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

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 analyse a specific class of QRACs, known as the 2 d → 1 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 non-projective 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 2 d → 1 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 characterise 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 uncharacterised measurements. For low dimensions the robustness of our bounds is comparable to that of currently available technology, which makes them relevant for existing experiments. Lastly, our results provide essential support for a recently proposed method for solving the long-standing existence problem of MUBs.

Self-testing of quantum systems: a review

“…The authors also use A 2 2 →1 to self-test a non-trivial set of qutrit preparations and measurements, and implement an adaptation of the numerical Swap method (see Section 7.1.4) to deal with the prepare-and-measure scenario. In [FK19], the authors study the 2 d → 1 RAC. It is proven that this game provides a robust self-test of a pair of measurements that correspond to two mutually unbiased bases in dimension d. Further to this, the authors show how the score of the RAC can also be used to bound both the incompatibility robustness [HKR15] of the pair and the randomness of the measurement outputs.…”

Simultaneous certification of entangled states and measurements in bounded dimensional semiquantum games

“…Apart from the route to a "standard self-testing" result, it is also intriguing to consider the certification problem in a higher bounded dimension. Some recent works on the prepare-andmeasure semi-device-independent protocol have shown novel certification results with a freedom beyond local unitary operations in higher-dimensional systems [53,54]. It would be interesting if similar phenomena also exist in the semiquantum games.…”

Weak form of self-testing

“…Self-testing is an active research field and a particularly interesting direction is to explore its powers and limitations by deriving new types of self-testing statements or impossibility results. For instance we have recently learned that one can self-test quantum channels [49], entangled measurements [50,51], and quantum instruments [52], or that one can extend the concept of self-testing to prepare-and-measure scenarios [53][54][55][56][57][58]. In this work we derive a new type of self-testing statement which allows us to certify the state but not the measurements.…”