Causal inequality

The general setting that we consider involves a number of experimenters—Alice, Bob and others—who reside in separate laboratories. At a given run of the experiment, each of them receives a physical system (for instance, a spin-1/2 particle) and performs operations on it (for example, measurements or rotations of the spin), after which she/he sends the system out of the laboratory. We assume that during the operations of each experimenter, the respective laboratory is isolated from the rest of the world—it is only opened for the system to come in and to go out, but between these two events it is kept closed. It is easy to see that, under this assumption, causal order puts a restriction on the way in which the parties can communicate during a given run. For instance, imagine that Alice can send a signal to Bob. (Formally, sending a signal (or signalling) is the existence of statistical correlations between a random variable that can be chosen by the sender and another one observed by the receiver.) As Bob can only receive a signal through the system entering his laboratory, this means that Alice must act on her system before that. But this implies that Bob cannot send a signal to Alice as each party receives a system only once. Therefore, bidirectional signalling is forbidden.

Consider, in particular, the following communication task to be performed by two parties, Alice and Bob. After a given party receives the system in her/his laboratory, she/he will have to toss a coin (or use any other means) to obtain a random bit. Denote the bits generated by Alice and Bob in this way by a and b, respectively. In addition, Bob will have to generate another random bit b′, whose value, 0 or 1, will specify their goal: if b′=0, Bob will have to communicate the bit b to Alice, whereas if b′=1, he will have to guess the bit a. Without loss of generality, we will assume that the parties always produce a guess, denoted by x and y for Alice and Bob, respectively, for the bit of the other (although the guess may not count depending on the value of b′). Their goal is to maximize the probability of success

If all events obey causal order, no strategy can allow Alice and Bob to exceed the bound

Indeed, as argued above, in any particular order of events, there can be at most unidirectional signalling between the parties, which means that at least one of the following must be true: Alice cannot signal to Bob or Bob cannot signal to Alice. Consider, for example, a case where Bob cannot signal to Alice. Then, if b−=1, they could in principle achieve up to P(y=a|b′−=1)=1 (for instance, if Alice operates on her system before Bob, she could encode information about the bit a in the system and send it to him). However, if b′−=0, the best guess that Alice can make is a random one, resulting in P(x=b|b′−=0)=1/2 (see Fig. 1a). Hence, the overall probability of success in this case will satisfy p succ ≤3/4. The same holds if Alice cannot signal to Bob. It is easy to see that no probabilistic strategy can increase the probability of success.

Figure 1: Strategy for accomplishing communication task by using processes with definite and indefinite causal order. (a) There exists a global background time according to which Alice's actions are strictly before Bob's. She sends her input a to Bob, who can read it out at some later time and give his estimate y=a. However, Bob cannot send his bit b to Alice as the system passes through her laboratory at some earlier time. Consequently, she can only make a random guess of Bob's bit. This results in a probability of success of 3/4. (b) If the assumption of a definite order is dropped, it is possible to devise a resource (that is, a process matrix W) and a strategy that enables a probability of success (see text). Full size image

Formally, the assumptions behind the causal inequality (2) can be summarized as follows:

Causal structure

The main events in the task (a system entering Alice's/Bob's laboratory, the parties obtaining the bits a, b and b′, and producing the guesses x and y) are localized in a causal structure. A causal structure (such as space-time) is a set of event locations equipped with a partial order that defines the possible directions of signalling. If A B, we say that A is in the causal past of B (or B is in the causal future of A). In this case, signalling from A to B is possible, but not from B to A. For more details on causal structures, see Supplementary Methods.

Free choice

Each of the bits a, b and b can only be correlated with events in its causal future (this concerns only events relevant to the task). We assume also that each of them takes values 0 or 1 with probability 1/2.

Closed laboratories

Alice's guess x can be correlated with Bob's bit b only if the latter is generated in the causal past of the system entering Alice's laboratory. Analogously, y can be correlated with a only if a is generated in the causal past of the system entering Bob's laboratory.

In the Supplementary Methods, we present a formal derivation of the inequality from these assumptions.Interestingly, we find that if the local laboratories are described by quantum mechanics, but no assumption about a global causal structure is made (Fig. 1b), it is in principle possible to violate the causal inequality in physical situations in which one would have all the reasons to believe that the bits are chosen freely and the laboratories are closed. This would imply that the assumption Causal structure does not hold.

Framework for local quantum mechanics

The most studied, almost epitomical, quantum correlations are the non-signalling ones, such as those obtained when Alice and Bob perform measurements on two entangled systems. Signalling quantum correlations exist as well, such as those arising when Alice operates on a system that is subsequently sent through a quantum channel to Bob who operates on it after that. The usual quantum formalism does not consider more general possibilities, as it does assume a global causal structure. Here, we want to drop the latter assumption while retaining the validity of quantum mechanics locally. For this purpose, we consider a multipartite setting of the type outlined earlier, where each party performs an operation on a system passing once through her/his laboratory, but we make no assumption about the spatio-temporal location of these experiments, not even that there exists a space-time or any causal structure in which they could be positioned (see Fig. 2). Our framework is thus based on the central premise of local quantum mechanics, which is to say the local operations of each party are described by quantum mechanics.

Figure 2: Local quantum experiments with no assumption of a pre-existing background time or global causal structure. Although the global causal order of events in the two laboratories is not fixed in advance and in general not even definite (here illustrated by the 'shifted' relative orientation of the two laboratories), the two agents, Alice and Bob, are each certain about the causal order of events in their respective laboratories. Full size image

More specifically, we assume that one party, say Alice, can perform all the operations she could perform in a closed laboratory, as described in the standard space-time formulation of quantum mechanics. These are defined as the set of quantum instruments26 with an input Hilbert space (the system coming in) and an output Hilbert space (the system going out). (The set of allowed quantum operations can be used as a definition of 'closed quantum laboratory' with no reference to a global causal structure.) A quantum instrument can most generally be realized by applying a joint unitary transformation on the input system plus an ancilla, followed by a projective measurement on part of the resulting joint system, which leaves the other part as an output. (From the point of view of each party, the input/output systems most generally correspond to two subsystems of the Hilbert space associated with the local laboratory, each considered at a different instant—the time of entrance and the time of exit, respectively—where the subsystems and the respective instants are independent of the choice of operation that connects them.) When Alice uses a given instrument, she registers one out of a set of possible outcomes, labelled by j=1,...,n. Each outcome induces a specific transformation from the input to the output, which corresponds to a completely positive (CP) trace-non-increasing map27 , where , is the space of matrices over a Hilbert space of dimension d X . The action of each on any matrix can be written as27 , where the matrices satisfy , ∀j. If the operation is performed on a quantum state described by a density matrix ρ, describes the updated state after the outcome j up to normalization, whereas the probability to observe this outcome is given by . The set of CP maps corresponding to all the possible outcomes of a quantum instrument has the property that is CP and trace-preserving (CPTP) or equivalently , which reflects the fact that the probability to observe any of the possible outcomes is unity. A CPTP map itself corresponds to an instrument with a single outcome that occurs with certainty.

In the case of more than one party, the set of local outcomes corresponds to a set of CP maps . A complete list of probabilities for all possible local outcomes will be called process. (It is implicitly assumed that the joint probabilities are non-contextual, namely that they are independent of any variable concerning the concrete implementation of the local CP maps. For example, the probability for a pair of maps to be realized should not depend on the particular set of possible CP maps associated with Alice's operation.) A process can be seen as an extension of the notion of state as a list of probabilities for detection results3 described by a positive operator-valued measure (POVM), which takes into account the transformation of the system after the measurement and can thus capture more general scenarios than just detection. Here, we will consider explicitly only the case of two parties (the generalization to arbitrarily many parties is straightforward). We want to characterize the most general probability distributions for a pair of outcomes i, j, corresponding to CP maps , to be observed, that is, to characterize all bipartite processes.

In quantum mechanics, operations obey a specific algebraic structure that reflects the operational relations between laboratory procedures3. For example, a probabilistic mixture of operations is expressed as a linear convex combination of CP maps. It can be shown (see Methods) that the only probabilities consistent with the algebraic structure of local quantum operations are bilinear functions of the CP maps and . Thus, the study of the most general bipartite quantum correlations reduces to the study of bilinear functions of CP maps.

It is convenient to represent CP maps by positive semi-definite matrices via the Choi-Jamiołkowsky (CJ) isomorphism28,29. The CJ matrix coresponding to a linear map is defined as , where is a (not normalized) maximally entangled state, the set of states is an orthonormal basis of is the identity map and T denotes matrix transposition (the transposition, absent in the original definition, is introduced for later convenience). Using this correspondence, the probability for two measurement outcomes can be expressed as a bilinear function of the corresponding CJ operators as follows:

where is a matrix in .

The matrix W should be such that probabilities are non-negative for any pair of CP maps . We require that this be true also for measurements in which the system interacts with any system in the local laboratory, including systems entangled with the other laboratory. This implies that must be positive semidefinite (see Methods). Furthermore, the probability for any pair of CPTP maps to be realized must be unity (they correspond to instruments with a single outcome). As a map is CPTP if and only if its CJ operator satisfies and (similarly for ), we conclude that all bipartite probabilities compatible with local quantum mechanics are generated by matrices W that satisfy

We will refer to a matrix that satisfies these conditions as a process matrix. Conditions equivalent to equations (4) and (5) were first derived as part of the definition of a 'quantum comb'30, an object that formalizes quantum networks. Combs, however, are subject to additional conditions fixing a definite causal order, which are not assumed here.

A process matrix can be understood as a generalization of a density matrix and equation (3) can be seen as a generalization of Born's rule. In fact, when the output systems A 2 , B 2 are taken to be one-dimensional (that is, each party performs a measurement after which the system is discarded), the expression above reduces to , where now are elements of local POVMs and is a quantum state. This implies that a quantum state shared by Alice and Bob is generally represented by the process matrix . Signalling correlations can also be expressed in terms of process matrices. For instance, the situation where Bob is given a state and his output is sent to Alice through a quantum channel , which gives , is described by , where is the CJ matrix of the channel from B 2 to A 1 .

The most general bipartite situation typically encountered in quantum mechanics (that is, one that can be expressed in terms of a quantum circuit) is a quantum channel with memory, where, say, Bob operates on one part of an entangled state and his output plus the other part is transferred to Alice through a channel. This is described by a process matrix of the form . Conversely, all process matrices of this form represent channels with memory30. This is the most general situation in which signalling from Alice to Bob is not possible, a relation that we will denote by in accord with the causal notation introduced earlier. Process matrices of this kind will be denoted by (note that for non-signalling processes, both and are true). As argued earlier, if all events are localized in a causal structure, and Alice and Bob perform their experiments inside closed laboratories, at most unidirectional signalling between the laboratories is allowed. In a definite causal structure, it may still be the case that the location of each event, and thus the causal relation between events, is not known with certainty. A situation where with probability 0≤q≤1 and with probability 1−q is represented by a process matrix of the form

We will call the processes of this kind causally separable (note that the decomposition (6) need not be unique as non-signalling processes can be included either in or in ). They represent the most general bipartite quantum processes for which the local experiments are performed in closed laboratories embedded in a definite causal structure. In particular, they generate the most general quantum correlations between measurements that take place at definite (though possibly unknown) instants of time. Clearly, according to the argument presented earlier, causally separable processes cannot be used by Alice an Bob to violate the causal inequality (2).

In the Supplementary Methods, we provide a complete characterization of process matrices via the terms allowed in their expansion in a Hilbert–Schmidt basis, which we relate to the possible directions of signalling they allow (see Fig. 3). We also provide possible interpretations of the terms that are not allowed in a process matrix (see Fig. 4 and Supplementary Fig. S1).

Figure 3: Terms appearing in a process matrix. A matrix satisfying condition (4) can be expanded as , where the set of matrices , with and for , provide a basis of . We refer to terms of the form as of the type A 1 , terms of the form as of the type A 1 A 2 and so on. In the Supplementary Information, we prove that a matrix satisfies condition (5) if it contains the terms listed in this table. Each of the terms can allow signalling in at most one direction and can be realized in a situation in which either Bob's actions are not in the causal past of Alice's or vice versa . The most general unidirectional process is a quantum channel with memory. Measurements of bipartite states that lead to non-signalling probabilities can be realized in both situations. The most general process matrix can contain terms from both rows and may not be decomposable into a mixture of quantum channels from Alice to Bob and from Bob to Alice. Full size image

Figure 4: Terms not appearing in a process matrix. These terms are not compatible with local quantum mechanics because they yield non-unit probabilities for some CPTP maps. A possible interpretation of these terms within our framework is that they correspond to statistical sub-ensembles of possible processes. For example, terms of the type A 2 can be understood as postselection. One specific case is when a system enters a laboratory in a maximally mixed state, is subject to the map M and, after going out of the laboratory, is measured to be in some state |ψ . The corresponding probability is given by , generated in our formalism by . Notably, correlations of the type A 1 A 2 have been exploited in models for describing CTCs43,45. The pictures are only suggestive of the possible interpretations. Full size image

A causally non-separable process

The question whether all local quantum experiments can be embedded in a global causal structure corresponds to the question whether all process matrices are causally separable. Note that this is not a question about entanglement: all possible entangled states, and more generally all quantum circuits, correspond to matrices of the form or , whereas the non-separable processes we are looking for cannot be written as quantum circuits or even as probabilistic mixtures of different circuits. Surprisingly, an example of such a kind exists. Consider the process matrix

where A 1 , A 2 , B 1 and B 2 are two-level systems (for example, the spin degrees of freedom of a spin- particle) and σ x and σ z are the Pauli spin matrices. It can be verified straightforwardly that conditions (4) and (5) are satisfied, hence (7) is a valid bipartite process. Having such a resource, Alice and Bob can play the game described above and exceed the bound on the probability of success (2) imposed by causal order. Indeed, if Bob measures in the z basis and detects one of the states |z ± 〉, the corresponding CJ operator contains the factor . Inserting this, together with equation (7), into the expression (3) for the probabilities, the term containing in the process matrix is annihilated and what remains corresponds to a noisy channel from Alice to Bob. If Alice encodes her bit in the z basis with the CJ operator , this channel allows Bob to guess Alice's bit with probability . If, on the other hand, Bob measures in the x basis, equation (7) is reduced to a similar noisy channel from Bob to Alice. Bob is thus able to activate a channel in the desired direction by choosing the measurement basis (see Methods for a detailed calculation and analysis of the protocol). In this way they can achieve

which proves that (7) is not causally separable. We see that, depending on his choice, Bob can effectively end up 'before' or 'after' Alice, each possibility with a probability . This is remarkable, because if Alice and Bob perform their experiments inside laboratories that they believe are isolated from the outside world for the duration of their operations (for example, by walls made of impenetrable material), and if they believe that they are able to freely choose the bits a, b and b' (for example, by tossing a coin), they will have to conclude that the events in their experiment do not take place in a causal sequence. Indeed, the framework only assumes that the local operations from the input to the output system of each party are correctly described by quantum mechanics, and it is compatible with any physical situation in which one would have all the reasons to believe that each party's operations are freely chosen in a closed laboratory.

Interestingly, both the classical bound (2) and the quantum violation (8) match the corresponding numbers in the CHSH-Bell inequality31, which strongly resembles inequality (2). However, the physical situations to which these inequalities correspond is very different: Bell inequalities can be violated in space-like separated laboratories, while (8) cannot be achieved neither with space-like nor with time-like separated laboratories. It is an open question whether (8) is the maximal possible violation allowed by quantum mechanics.

Classical processes are causally separable