a cycle as in the process matrix framework (i.e., as in Fig. 1), when one changes to a description in terms of specific timedelocalised subsystems—whose choice, broadly speaking, formalises the intuition that the input system is T_{1} when the control system is in state \(\left 0\right\rangle \) and T_{2} when the control system is in state \(\left 1\right\rangle \), and similarly for the output systems^{29}. In other words, when these experiments are realised physically, what happens on these alternative systems is precisely the structure described in the process matrix framework. It is in that sense that these experiments can be considered realisations of the abstract mathematical concept.
It was then shown that this argument can be generalised, and that there exist other types of processes which have a realisation in this sense. Notably, this is the case for the entire class of unitary extensions of bipartite processes, of which the quantum switch is a particular example. It was subsequently shown in refs. ^{40,41} that all such processes are variations of the quantum switch, but the proof of Ref. ^{29} did not rely on this knowledge. It is the idea behind the original proof from ref. ^{29}, together with the subsequent result of refs. ^{40,41}, that will allow us to generalise the proof to the tripartite case. We therefore recall the bipartite result from ref. ^{29}, in the language and conventions we use in this paper (notably employing the Choi representation and the link product), in Methods, and the corresponding proofs in Supplementary Note 2.
Unitary extensions of tripartite processes on timedelocalised subsystems
For unitary extensions of processes with more than two parties, it is a priori unclear whether and how a realisation on timedelocalised subsystems can be found. In the following, we will establish the result for unitary extensions of tripartite processes. Briefly summarised, we show that for any unitarily extended tripartite process, there exists a standard, temporally ordered quantum circuit, with operations that depend on the local operations U_{A}, U_{B} and U_{C} applied in the process, which precisely corresponds to the situation considered in the process matrix framework, with one instance of each U_{A}, U_{B} and U_{C} composed with the process matrix in a circuit with a cycle, when described in terms of a suitable choice of timedelocalised subsystems.
Formally, we prove the following proposition.
Proposition 1
Consider a unitary extension of a tripartite process, described by a process vector \(\left.\left U\right\rangle \right\rangle \in {{{{{{\mathcal{H}}}}}}}^{{P}_{O}{A}_{IO}{B}_{IO}{C}_{IO}{F}_{I}}\), composed with unitary local operations \({U}_{A}:{{{{{{\mathcal{H}}}}}}}^{{A}_{I}{A}_{I}^{{\prime} }}\to {{{{{{\mathcal{H}}}}}}}^{{A}_{O}{A}_{O}^{{\prime} }},{U}_{B}:{{{{{{\mathcal{H}}}}}}}^{{B}_{I}{B}_{I}^{{\prime} }}\to {{{{{{\mathcal{H}}}}}}}^{{B}_{O}{B}_{O}^{{\prime} }}\) and \({U}_{C}:{{{{{{\mathcal{H}}}}}}}^{{C}_{I}{C}_{I}^{{\prime} }}\to {{{{{{\mathcal{H}}}}}}}^{{C}_{O}{C}_{O}^{{\prime} }}\). For any such process, the following exist.

1.
A temporal circuit of the form shown in Fig. 3, in which U_{A} and U_{B} are applied on the target input and output systems \({T}_{1}^{({\prime} )}\) or \({T}_{2}^{({\prime} )}\), coherently conditioned on the state of the control systems \({Q}_{1}^{({\prime} )}\) and \({Q}_{2}^{({\prime} )}\), and which is composed of circuit operations that depend on U_{C}.

2.
Isomorphisms \({J}_{{{{{{\rm{in}}}}}}}:{{{{{{\mathcal{H}}}}}}}^{{A}_{I}{B}_{I}{C}_{I}YZ}\to {{{{{{\mathcal{H}}}}}}}^{{T}_{1}{T}_{2}{\bar{T}}_{1}^{{\prime} }{\bar{T}}_{2}^{{\prime} }{Q}_{1}{P}_{O}}\) and \({J}_{{{{{{\rm{out}}}}}}}:{{{{{{\mathcal{H}}}}}}}^{{T}_{1}^{{\prime} }{T}_{2}^{{\prime} }{\bar{T}}_{1}{\bar{T}}_{2}{Q}_{2}^{{\prime} }{F}_{I}}\to {{{{{{\mathcal{H}}}}}}}^{{A}_{O}{B}_{O}{C}_{O}\bar{Y}\bar{Z}}\), such that, with respect to the subsystems A_{I}, B_{I} and C_{I} of \({T}_{1}{T}_{2}{\bar{T}}_{1}^{{\prime} }{\bar{T}}_{2}^{{\prime} }{Q}_{1}{P}_{O}\) and the subsystems A_{O}, B_{O} and C_{O} of \({T}_{1}^{{\prime} }{T}_{2}^{{\prime} }{\bar{T}}_{1}{\bar{T}}_{2}{Q}_{2}^{{\prime} }{F}_{I}\) that these isomorphisms define, the circuit in Fig. 3 takes the form of a cyclic circuit composed of U, U_{A}, U_{B} and U_{C} as in the process matrix framework (Fig. 4).
In the following, we outline the proof. All technical proofs and calculations for this tripartite construction are given in Supplementary Note 3.
Outline of proof
The existence of a temporal circuit as in Fig. 3 is shown in Supplementary Note 3A. It follows from the result that all unitary extensions of bipartite processes can be implemented as variations of the quantum switch^{40,41}, in which the time of the two local operations is controlled coherently. Any unitary extension of a tripartite process can thus be implemented as a variation of the quantum switch, with two local operations whose time is controlled coherently, and which is composed of circuit operations that depend on the third local operation. The isomorphisms \({J}_{{{{{{\rm{in}}}}}}}:{{{{{{\mathcal{H}}}}}}}^{{A}_{I}{B}_{I}{C}_{I}YZ}\to {{{{{{\mathcal{H}}}}}}}^{{T}_{1}{T}_{2}{\bar{T}}_{1}^{{\prime} }{\bar{T}}_{2}^{{\prime} }{Q}_{1}{P}_{O}}\) and \({J}_{{{{{{\rm{out}}}}}}}:{{{{{{\mathcal{H}}}}}}}^{{T}_{1}^{{\prime} }{T}_{2}^{{\prime} }{\bar{T}}_{1}{\bar{T}}_{2}{Q}_{2}^{{\prime} }{F}_{I}}\to {{{{{{\mathcal{H}}}}}}}^{{A}_{O}{B}_{O}{C}_{O}\bar{Y}\bar{Z}}\) (where \(Y,Z,\bar{Y}\) and \(\bar{Z}\) are appropriate complementary subsystems) are defined in Supplementary Note 3B, based on a specific decomposition of unitarily extended process vectors which plays a central role in the bipartite proof (cf. Supplementary Equation (3)), and which generalises to the multipartite case (cf. Supplementary Equation (24)).
In Supplementary Note 3C, we change to the description of the circuit in terms of the corresponding timedelocalised subsystems. For that purpose, we decompose the circuit into the red and blue circuit fragment shown in Fig. 4. By construction, when composed with J_{in} and J_{out}, the red circuit fragment shown in Fig. 4a consists of precisely one application of U_{A} and U_{B}, in parallel to a unitary operation \(R({U}_{C}):{{{{{{\mathcal{H}}}}}}}^{{C}_{I}^{{\prime} }{C}_{I}YZ{\bar{Q}}_{2}^{{\prime} }}\to {{{{{{\mathcal{H}}}}}}}^{{C}_{O}^{{\prime} }{C}_{O}\bar{Y}\bar{Z}{\bar{Q}}_{1}}\). Under that change of subsystems, the complementary blue fragment needs to be composed with the inverse isomorphisms \({J}_{{{{{{\rm{in}}}}}}}^{{{\dagger}} }\) and \({J}_{{{{{{\rm{out}}}}}}}^{{{\dagger}} }\), which results in an operation \({R}^{{\prime} }:{{{{{{\mathcal{H}}}}}}}^{{P}_{O}{A}_{O}{B}_{O}{C}_{O}\bar{Y}\bar{Z}{\bar{Q}}_{1}}\to {{{{{{\mathcal{H}}}}}}}^{{F}_{I}{A}_{I}{B}_{I}{C}_{I}YZ{\bar{Q}}_{2}^{{\prime} }}\) (Fig. 4b). R(U_{C}) and \({R}^{{\prime} }\) cannot be further decomposed for now.
At this point, we thus have a cyclic circuit which consists of the four boxes U_{A}, U_{B}, R(U_{C}) and \({R}^{{\prime} }\), and which involves the systems \({P}_{O},{A}_{IO}^{({\prime} )},{B}_{IO}^{({\prime} )},{C}_{IO}^{({\prime} )},{F}_{I}\), as well as \(Y,\bar{Y},Z,\bar{Z},{\bar{Q}}_{1},{\bar{Q}}_{2}^{{\prime} }\) (see the lefthand side of Fig. 4c). In order to obtain a description with respect to only the systems \({P}_{O},{A}_{IO}^{({\prime} )},{B}_{IO}^{({\prime} )},{C}_{IO}^{({\prime} )},{F}_{I}\), we need to evaluate the composition of R(U_{C}) and \({R}^{{\prime} }\) over the systems \(Y,\bar{Y},Z,\bar{Z},{\bar{Q}}_{1},{\bar{Q}}_{2}^{{\prime} }\) (but not over the systems C_{I} and C_{O}, which we wish to maintain in the description). The isomorphisms J_{in} and J_{out} are constructed in precisely such a way (based on the abstract relation between the systems in the process that is also used in the bipartite proof) that, when this composition of R(U_{C}) and \({R}^{{\prime} }\) over \(Y,\bar{Y},Z,\bar{Z},{\bar{Q}}_{1},{\bar{Q}}_{2}^{{\prime} }\) is evaluated, the result is a cyclic circuit fragment consisting of the unitary operation \(U:{{{{{{\mathcal{H}}}}}}}^{{P}_{O}{A}_{O}{B}_{O}{C}_{O}}\to {{{{{{\mathcal{H}}}}}}}^{{F}_{I}{A}_{I}{B}_{I}{C}_{I}}\) that defines the process, composed with the operation \({U}_{C}:{{{{{{\mathcal{H}}}}}}}^{{C}_{I}{C}_{I}^{{\prime} }}\to {{{{{{\mathcal{H}}}}}}}^{{C}_{O}{C}_{O}^{{\prime} }}\) (see the middle of Fig. 4c). (Note the particularity that U_{C} only appears as an explicit part of the cyclic circuit after this composition of R(U_{C}) with \({R}^{{\prime} }\), and is not a tensor product factor of R(U_{C})).
Therefore, in its description with respect to the systems \({P}_{O},{A}_{IO}^{({\prime} )},{B}_{IO}^{({\prime} )},{C}_{IO}^{({\prime} )},{F}_{I}\), the circuit in Fig. 3 indeed consists of the four operations \({U}_{A}:{{{{{{\mathcal{H}}}}}}}^{{A}_{I}{A}_{I}^{{\prime} }}\to {{{{{{\mathcal{H}}}}}}}^{{A}_{O}{A}_{O}^{{\prime} }},{U}_{B}:{{{{{{\mathcal{H}}}}}}}^{{B}_{I}{B}_{I}^{{\prime} }}\to {{{{{{\mathcal{H}}}}}}}^{{B}_{O}{B}_{O}^{{\prime} }},{U}_{C}:{{{{{{\mathcal{H}}}}}}}^{{C}_{I}{C}_{I}^{{\prime} }}\to {{{{{{\mathcal{H}}}}}}}^{{C}_{O}{C}_{O}^{{\prime} }}\) and \(U:{{{{{{\mathcal{H}}}}}}}^{{P}_{O}{A}_{O}{B}_{O}{C}_{O}}\to {{{{{{\mathcal{H}}}}}}}^{{F}_{I}{A}_{I}{B}_{I}{C}_{I}}\), connected in a cyclic circuit as in the process matrix framework (see the righthand side of Fig. 4c). This establishes the tripartite result.
Note that a similar construction is possible when one considers an asymmetric tripartite temporal circuit where U_{A} is applied at a given, welldefined time, and U_{B} either before or after it, coherently depending on the control systems (or vice versa, with the roles of A and B exchanged).
A process that violates causal inequalities on timedelocalised subsystems
In ref. ^{10}, it was shown that, for three and more parties, there exist process matrices that violate causal inequalities and that can be interpreted as classical process matrices, since they are diagonal in the computational basis. An example, first found by Araújo and Feix and further studied by Baumeler and Wolf in refs. ^{11,42}, is the process matrix
where a_{O}, b_{O}, c_{O} ∈ {0, 1} and where ¬ is the negation. It was then shown by Baumeler and Wolf^{42} (cf. also refs. ^{34,43}) that W_{AF} has a unitary extension \({W}_{{{\mbox{BW}}}}=\left.\left {U}_{{{\mbox{BW}}}}\right\rangle \right\rangle \left\langle \left\langle {U}_{{{\mbox{BW}}}}\right \right.\), with
(with p_{1}, p_{2}, p_{3} ∈ {0, 1}, i.e., \({{{{{{\mathcal{H}}}}}}}^{{P}_{O}}={{{{{{\mathcal{H}}}}}}}^{{P}_{1}{P}_{2}{P}_{3}}\) and \({{{{{{\mathcal{H}}}}}}}^{{F}_{I}}={{{{{{\mathcal{H}}}}}}}^{{F}_{1}{F}_{2}{F}_{3}}\) consisting of three qubits each, and with ⊕ denoting addition modulo 2). W_{AF} is recovered from \( {U}_{{{\mbox{BW}}}}\rangle \rangle \langle \langle {U}_{{{\mbox{BW}}}} \) when the global past party prepares the state \(\left 0,0,0\right\rangle {\left\langle 0,0,0\right }^{{P}_{1}{P}_{2}{P}_{3}}\), and the global future party is traced out. What kind of temporal circuit do we obtain when we apply the general tripartite considerations from the previous section to this particular example? A possible such realisation of this process on timedelocalised subsystems is given by the circuit shown in Fig. 5 (similar circuits corresponding to this process have also been studied in other contexts in refs. ^{43,44,45}).
In Supplementary Note 4A, we give the explicit expressions of the circuit operations in Fig. 5, as well as for the isomorphisms that define the description in terms of timedelocalised subsystems for this particular case, and we sketch how to apply the general tripartite proof to this example.
The abstract process W_{AF} in Eq. (4) violates causal inequalities when each party performs a computational basis measurement on its incoming Hilbert space (and outputs the measurement result o_{X}), and prepares the computational basis state \(\left {i}_{X}\right\rangle \) (corresponding to its classical input i_{X}) on its outgoing Hilbert space. The corresponding unitary operations that need to be applied in the pure description of the process (and therefore in the circuit of Fig. 5) are \({U}_{X}={{\mathbb{1}}}^{{X}_{I}\to {X}_{O}^{{\prime} }}\otimes {{\mathbb{1}}}^{{X}_{I}^{{\prime} }\to {X}_{O}}\), with each incoming ancillary system being prepared in the state \({\left {i}_{X}\right\rangle }^{{X}_{I}^{{\prime} }}\) and the outgoing ancillary systems being measured in the computational bases. One obtains the deterministic correlation \(P({o}_{A},{o}_{B},{o}_{C}  {i}_{A},{i}_{B},{i}_{C})={\delta }_{{o}_{A},\neg {i}_{B}\wedge {i}_{C}}{\delta }_{{o}_{B},\neg {i}_{C}\wedge {i}_{A}}{\delta }_{{o}_{C},\neg {i}_{A}\wedge {i}_{B}}\), which was shown to violate causal inequalities in ref. ^{11}.
An example of a causal inequality that is violated by this correlation is
which was derived in ref. ^{8}. (It corresponds to Eq. (26) given there, with 0 and 1 exchanged for all inputs and outputs). Here, we find that I_{1} = −1.
Interestingly, for that particular process with these particular local operations, all operations involved in the tripartite construction simply take computational basis states to computational basis states. These can be understood as deterministic operations between classical random variables, rather than unitary operations between quantum systems. In Supplementary Note 4B, we explain this in more detail.
All things considered, our main result is thus that there exist classical, deterministic circuits, composed of operations between timelocal variables, which, when described in terms of suitable timedelocalised variables, correspond to classical, deterministic processes that violate causal inequalities.
Noncausal correlations between timedelocalised variables
After having established that this realisation of a noncausal process exists, we now turn to the question of what we should conclude from the fact that a causal inequality can be violated in such a situation. The general reasoning behind causal inequalities is similar to that behind Bell inequalities—one considers certain assumptions which restrict the correlations that can arise from some experiment, and their violation then implies that not all of these assumptions are satisfied. To determine whether a causal inequality violation is a meaningful deviceindependent witness of causal indefiniteness, one must therefore clarify whether the assumptions underlying causal inequalities are plausible or compelling in the setting under consideration—a question that is subtle, notably in regimes of relativistic quantum information and quantum gravity^{46,47}, but, as it will turn out, also in the standard quantum situations we consider here. In the following, we will therefore analyse our result in this regard, and argue that causal inequalities are indeed a meaningful concept to show the absence of a definite causal order between the timedelocalised variables we identified.
In the original approach developed in ref. ^{2}, one firstly assumes that the events involved in the experiment take place in a causal order (which, in general, can be dynamical and subject to randomness^{6,8}). With respect to this causal order, there are two further assumptions that enter the derivation of causal inequalities. Firstly, the classical inputs which the parties receive are subject to free choice. Technically speaking, this means that they cannot be correlated with any properties pertaining to their causal past or elsewhere (see Methods). Secondly, the parties operate in closed laboratories. That is, intuitively speaking, they open their laboratory once to let a physical system enter, interact with it and open their laboratory once again to send out a physical system, which provides the sole means of information exchange between the local variables and the rest of the experiment. More formally, the closed laboratory assumption says that, for each party X, any causal influence from the setting variable I_{X}, which describes its classical input, to any other variable, except the variable O_{X} which describes its classical outcome, has to pass through the outgoing variable X_{O}. Similarly, any causal influence to O_{X} from any other variable except I_{X} has to pass through X_{I}. Furthermore, X_{I} is in the causal past of X_{O} (see Methods). In order to clarify whether the violation of a causal inequality discovered here is meaningful and interesting, the question that we need to address is whether one would naturally expect the free choice and closed laboratory assumptions to be satisfied in our scenario with timedelocalised (classical) variables, or whether one of them is manifestly violated.
In the Methods section “Causal inequality assumptions”, we formulate these assumptions, for the multipartite case, in a way that is suitable for our timedelocalised setting, namely directly in terms of the variables involved (rather than in terms of events as in ref. ^{2}), and show that they indeed imply that causal inequalities must be respected. Our formulation provides a strengthening of the original derivation in ref. ^{2} by relaxing the closed laboratories assumption—rather than imposing that the incoming variable X_{I} is always in the causal past of the outgoing variable X_{O}, we only require this constraint to hold for at least one particular value of the corresponding setting variable I_{X} (see Methods). As we discuss in the following, this formulation of the assumptions is directly motivated by the observable causal relations between the variables of interest. Thus, the violation of a causal inequality in the experiment can be seen as a compelling, deviceindependent demonstration of the nonexistence of a possibly dynamical and random causal order between the variables.
The causal relations between the incoming and outgoing variables X_{I} and X_{O}, as well as the setting and outcome variables I_{X} and O_{X}, X = A, B, C, can be graphically represented by a directed graph as in Fig. 6, where the arrows describe direct causal influences.
In the causal structure in Fig. 6, the variables I_{X} are root variables and hence they can only be correlated with other variables as a result of causal influence from them to these other variables. It is thus natural to assume the same would be true if there existed an explanation of the correlations in terms of a definite causal order, which legitimates the free choice assumption.
Regarding the closed laboratory assumption, in the graph of Fig. 6, any causal influence from I_{X} to variables other than O_{X} and X_{O} is mediated, or screened off, by X_{O}. Similarly, any influence onto O_{X} by variables other than I_{X} and X_{I} is mediated by X_{I}. It is natural to assume that these constraints would also hold in any potential explanation of the correlations in terms of a definite causal order. Finally, the causal diagram displays causal influence from X_{I} to X_{O}. Note that this causal influence from X_{I} to X_{O} can be turned on or off depending on the value of the setting variable I_{X}. This is precisely the reason why we introduced the weakened form of the closed laboratory assumption described above, which indeed allows for X_{O} to be inside or outside of the causal future of X_{I}, depending on the value of I_{X}.
To summarise, we have shown that there is a set of natural assumptions about the possible underlying causal orders between the variables of interest in our experiment, which are directly motivated by the observable causal relations between these variables, and which imply that the correlations in the experiment would need to respect causal inequalities. The observable violation of a causal inequality in the experiment thus implies that an underlying causal order compatible with these assumptions cannot exist.
Are there any considerations that would lead us to drop one assumption over another in this type of experiment? In particular, could it be that, in spite of the outlined considerations about the observable causal relations, a more careful inspection of the temporal description of the experiment would reveal that it is in fact the free choice or closed laboratory assumptions that is violated, as opposed to the existence of a causal order per se? In the discussion below and in Supplementary Note 6, we analyse this question and argue that if the hypothetical causal order is expected to be imposed by spatiotemporal relations, it is the existence of causal order per se that seems violated, since the variables of interest do not admit an effective localisation in spacetime.
Discussion
A central question in the study of quantum causality is which processes with indefinite causal order have a realisation within standard quantum theory. In order to address this question, it is first of all necessary to clarify what it means for a causally indefinite process to have a standard quantum theoretical realisation, a question that is subtle and has led to a lot of controversy. An answer to this question is provided by the concept of timedelocalised subsystems, which establishes a bridge between the standard quantum theoretical description of the scenarios under consideration and their description in the process matrix framework, in which the notion of indefinite causal order is formalised. Prior to our work, it had been known that indefinite causal order can be realised on systems that are timedelocalised in a coherently controlled manner—that is, intuitively speaking, the input and output systems of each party effectively reduce to one or another timelocal system, conditionally on the state of a control quantum system. Here, we showed that this paradigm does not encompass all possibilities, and that standard quantum theory also allows for more radical ways to realise indefinite causal order processes. Notably, there exist processes that have realisations on timedelocalised subsystems and that violate causal inequalities, a feature that is generally believed to be impossible within standard (quantum) physics^{14}. We analysed a concrete tripartite example, for which it turned out that the situation can entirely be understood in terms of classical variables, rather than quantum systems. There, Alice’s and Bob’s input and output variables are timedelocalised in a classically controlled way, while the situation for Charlie is quite different. From the point of view of the temporal description of the experiment, one timelocal instance of Charlie’s operation is applied in the beginning of the circuit, which may be reversed and reapplied at the end of the circuit, conditionally on the output of Alice and Bob. We then analysed this causal inequality violation with regard to the assumptions that underlie the derivation of causal inequalities, and found that the free choice and closed laboratory assumptions are not manifestly violated, which makes causal inequalities a meaningful deviceindependent concept to qualify these realisations as incompatible with a definite causal order.
Let us further elaborate on the subtleties that this analysis involves, in particular with respect to the closed laboratory assumption (see a more detailed discussion in Supplementary Note 6). From an intuitive reading of the circuit in Fig. 5, one may be tempted to say that Charlie acts multiple times or receives several inputs, and sends out several outputs. At first sight, this seems to violate the closed laboratory assumption, which essentially stipulates that each party is involved in a single round of information exchange, where they receive information about the past through the input variable X_{I} and subsequently send out information into the future through the output variable X_{O}. However, it is crucial to realise that the causal inequality assumptions concern concrete variables (or quantum systems), which in our case we have explicitly specified, and which are not the same as what one might intuitively assume if one thinks of this experiment as involving three laboratories existing through time that exchange information with each other. In particular, the parties Alice, Bob and Charlie must be understood abstractly as agents who control the parameters that describe the operations taking place on the timedelocalised variables. As such, they indeed apply their operations once and only once on the pairs of input and output variables we have identified. To say that the closed laboratory assumption is violated, one would need to come up with an account for the process in terms of variables which are embedded into a causal order, but for which the closed laboratory assumption fails. We are not aware of any explanation in terms of the timelocal variables in the temporal circuit and the causal order defined by their spatiotemporal relations (or any other operationally meaningful variables) where this is the case. In particular, the aboveoutlined intuitive reading of the circuit, with the operations being effectively localised in time, conditioned on other variables in the process, while meaningful for quantumly controlled timedelocalised operations, does not make operational sense in our case (as it would mean that some future parties can influence what has happened in the past, see Supplementary Note 6). In Supplementary Note 6, we show that, for some of the timedelocalised variables we identified, there do not exist timelocal variables that take their value, meaning that they do not admit any effective localisation in time.
The further implications of this finding are yet to be unravelled, and raise various open questions. In a more general sense, there is a causal explanation for how these correlations in our process come aboutnamely, precisely the tripartite circuit realisation we found. This raises the question of whether and how the concept of causal inequalities in itself could be revised or modified. For instance, could there be a notion of causal process which is more relaxed, and which includes such possibilities?
What other processes beyond the classes considered here have a realisation on timedelocalised subsystems, and what other types of timedelocalisation would this involve? Could it be that any indefinite causal order process admits such a realisation, or are there counterexamples? The proof for unitarily extended tripartite processes is crucially based on the fact that the bipartite unitarily extended process resulting from fixing one of the operations has a particular standard form—namely, a variation of the quantum switch^{40,41}. Establishing whether a similar standard form exists for unitarily extended processes with more than two parties could give insight into whether the constructions presented here can be generalised to more parties.
Note that there are also unitary extensions of bipartite processes—i.e., variations of the quantum switch—that have realisations of the kind considered here, with one of the operations being reversed and reapplied (for instance, one obtains such a realisation when one fixes Alice’s or Bob’s operation in the circuit of Fig. 5). This raises the question of whether, conversely, the process considered in this work could have an alternative, more intuitive interpretation as a superposition of processes with different definite causal orders in some sense (although it cannot be achieved by direct multipartite generalisations of the quantum switch^{13}). The decomposition of this process into a direct sum of causal unitary processes shown in^{40} may offer insights into this question.
Finally, in the way the process framework was originally conceived, the operations performed by the parties were imagined to be local from the point of view of some local notion of time for each party. Can we conceive of a notion of a quantum temporal reference frame with respect to which the timedelocalised variables considered here would look local, and what implications would this have for our understanding of the spacetime causal structure in which these experiments are embedded? In view of the fact that the example considered here is purely classical, the question arises of which part of a noncausal process is actually related to the quantumness of causal relations. On the practical side, an obvious question is whether our finding could unveil new applications. For instance, could we use such timedelocalised variables for new cryptographic or other informationprocessing protocols?
Methods
The Choi isomorphism and the link product
The Choi isomorphism^{48} is a convenient way to represent linear maps between vector spaces as vectors themselves, and linear maps between spaces of operators as operators themselves. In order to define it, we choose for each Hilbert space \({{{{{{\mathcal{H}}}}}}}^{Y}\) a fixed orthonormal, socalled computational basis \({\{{\left i\right\rangle }^{Y}\}}_{i}\). For a Hilbert space \({{{{{{\mathcal{H}}}}}}}^{YZ}={{{{{{\mathcal{H}}}}}}}^{Y}\otimes {{{{{{\mathcal{H}}}}}}}^{Z}\), with computational bases \({\{{\left i\right\rangle }^{Y}\}}_{i}\) of \({{{{{{\mathcal{H}}}}}}}^{Y}\) and \({\{{\left j\right\rangle }^{Z}\}}_{j}\) of \({{{{{{\mathcal{H}}}}}}}^{Z}\), respectively, the computational basis is taken to be \({\{{\left i,j\right\rangle }^{YZ}: = {\left i\right\rangle }^{Y}\otimes {\left j\right\rangle }^{Z}\}}_{i,j}\). We then define the pure Choi representation of a linear operator \(V:{{{{{{\mathcal{H}}}}}}}^{Y}\to {{{{{{\mathcal{H}}}}}}}^{Z}\) as
with \({\left.\left {\mathbb{1}}\right\rangle \right\rangle }^{YY}: = {\sum }_{i}{\left i\right\rangle }^{Y}\otimes {\left i\right\rangle }^{Y}\in {{{{{{\mathcal{H}}}}}}}^{Y}\otimes {{{{{{\mathcal{H}}}}}}}^{Y}\). Similarly, we define the (mixed) Choi representation of a linear map \({{{{{\mathcal{M}}}}}}:{{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{Y})\to {{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{Z})\) as
where \({{{{{{\mathcal{I}}}}}}}^{Y}\) denotes the identity map on \({{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{Y})\).
The link product^{32,33} is a tool which allows one to compute the Choi representation of a composition of maps in terms of the Choi representation of the individual maps. Consider two tensor product Hilbert spaces \({{{{{{\mathcal{H}}}}}}}^{XY}={{{{{{\mathcal{H}}}}}}}^{X}\otimes {{{{{{\mathcal{H}}}}}}}^{Y}\) and \({{{{{{\mathcal{H}}}}}}}^{YZ}={{{{{{\mathcal{H}}}}}}}^{Y}\otimes {{{{{{\mathcal{H}}}}}}}^{Z}\) which share the same (possibly trivial) space factor \({{{{{{\mathcal{H}}}}}}}^{Y}\), and with nonoverlapping \({{{{{{\mathcal{H}}}}}}}^{X},{{{{{{\mathcal{H}}}}}}}^{Z}\). The link product of any two vectors \(\left a\right\rangle \in {{{{{{\mathcal{H}}}}}}}^{XY}\) and \(\left b\right\rangle \in {{{{{{\mathcal{H}}}}}}}^{YZ}\) is defined (with respect to the computational basis \({\{{\left i\right\rangle }^{Y}\}}_{i}\) of \({{{{{{\mathcal{H}}}}}}}^{Y}\)) as^{13}
with \({\left {a}_{i}\right\rangle }^{X}: = ({{\mathbb{1}}}^{X}\otimes {\left\langle i\right }^{Y})\left a\right\rangle \in {{{{{{\mathcal{H}}}}}}}^{X}\) and \({\left {b}_{i}\right\rangle }^{Z}: = ({\left\langle i\right }^{Y}\otimes {{\mathbb{1}}}^{Z})\left b\right\rangle \in {{{{{{\mathcal{H}}}}}}}^{Z}\). Similarly, the link product of any two operators \(A\in {{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{XY})\) and \(B\in {{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{YZ})\) is defined as^{32,33}
with \({A}_{i{i}^{{\prime} }}^{X}: = ({{\mathbb{1}}}^{X}\otimes {\left\langle i\right }^{Y})A({{\mathbb{1}}}^{X}\otimes {\left {i}^{{\prime} }\right\rangle }^{Y})\in {{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{X})\) and \({B}_{i{i}^{{\prime} }}^{Z}: = ({\left\langle i\right }^{Y}\otimes {{\mathbb{1}}}^{Z})A({\left {i}^{{\prime} }\right\rangle }^{Y}\otimes {{\mathbb{1}}}^{Z})\in {{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{Z})\).
The link products thus defined are commutative (up to a reordering of the tensor products), and associative provided that each constituent Hilbert space appears at most twice^{13,33}. For \(\left a\right\rangle \in {{{{{{\mathcal{H}}}}}}}^{X}\) and \(\left b\right\rangle \in {{{{{{\mathcal{H}}}}}}}^{Z}\), or \(A\in {{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{X})\) and \(B\in {{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{Z})\) in distinct, nonoverlapping spaces, they reduce to tensor products (\(\left a\right\rangle * \left b\right\rangle=\left a\right\rangle \otimes \left b\right\rangle \) or A*B = A ⊗ B). For \(\left a\right\rangle,\left b\right\rangle \in {{{{{{\mathcal{H}}}}}}}^{Y}\), or \(A,B\in {{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{Y})\) in the same spaces, they reduce to scalar products (\(\left a\right\rangle * \left b\right\rangle={\sum }_{i}\langle i  a\rangle \langle i  b\rangle={\left a\right\rangle }^{T}\left b\right\rangle \) or \(A * B={{{{{\rm{Tr}}}}}}[{A}^{T}B]\)).
For two linear operators \({V}_{1}:{{{{{{\mathcal{H}}}}}}}^{X}\to {{{{{{\mathcal{H}}}}}}}^{{X}^{{\prime} }Y}\) and \({V}_{2}:{{{{{{\mathcal{H}}}}}}}^{YZ}\to {{{{{{\mathcal{H}}}}}}}^{{Z}^{{\prime} }}\), the pure Choi representation of the composition \(V: = ({{\mathbb{1}}}^{{X}^{{\prime} }}\otimes {V}_{2})({V}_{1}\otimes {{\mathbb{1}}}^{Z}):{{{{{{\mathcal{H}}}}}}}^{XZ}\to {{{{{{\mathcal{H}}}}}}}^{{X}^{{\prime} }{Z}^{{\prime} }}\) is obtained, in terms of the pure Choi representations \(\left.\left {V}_{1}\right\rangle \right\rangle \in {{{{{{\mathcal{H}}}}}}}^{X{X}^{{\prime} }Y}\) and \(\left.\left {V}_{2}\right\rangle \right\rangle \in {{{{{{\mathcal{H}}}}}}}^{YZ{Z}^{{\prime} }}\) of the individual operators V_{1} and V_{2}, as
Similarly, for two linear maps \({{{{{{\mathcal{M}}}}}}}_{1}:{{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{X})\to {{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{{X}^{{\prime} }Y})\) and \({{{{{{\mathcal{M}}}}}}}_{2}:{{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{YZ})\to {{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{{Z}^{{\prime} }})\) the Choi representation of the composition \({{{{{\mathcal{M}}}}}}: = ({{{{{{\mathcal{I}}}}}}}^{{X}^{{\prime} }}\otimes {{{{{{\mathcal{M}}}}}}}_{2})\circ ({{{{{{\mathcal{M}}}}}}}_{1}\otimes {{{{{{\mathcal{I}}}}}}}^{Z}):{{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{XZ})\to {{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{{X}^{{\prime} }{Z}^{{\prime} }})\) is obtained, in terms of the Choi representations of the individual maps \({M}_{1}\in {{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{X{X}^{{\prime} }Y})\) and \({M}_{2}\in {{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{YZ{Z}^{{\prime} }})\) of \({{{{{{\mathcal{M}}}}}}}_{1}\) and \({{{{{{\mathcal{M}}}}}}}_{2}\), as
Another property of the link product, which can easily be verified from its definition, is that for any \(\left a\right\rangle \in {{{{{{\mathcal{H}}}}}}}^{XY},\left b\right\rangle \in {{{{{{\mathcal{H}}}}}}}^{YZ}\) and any unitary \(U:{{{{{{\mathcal{H}}}}}}}^{Y}\to {{{{{{\mathcal{H}}}}}}}^{{Y}^{{\prime} }}\), it holds that
Similarly, for any \(A\in {{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{XY}),B\in {{{{{\mathcal{L}}}}}}({{{{{{\mathcal{H}}}}}}}^{YZ})\) and any unitary \(U:{{{{{{\mathcal{H}}}}}}}^{Y}\to {{{{{{\mathcal{H}}}}}}}^{{Y}^{{\prime} }}\), it holds that
This is precisely the property we use in the main text when changing the subsystem description of a circuit. Namely, it is due to this property that the overall composition of two circuit fragments remains the same when we compose one fragment with certain isomorphisms (i.e., unitary transformations) defining new subsystems, and the complementary fragment with the inverses of these isomorphisms.
Unitary extensions of bipartite processes on timedelocalised subsystems
In summary, the bipartite result says that for any unitarily extended bipartite process, there exists a temporally ordered quantum circuit, with operations that depend on the local operations U_{A} and U_{B} applied in the process, which precisely corresponds to the situation considered in the process matrix framework, with one instance of each U_{A} and U_{B} composed with the process matrix in a cyclic circuit, when described in terms of a suitable choice of timedelocalised subsystems.
Formally, the bipartite result can be stated as follows.
Proposition 2
Consider a unitary extension of a bipartite process, described by a process vector \(\left.\left U\right\rangle \right\rangle \in {{{{{{\mathcal{H}}}}}}}^{{P}_{O}{A}_{IO}{B}_{IO}{F}_{I}}\), composed with unitary local operations \({U}_{A}:{{{{{{\mathcal{H}}}}}}}^{{A}_{I}{A}_{I}^{{\prime} }}\to {{{{{{\mathcal{H}}}}}}}^{{A}_{O}{A}_{O}^{{\prime} }}\) and \({U}_{B}:{{{{{{\mathcal{H}}}}}}}^{{B}_{I}{B}_{I}^{{\prime} }}\to {{{{{{\mathcal{H}}}}}}}^{{B}_{O}{B}_{O}^{{\prime} }}\). For any such process, the following exist.

1.
A temporal circuit as in Fig. 7, in which U_{A} is applied on some systems A_{I} and A_{O} at a definite time, preceded and succeded respectively by two unitary circuit operations \({\omega }_{1}({U}_{B}):{{{{{{\mathcal{H}}}}}}}^{{B}_{I}^{{\prime} }{P}_{O}}\to {{{{{{\mathcal{H}}}}}}}^{{A}_{I}E}\) and \({\omega }_{2}({U}_{B}):{{{{{{\mathcal{H}}}}}}}^{{A}_{O}E}\to {{{{{{\mathcal{H}}}}}}}^{{B}_{O}^{{\prime} }{F}_{I}}\) that depend on U_{B}.

2.
Isomorphisms \({J}_{{{{{{\rm{in}}}}}}}:{{{{{{\mathcal{H}}}}}}}^{{B}_{I}Z}\to {{{{{{\mathcal{H}}}}}}}^{{A}_{O}{P}_{O}}\) and \({J}_{{{{{{\rm{out}}}}}}}:{{{{{{\mathcal{H}}}}}}}^{{A}_{I}{F}_{I}}\to {{{{{{\mathcal{H}}}}}}}^{{B}_{O}\bar{Z}}\), such that, with respect to the subsystem B_{I} of A_{O}P_{O} and the subsystem B_{O} of A_{I}F_{I} that these isomorphisms define, the circuit in Fig. 7 takes the form of a cyclic circuit composed of U, U_{A} and U_{B}, as in the process matrix framework (Fig. 8).
Here, we outline the main points of the proof. All technical details and calculations are given in Supplementary Note 2.
Outline of proof
The existence of a temporal circuit with the form of Fig. 7 is shown in Supplementary Note 2A. It follows from the fact that any unitary extension of a oneparty process can be implemented as a fixedorder circuit or quantum comb^{32,33}, in which the party applies its operation at a definite time. For a unitary extension of a bipartite process, one can therefore find a fixedorder circuit in which one of the parties acts at a definite time, and which is composed of circuit operations that depend on the operation of the other party.
In Supplementary Note 2B, we show that the unitary U which defines the process isomorphically maps some subsystem of A_{O}P_{O} to B_{I}, and B_{O} to some subsystem of A_{I}F_{I}. The corresponding isomorphisms \({J}_{{{{{{\rm{in}}}}}}}:{{{{{{\mathcal{H}}}}}}}^{{B}_{I}Z}\to {{{{{{\mathcal{H}}}}}}}^{{A}_{O}{P}_{O}}\) and \({J}_{{{{{{\rm{out}}}}}}}:{{{{{{\mathcal{H}}}}}}}^{{A}_{I}{F}_{I}}\to {{{{{{\mathcal{H}}}}}}}^{{B}_{O}\bar{Z}}\) (where Z and \(\bar{Z}\) are appropriate complementary subsystems) can be taken to define an alternative description of the circuit in Fig. 7 in terms of timedelocalised subsystems, since there, P_{O}, A_{I}, A_{O} and F_{I} are timelocal wires.
In Supplementary Note 2C, we change to the description of the circuit in terms of these timedelocalised subsystems. For that purpose, we decompose the circuit into the red and blue circuit fragment shown in Fig. 8. By construction, when composed with J_{in} and J_{out}, the red fragment consists of precisely one application of \({U}_{B}:{{{{{{\mathcal{H}}}}}}}^{{B}_{I}{B}_{I}^{{\prime} }}\to {{{{{{\mathcal{H}}}}}}}^{{B}_{O}{B}_{O}^{{\prime} }}\), in parallel to an identity channel from Z to \(\bar{Z}\) (Fig. 8a). The blue fragment, which is just the operation U_{A}, needs to be composed with the inverse isomorphisms \({J}_{{{{{{\rm{in}}}}}}}^{{{\dagger}} }\) and \({J}_{{{{{{\rm{out}}}}}}}^{{{\dagger}} }\) so that the overall, global transformation implemented by the circuit remains the same (Fig. 8b). In the new description of the circuit of Fig. 7 in terms of these subsystems, one thus obtains a cyclic circuit as on the lefthand side of Fig. 8c).
The final step is to note that the composition of the inverse isomorphisms \({J}_{\,{{\mbox{in}}}\,}^{{{\dagger}} }\) and \({J}_{\,{{\mbox{out}}}\,}^{{{\dagger}} }\) with the identity channel \({{\mathbb{1}}}^{Z\to \bar{Z}}\) over the systems Z and \(\bar{Z}\) is precisely the unitary operation U that defines the process. Therefore, in this coarsegrained description with respect to the systems \({P}_{O},{A}_{IO}^{({\prime} )},{B}_{IO}^{({\prime} )}\), and F_{I}, the circuit indeed consists of three transformations \({U}_{A}:{{{{{{\mathcal{H}}}}}}}^{{A}_{I}{A}_{I}^{{\prime} }}\to {{{{{{\mathcal{H}}}}}}}^{{A}_{O}{A}_{O}^{{\prime} }},{U}_{B}:{{{{{{\mathcal{H}}}}}}}^{{B}_{I}{B}_{I}^{{\prime} }}\to {{{{{{\mathcal{H}}}}}}}^{{B}_{O}{B}_{O}^{{\prime} }}\) and \(U:{{{{{{\mathcal{H}}}}}}}^{{P}_{O}{A}_{O}{B}_{O}}\to {{{{{{\mathcal{H}}}}}}}^{{F}_{I}{A}_{I}{B}_{I}}\) that are composed in a cyclic circuit as in the process matrix picture (see the righthand side of Fig. 8c). In other words, it is precisely that structure that happens on the subsystems with respect to which we chose to describe the circuit. This establishes the bipartite result.
Applying the bipartite constructions presented here to the particular case of the quantum switch leads to an asymmetric implementation with Alice performing a timelocal operation and Bob’s operation being timedelocalised through coherent control of the times at which it is applied. For symmetric implementations in which both Alice’s and Bob’s operation are timedelocalised, a similar argument can be made^{29}.
Causal inequality assumptions
A causal order between the elements of some set \({{{{{\mathcal{S}}}}}}\) is formally described by a strict partial order (SPO) on \({{{{{\mathcal{S}}}}}}\)^{2,6}. A SPO is a binary relation ≺ , which, for all \(X,Y,Z\in {{{{{\mathcal{S}}}}}}\), satisfies irreflexivity (not X ≺ X) and transitivity (if X ≺ Y and Y ≺ Z, then X ≺ Z). (Note that irreflexivity and transitivity together imply asymmetry, i.e., if X ≺ Y, then not Y ≺ X.) If X ≺ Y, we will say that X is in the causal past of Y (equivalently, Y is in the causal future of X). For X ≠ Y and not X ≺ Y, we will use the notation X ⋠ Y, and the terminology X is not in the causal past of Y (equivalently, Y is not in the causal future of X). If X ⋠ Y and Y ⋠ X, we will say that X is in the causal elsewhere of Y^{49} (sometimes also termed X is not causally connected to Y, or X is causally disconnected from Y). For subsets \({{{{{{\mathcal{S}}}}}}}^{{\prime} }\subset {{{{{\mathcal{S}}}}}}\), we will use the shorthand notation \(X\preceq {{{{{{\mathcal{S}}}}}}}^{{\prime} }\) to denote that \(\forall \,Y\in {{{{{{\mathcal{S}}}}}}}^{{\prime} },X\preceq Y\). We furthermore define the causal past of X as the set \({{{{{{\mathcal{P}}}}}}}_{X}: = \{Y\in {{{{{\mathcal{S}}}}}}  Y\prec X\}\), the causal future of X as \({{{{{{\mathcal{F}}}}}}}_{X}: = \{Y\in {{{{{\mathcal{S}}}}}}  X\prec Y\}\) and the causal elsewhere of X as \({{{{{{\mathcal{E}}}}}}}_{X}: = \{Y\in {{{{{\mathcal{S}}}}}}  Y\preceq X\,\,{{\mbox{and}}}\,\,X\preceq Y\}\). Also, note that a SPO on \({{{{{\mathcal{S}}}}}}\) naturally induces a SPO on any subset of \({{{{{\mathcal{S}}}}}}\).
The variables involved in the process under consideration are the timedelocalised incoming and outgoing variables A_{I}, A_{O}, B_{I}, B_{O}, C_{I}, C_{O}, as well as the settings and outcomes, which can be described by random variables I_{A}, I_{B}, I_{C} (with values i_{A}, i_{B}, i_{C}, respectively) and O_{A}, O_{B}, O_{C} (with values o_{A}, o_{B}, o_{C}, respectively). We will abbreviate the set of all these variables to Γ:= {A_{I}, A_{O}, B_{I}, B_{O}, C_{I}, C_{O}, I_{A}, O_{A}, I_{B}, O_{B}, I_{C}, O_{C}}. The assumption that the correlations P(o_{A}, o_{B}, o_{C}∣i_{A}, i_{B}, i_{C}) arise from a situation in which these variables occur in a (generally probabilistic and dynamical) causal order can be formalised as follows.
Causal order assumption. There exists a random variable which takes values κ(Γ) in the possible strict partial orders on the set Γ, and a joint probability distribution P(o_{A}, o_{B}, o_{C}, κ(Γ)∣i_{A}, i_{B}, i_{C}), which, when marginalised over that variable, yields the correlations P(o_{A}, o_{B}, o_{C}, ∣i_{A}, i_{B}, i_{C}) observable in the process, i.e.,
This probability distribution satisfies the following two conditions.

1.
Free choice. The settings I_{A}, I_{B} and I_{C} are assumed to be freely chosen, which means that they cannot be correlated with any properties pertaining to their causal past or elsewhere. That is, the probability for their causal past and elsewhere to consist of certain variables, for the variables in these sets to have a certain causal order, and for the outcome variables in these sets to take certain values, cannot depend on the respective setting. Formally, with respect to I_{A}, for any (disjoint) subsets \({{{{{\mathcal{Y}}}}}}\) and \({{{{{\mathcal{Z}}}}}}\) of Γ\{I_{A}}, and any causal order \(\kappa ({{{{{\mathcal{Y}}}}}}\cup {{{{{\mathcal{Z}}}}}})\) on the variables in \({{{{{\mathcal{Y}}}}}}\cup {{{{{\mathcal{Z}}}}}}\), the following must hold:
$$P({o}^{{{{{{\mathcal{Y}}}}}}},{o}^{{{{{{\mathcal{Z}}}}}}},{{{{{{\mathcal{P}}}}}}}_{{I}_{A}} ={{{{{\mathcal{Y}}}}}},{{{{{{\mathcal{E}}}}}}}_{{I}_{A}}={{{{{\mathcal{Z}}}}}},\kappa ({{{{{\mathcal{Y}}}}}}\cup {{{{{\mathcal{Z}}}}}})  {i}_{A},{i}_{B},{i}_{C}) \\ =P({o}^{{{{{{\mathcal{Y}}}}}}},{o}^{{{{{{\mathcal{Z}}}}}}},{{{{{{\mathcal{P}}}}}}}_{{I}_{A}}={{{{{\mathcal{Y}}}}}},{{{{{{\mathcal{E}}}}}}}_{{I}_{A}}={{{{{\mathcal{Z}}}}}},\kappa ({{{{{\mathcal{Y}}}}}}\cup {{{{{\mathcal{Z}}}}}})  {i}_{B},{i}_{C}).$$(16)Here, by \(P({o}^{{{{{{\mathcal{Y}}}}}}},{o}^{{{{{{\mathcal{Z}}}}}}},{{{{{{\mathcal{P}}}}}}}_{{I}_{A}}={{{{{\mathcal{Y}}}}}},{{{{{{\mathcal{E}}}}}}}_{{I}_{A}}={{{{{\mathcal{Z}}}}}},\kappa ({{{{{\mathcal{Y}}}}}}\cup {{{{{\mathcal{Z}}}}}})  {i}_{A},{i}_{B},{i}_{C})\), we denote the probability that is obtained from P(o_{A}, o_{B}, o_{C}, κ(Γ)∣i_{A}, i_{B}, i_{C}) by marginalising over all \({O}_{X}\,\notin \,{{{{{\mathcal{Y}}}}}}\cup {{{{{\mathcal{Z}}}}}}\), and by summing over all κ(Γ) that satisfy the specified constraints—that is, all κ(Γ) for which the causal past \({{{{{{\mathcal{P}}}}}}}_{{I}_{A}}\) of I_{A} is \({{{{{\mathcal{Y}}}}}}\), the causal elsewhere \({{{{{{\mathcal{E}}}}}}}_{{I}_{A}}\) of I_{A} is \({{{{{\mathcal{Z}}}}}}\), and the causal order on the subset \({{{{{\mathcal{Y}}}}}}\cup {{{{{\mathcal{Z}}}}}}\) is \(\kappa ({{{{{\mathcal{Y}}}}}}\cup {{{{{\mathcal{Z}}}}}})\). The free choice assumption is that this probability is independent of the value of I_{A}. The analogous conditions must hold with respect to I_{B} and I_{C}.

2.
Closed laboratories. The second constraint is the closed laboratory assumption, which says, intuitively speaking, that causal influence from I_{A} to any other variable except O_{A} has to pass through A_{O}; that, similarly, any causal influence to O_{A} from any other variable except I_{A} has to pass through A_{I}; and that A_{I} is in the causal past of A_{O} (and analogously for B and C). Note that, in the original derivation of causal inequalities^{2}, it was assumed that X_{I} ≺ X_{O} always holds. Here, we weaken this assumption by requiring that this constraint only holds for at least one particular value of the corresponding setting variable I_{X}. The reason is that this weakened form of the assumption (unlike the stronger assumption of X_{I} ≺ X_{O} regardless of the value of I_{X}) is directly motivated by the observable causal relations in our situation with timedelocalised variables (see the discussion in the main text).
This closed laboratory assumption can be formalised as a constraint on the possible causal orders as follows.
Furthermore, there exists at least one value \({i}_{A}^{ * }\) of I_{A} for which A_{I} ≺ A_{O} with certainty, that is
The analogous conditions must be satisfied for B and C.
We show in Supplementary Note 5 that this causal order assumption—notably, even with the weakened form of the closed laboratory condition we introduced—implies that the correlations P(o_{A}, o_{B}, o_{C}∣i_{A}, i_{B}, i_{C}) that are established in the process must be causal^{6,7,8}. Such correlations form a polytope, whose facets precisely define causal inequalities^{6,7,8}.
(Note furthermore that we could similarly weaken the assumption that O_{A} is always in the causal future of A_{I}. This would however change nothing about the argument, and the proof from Supplementary Note 5 would go through in the same way).
Here, we presented the argument in the classical case for concreteness, but it can be readily extended to a quantum process, or even an abstract process^{6} possibly compatible with more general operational probabilistic theories (OPTs)^{37,38}, where there is no analogue of the classical variables X_{I} and X_{O}. Indeed, in the general case all elements of the argument remain the same, except that the objects X_{I} and X_{O} over which the partial order is assumed would be general systems rather than classical variables (I_{X} and O_{X} will remain classical). Moreover, the argument applies analogously for any number of parties, so we have assumptions applicable to the most general case of a process.
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Acknowledgements
This publication was made possible through the support of the ID# 61466 grant and ID# 62312 grant from the John Templeton Foundation, as part of the project https://www.templeton.org/grant/thequantuminformationstructureofspacetimeqisssecondphase ‘The Quantum Information Structure of Spacetime’ (QISS). The opinions expressed in this project/publication are those of the author(s) and do not necessarily reflect the views of the John Templeton Foundation. This work was supported by the Program of Concerted Research Actions (ARC) of the Université libre de Bruxelles and by the French National Research Agency through its “Investissements d’avenir” (ANR15IDEX02) program and the ANR22CE470012 project. J.W. is supported by the Chargé de Recherche fellowship of the Fonds de la Recherche Scientifique FNRS (F.R.S.FNRS). O.O. is a Research Associate of the Fonds de la Recherche Scientifique (F.R.S.FNRS). Published with the support of the University Foundation of Belgium.
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Wechs, J., Branciard, C. & Oreshkov, O. Existence of processes violating causal inequalities on timedelocalised subsystems. Nat Commun 14, 1471 (2023). https://doi.org/10.1038/s41467023368933
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DOI: https://doi.org/10.1038/s41467023368933
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