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.
Highlights
Understanding physical transformations between quantum systems is one of the central pillars of quantum mechanics
We show that pure quantum superchannels with two slots can be divided into two cases: 1) quantum combs realizable only with unitary operations; 2) coherent superposition of two pure combs of which input operations are differently ordered, which may be seen as a generalization of the quantum switch1
Quantum superchannels with indefinite causal order have revealed fundamental properties of quantum theory and have proved useful to enhance performance in several informationtheoretic tasks
Summary
Understanding physical transformations between quantum systems is one of the central pillars of quantum mechanics. Deterministic higher-order operations are named quantum superchannels or process matrices [6, 7] They are the most general deterministic transformations between multiple independent operations allowed by quantum mechanics. Quantum theory admits superchannels which make use of input operations in an indefinite causal order. The set of superchannels with indefinite causal order is in principle compatible with quantum mechanics and is not restricted to elements equivalent to the quantum switch This set includes superchannels which allow device-independent indefinite causal order certification [6] and cannot be decomposed into a simple coherent superposition of ordered circuits. The quantum switch has found limitations in the tasks of device-independent indefinite causal order certification [6, 19] and transforming unitary operations [20, 21] where only non-switch indefinite quantum superchannels display an advantage. Appendix C and Appendix D present a proof of Thm. 5
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