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 QRAC, in which two classical dits are encoded in a -dimensional quantum system. It is known that among rank-1 projective measurements MUBs give the best performance. We show (for every ) 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 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.
- 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)
Corrections
16 March 2020
Correction: Missing support information in the Acknowledgment section has been inserted.