Quantum correlations with no causal order

Abstract

The idea that events obey a definite causal order is deeply rooted in our understanding of the world and at the basis of the very notion of time. But where does causal order come from, and is it a necessary property of nature? Here, we address these questions from the standpoint of quantum mechanics in a new framework for multipartite correlations that does not assume a pre-defined global causal structure but only the validity of quantum mechanics locally. All known situations that respect causal order, including space-like and time-like separated experiments, are captured by this framework in a unified way. Surprisingly, we find correlations that cannot be understood in terms of definite causal order. These correlations violate a 'causal inequality' that is satisfied by all space-like and time-like correlations. We further show that in a classical limit causal order always arises, which suggests that space-time may emerge from a more fundamental structure in a quantum-to-classical transition.

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).

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.

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 A₁, terms of the form as of the type A₁A₂ 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.

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 A₂ 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 A₁A₂ have been exploited in models for describing CTCs. The pictures are only suggestive of the possible interpretations.