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Quantum Physics

Title: Union bound for quantum information processing

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.
Comments: v2: 23 pages, includes proof, based on arXiv:1208.1400 and arXiv:1510.04682, for a lower bound on the second-order asymptotics of hypothesis testing for i.i.d. quantum states acting on a separable Hilbert space
Journal reference: Proceedings of the Royal Society A, vol. 475, no. 2221, id 20180612, January 2019
DOI: 10.1098/rspa.2018.0612
Cite as: arXiv:1804.08144 [quant-ph]
  (or arXiv:1804.08144v2 [quant-ph] for this version)

Submission history

From: Mark Wilde [view email]
[v1] Sun, 22 Apr 2018 17:39:41 GMT (47kb,D)
[v2] Sun, 18 Nov 2018 13:50:57 GMT (24kb)

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