## Quantum correlations with no causal order

**Ognyan Oreshkov, Fabio Costa & Časlav Brukner**

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.

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**Discussion**

We have seen that by relaxing the assumption of definite global causal order and requiring that the standard quantum formalism holds only locally, we obtain the possibility for global causal relations that are not included in the usual formulation of quantum mechanics. The latter is reminiscent of the situation in general relativity, where by requiring that locally the geometry is that of flat Minkowski space-time, one obtains the possibility of having more general, curved space-times.

The natural question is whether ‘non-causal’ quantum correlations of the kind described by our formalism can be found in nature. One can speculate that they may exist in unprobed physical regimes, such as, for example, those in which quantum mechanics and general relativity become relevant. Indeed, our result that classical theories can always be understood in terms of a global causal structure suggests the possibility that the observed causal order of space-time might not be a fundamental property of nature but rather emerge from a more fundamental theory^{32, 33, 34} in a quantum-to-classical transition due to, for example, decoherence^{35} or coarse-grained measurements^{36}. Once a causal structure is present, it is possible to derive relativistic space-time from it under appropriate conditions^{37, 38}. Furthermore, as the conformal space-time metric is a description of the causal relation between space-time points^{39, 40}, one can expect that an extension of general relativity to the quantum domain would involve situations where different causal orders could coexist ‘in superposition’. The formalism we presented may offer a natural route in this direction: based only on the assumption that quantum mechanics is valid locally, it yields causal relations that cannot be understood as arising from a definite, underlying order.

It is also worth noting that exotic causal structures already appear in the classical theory of general relativity. For example, there exist solutions to the Einstein equation containing closed time-like curves (CTCs)^{41}. In this context, it should be noted that any process matrix *W* in our framework can be interpreted as a CPTP map from the outputs, *A*_{2}, *B*_{2}, of the parties, to their inputs, *A*_{1}, *B*_{1}. In other words, any process can be thought of as having the form of a CTC, where information is sent back in time through a noisy channel (see also Fig. 1b). The existence of processes that do not describe definite causal order is therefore not incompatible with general relativity in principle. It is sometimes argued that CTCs should not exist as they generate logical paradoxes, such as an agent going back in time and killing his grandfather. The possible solutions that have been proposed^{42, 43, 44, 45, 46, 47}, in which quantum mechanics and CTCs might coexist, involve non-linear extensions of quantum theory that deviate from quantum mechanics already at the level of local experiments. Our framework, on the other hand, is by construction linear and in agreement with local quantum mechanics, and yet paradoxes are avoided, in accordance with the Novikov principle^{48}, due to the noise in the evolution ‘backward in time’.

Finally, we remark that instances of indefinite causal orders may also emerge in situations closer to possible laboratory implementations. As already noted, our formalism describes more general correlations than those that can be realized with a quantum circuit, that is, as a sequence of quantum gates. Recently, a new model of quantum computation that goes beyond the causal paradigm of quantum circuits by using superpositions of the ‘wires’ connecting different gates was proposed^{49}. This possibility may allow breaking the assumption that events are localized in a causal structure. As the instant when a system enters a device depends on how the device is wired with the rest of the computer’s architecture, superpositions of wires may allow creating situations in which events are not localized in time (similarly to the way in which a quantum particle may not be localized in space). Although it is an open question whether violating the causal inequality (2) can be achieved by similar means, the present work suggests that new quantum resources for information processing might be available—beyond entanglement, quantum memories and even ‘superpositions of wires’—and the formalism introduced provides a natural framework for exploring them.

Read more: nature.com – arxiv.org/pdf

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