Union bound for quantum information processing
Abstract
In this paper, we prove a quantum union bound that is relevant when performing a sequence of binaryoutcome 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 HayashiNagaoka inequality (Hayashi & Nagaoka 2003 IEEE Trans. Inf. Theory 49, 17531768. (doi:10.1109/TIT.2003.813556)), used often in quantum information theory when analysing the error probability of a squareroot measurement. An advantage of the proof delivered here is that it is elementary, relying only on basic properties of projectors, Pythagoras' theorem, and the CauchySchwarz inequality. As a nontrivial 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 secondorder coding rate. This demonstrates the advantage of our quantum union bound in the nonasymptotic 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.
 Publication:

Proceedings of the Royal Society of London Series A
 Pub Date:
 January 2019
 DOI:
 10.1098/rspa.2018.0612
 arXiv:
 arXiv:1804.08144
 Bibcode:
 2019RSPSA.47580612K
 Keywords:

 Quantum Physics;
 Computer Science  Information Theory;
 Mathematical Physics
 EPrint:
 v2: 23 pages, includes proof, based on arXiv:1208.1400 and arXiv:1510.04682, for a lower bound on the secondorder asymptotics of hypothesis testing for i.i.d. quantum states acting on a separable Hilbert space