Máté Farkas (University of Gdańsk): Mutually unbiased measurements and Bell nonlocality
Channel: Centrum Fizyki Teoretycznej PAN
Date: 2021-04-11

Máté Farkas (University of Gdańsk).
Mutually unbiased measurements and Bell nonlocality
Mutually unbiased bases (MUBs) correspond to quantum measurements with myriad applications in quantum information processing. Among other tasks, they are optimal for quantum state determination, quantum key distribution, secret sharing and quantum random access coding. The underlying property that makes them so powerful is complementarity: if the outcome of one of the measurements on a quantum state is certain, then the outcome of the another measurement, unbiased to the first one, is completely random. This is an operational characterisation of MUB pairs, however, the full definition also includes the Hilbert space dimension, which is not an observable quantity. For this reason, the usual MUB definition is not fully operational, and consequently, it is unclear how to characterise MUB measurements in a device-independent fashion. In our work, we introduce a fully operational definition of generalised MUBs by removing the dimension assumption from the above definition. We call measurements satisfying this condition mutually unbiased measurements (MUMs), and provide a simple algebraic characterisation of MUMs in terms of the measurement operators. Naturally, d-outcome MUMs have similar characteristics to those of d-dimensional MUBs. They exhibit precisely the same behaviour in the context of entropic uncertainty relations, and they are not jointly measurable to the same extent. Indeed, we show that 2- and 3-outcome MUMs are always isomorphic to a direct sum of 2- and 3-dimensional MUBs. However, we show explicit examples of MUMs with outcome numbers 4 and 5 that cannot be written as direct sums of MUBs, and there does not even exist a completely positive unital map that would map them to 4- and 5-dimensional MUBs. This shows that MUMs are a strictly more general concept than MUBs. Having this device-independent characterisation, we introduce a family of Bell inequalities, parametrised by an integer d larger or equal 2, that are maximally violated by d-outcome MUMs. Moreover, the maximum violation is achieved by a unique probability point, which must therefore be extremal in the set of quantum correlations. This has two implications: first, the maximal violation of these Bell inequalities certifies a pair of d-outcome MUMs. Second, since not even all pairs of MUBs are unique up to a unitary transformation and complex conjugation, this certification is not a self-test. That is, the maximal violation of the Bell inequality does not certify the measurements up to local unitaries and extra degrees of freedom. Up to our knowledge, this is the first example of an extremal point of the set of quantum correlations that is not a self-test. This raises the question: given an extremal point of the set of quantum correlations, to what extent does it determine the a priori unknown quantum realisation? 
CTP Quantum Information Days (online), 22-24 February 2021.
Center for Theoretical Physics PAS (Warsaw, Poland)