Union bound for quantum information processing

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...

Full description

Bibliographic Details
Published in:Proceedings. Mathematical, physical, and engineering sciences, Vol. 475, No. 2221 (2019), p. 20180612
Main Author: Khabbazi Oskouei, Samad (Author)
Other Involved Persons: Mancini, Stefano ; Wilde, Mark M
Format: electronic Article
Language:English
ISSN:1364-5021
Item Description:Date Revised 19.02.2019
published: Print-Electronic
Citation Status PubMed-not-MEDLINE
Copyright: From MEDLINE®/PubMed®, a database of the U.S. National Library of Medicine
Physical Description:Online-Ressource
DOI:10.1098/rspa.2018.0612
Subjects:
QR Code: Show QR Code
Description:
  • 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 (Hayashi & Nagaoka 2003 IEEE Trans. Inf. Theory 49, 1753-1768. (doi:10.1109/TIT.2003.813556)), used often in quantum information theory when analysing 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, Pythagoras' 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