Consequences of preserving reversibility in quantum superchannels

Abstract

Similarly to quantum states, quantum operations can also be transformed by means of quantum superchannels, also known as process matrices. Quantum superchannels with multiple slots are deterministic transformations which take independent quantum operations as inputs. While they are enforced to respect the laws of quantum mechanics, the use of input operations may lack a definite causal order, and characterizations of general superchannels in terms of quantum objects with a physical implementation have been missing. In this paper, we provide a mathematical characterization for pure superchannels with two slots (also known as bipartite pure processes), which are superchannels preserving the reversibility of quantum operations. We show that the reversibility preserving condition restricts all pure superchannels with two slots to be either a quantum circuit only consisting of unitary operations or a coherent superposition of two unitary quantum circuits where the two input operations are differently ordered. The latter may be seen as a generalization of the quantum switch, allowing a physical interpretation for pure two-slot superchannels. An immediate corollary is that purifiable bipartite processes cannot violate device-independent causal inequalities.

Existence of processes violating causal inequalities on time-delocalised subsystems

“…For unitary extensions of bipartite processes, it was shown in Refs. [31,32] that a similar standard form exists. Namely, all unitary extensions of bipartite processes are variations of the quantum switch.…”

“…It was subsequently shown in Refs. [31,32] that all such processes are variations of the quantum switch, but the proof of Ref. [19] did not rely on this knowledge.…”

“…[19], together with the subsequent result of Refs. [31,32], that will allow us to generalise the proof to the tripartite case. We will therefore in the following recall the bipartite result from Ref.…”

“…5 is shown in Appendix D 1. It follows from the result that all unitary extensions of bipartite processes can be implemented as "variations of the quantum switch" [31,32], in which the time of the two local operations is controlled coherently. Any unitary extension of a tripartite process (and the ancillary systems), depending coherently on the state of the two-dimensional control systems Q…”

A no-go theorem for superpositions of causal orders

“…. In particular, Lemma 2 can be deduced from Corollary 5 in Ref [48],. although the consequence as a no go theorem for superpositions of orders is not discussed there.…”