Union bound for quantum information processing

TL;DR

This paper introduces a simple, tunable quantum union bound for sequential measurements, with applications to classical communication over quantum channels, improving understanding of error bounds in quantum information processing.

Contribution

It presents an elementary proof of a quantum union bound with a tunable parameter, applicable to non-asymptotic quantum communication scenarios.

Findings

Proves a quantum union bound involving a tunable parameter.

Demonstrates the bound's application to sequential decoding in quantum communication.

Shows the bound achieves a lower bound on the second-order coding rate.

Abstract

In this paper, we prove a quantum union bound that is relevant when performing a sequence of binary-outcome quantum measurements on a quantum state. The quantum union bound proved here involves a tunable parameter that can be optimized, and this tunable parameter plays a similar role to a parameter involved in the Hayashi-Nagaoka inequality [IEEE Trans. Inf. Theory, 49(7):1753 (2003)], used often in quantum information theory when analyzing the error probability of a square-root measurement. An advantage of the proof delivered here is that it is elementary, relying only on basic properties of projectors, the Pythagorean theorem, and the Cauchy--Schwarz inequality. As a non-trivial application of our quantum union bound, we prove that a sequential decoding strategy for classical communication over a quantum channel achieves a lower bound on the channel's second-order coding rate. This demonstrates the advantage of our quantum union bound in the non-asymptotic regime, in which a communication channel is called a finite number of times. We expect that the bound will find a range of applications in quantum communication theory, quantum algorithms, and quantum complexity theory.