Self-testing mutually unbiased bases in the prepare-and-measure scenario

Máté Farkas and Jędrzej Kaniewski
Phys. Rev. A 99, 032316 – Published 12 March 2019

Abstract

Mutually unbiased bases (MUBs) constitute the canonical example of incompatible quantum measurements. One standard application of MUBs is the task known as quantum random access code (QRAC), in which classical information is encoded in a quantum system, and later part of it is recovered by performing a quantum measurement. We analyze a specific class of QRACs, known as the 2d1 QRAC, in which two classical dits are encoded in a d-dimensional quantum system. It is known that among rank-1 projective measurements MUBs give the best performance. We show (for every d) that this cannot be improved by employing nonprojective measurements. Moreover, we show that the optimal performance can only be achieved by measurements which are rank-1 projective and mutually unbiased. In other words, the 2d1 QRAC is a self-test for a pair of MUBs in the prepare-and-measure scenario. To make the self-testing statement robust we propose measures which characterize how well a pair of (not necessarily projective) measurements satisfies the MUB conditions and show how to estimate these measures from the observed performance. Similarly, we derive explicit bounds on operational quantities like the incompatibility robustness or the amount of uncertainty generated by the uncharacterized measurements. For low dimensions the robustness of our bounds is comparable to that of currently available technology, which makes them relevant for existing experiments. Last, our results provide essential support for a recently proposed method for solving the long-standing existence problem of MUBs.

  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
  • Received 10 April 2018
  • Revised 31 August 2018
  • Corrected 16 March 2020

DOI:https://doi.org/10.1103/PhysRevA.99.032316

©2019 American Physical Society

Physics Subject Headings (PhySH)

Quantum Information, Science & Technology

Corrections

16 March 2020

Correction: Missing support information in the Acknowledgment section has been inserted.

Authors & Affiliations

Máté Farkas1,* and Jędrzej Kaniewski2

  • 1Institute of Theoretical Physics and Astrophysics, National Quantum Information Centre, Faculty of Mathematics, Physics and Informatics, University of Gdansk, 80-952 Gdansk, Poland
  • 2Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland

  • *mate.farkas@phdstud.ug.edu.pl

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand
Issue

Vol. 99, Iss. 3 — March 2019

Reuse & Permissions
Access Options
Author publication services for translation and copyediting assistance advertisement

Authorization Required


×
×

Images

×

Sign up to receive regular email alerts from Physical Review A

Log In

Cancel
×

Search


Article Lookup

Paste a citation or DOI

Enter a citation
×