Abstract
Recent frameworks describing quantum mechanics in the absence of a global causal order admit the existence of causally indefinite processes, where it is impossible to ascribe causal order for events A and B. These frameworks even allow for processes that violate the so-called causal inequalities, which are analogous to Bell's inequalities. However, the physicality of these exotic processes is, in the general case, still under debate, bringing into question their foundational relevance. While it is known that causally indefinite processes can be probabilistically realised by means of a quantum circuit, along with an additional conditioning event C, concrete insights into the ontological meaning of such implementation schemes have heretofore been limited. Here, we show that causally indefinite processes can be realised with schemes where C serves only as a classical flag heralding which causally indefinite process was realised. We then show that there are processes where any pure conditioning measurement of C leads to a causally indefinite process for A and B, thus establishing causal indefiniteness as a basis-independent quantity. Finally, we demonstrate that quantum mechanics allows for phenomena where C can deterministically decide whether A comes before B or vice versa, without signalling to either. This is akin to Wheeler's famous delayed-choice experiment establishing definite causal order in quantum mechanics as instrument-\textit{dependent} property.
Highlights
Genuine quantum properties, like entanglement and coherence, play an important role in many protocols and current or near future technologies [1]
We demonstrate that causal order itself can be understood as a basis-dependent property; if the conditioning measurements are made in one basis, A occurs before B, but if they are made in another, B occurs before A
Besides satisfying some reasonable desiderata one would require from a measure of causal nonseparability, CR is amenable to efficient numerical evaluation, as it can be phrased as a semidefinite program (SDP) [39]
Summary
Like entanglement and coherence, play an important role in many protocols and current or near future technologies [1]. This absence of quantum correlations allows for the interpretation that each measurement outcome on the conditioning system merely reveals—but does not create—the causally nonseparable processes that was “realized” in the individual run, and establishes causality as a principle that holds on average, but not necessarily for individual runs of an experiment While this latter interpretation has the obvious objection that the causal ordering of an individual run of an experiment is not a meaningful notion per se, it raises the question of whether entanglement between the conditioning degrees of freedom and the rest is possible and/or enhances the conditioning scheme. We begin by introducing the process matrix formalism, which is designed to represent spatio-temporal processes, including those that do not have a definite causal order
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