Operational meaning of quantum measures of recovery

TL;DR

This paper provides operational interpretations for quantum recovery measures, linking fidelity and relative entropy of recovery to quantum decision problems and hypothesis testing, and establishing their computational complexity classes.

Contribution

It introduces new operational meanings for quantum recovery measures using quantum interactive proof systems and analyzes their computational complexity.

Findings

Fidelity of recovery equals maximum probability in a quantum verification protocol.

The proof system with superposition requires only two messages, placing the problem in QIP(2).

Regularized relative entropy of recovery relates to the optimal error exponent in quantum hypothesis testing.

Abstract

Several information measures have recently been defined which capture the notion of "recoverability." In particular, the fidelity of recovery quantifies how well one can recover a system $A$ of a tripartite quantum state, defined on systems $ABC$, by acting on system $C$ alone. The relative entropy of recovery is an associated measure in which the fidelity is replaced by relative entropy. In this paper, we provide concrete operational interpretations of the aforementioned recovery measures in terms of a computational decision problem and a hypothesis testing scenario. Specifically, we show that the fidelity of recovery is equal to the maximum probability with which a computationally unbounded quantum prover can convince a computationally bounded quantum verifier that a given quantum state is recoverable. The quantum interactive proof system giving this operational meaning requires four messages exchanged between the prover and verifier, but by forcing the prover to perform his actions in superposition, we construct a different proof system that requires only two messages. The result is that the associated decision problem is in QIP(2) and another argument establishes it as hard for QSZK (both classes contain problems believed to be difficult to solve for a quantum computer). We finally prove that the regularized relative entropy of recovery is equal to the optimal Type II error exponent when trying to distinguish many copies of a tripartite state from a recovered version of this state, such that the Type I error is constrained to be no larger than a constant.