Discrimination of POVMs with rank-one effects

Abstract

The main goal of this work is to provide an insight into the problem of discrimination of positive operator-valued measures with rank-one effects. It is our intention to study multiple-shot discrimination of such measurements, that is the case when we are able to use to unknown measurement a given number of times. Furthermore, we are interested in comparing two possible discrimination schemes: the parallel and adaptive ones. To this end, we construct a pair of symmetric informationally complete positive operator-valued measures which can be perfectly discriminated in a two-shot adaptive scheme but cannot be distinguished in the parallel scheme. On top of this, we provide an explicit algorithm which allows us to find this adaptive scheme.

Excluding false negative error in certification of quantum channels

“…We will provide an example of channels which can be certified in a finite number of queries in parallel, but cannot be distinguished unambiguously. Moreover, we will show that, in contrast to discrimination of quantum channels 19 , 20 , parallel certification scheme is always sufficient for certification, although the number of uses of the certified channel may not be optimal. On top of that, we will consider certification of quantum measurements and focus on the class of measurements with rank-one effects.…”

“…It appeared that parallel discrimination scheme is optimal in the case of distinguishability of unitary channels 17 and von Neumann measurements 18 . In some cases however, the use of adaptive discrimination scheme can significantly improve the certification 19 , 20 . Advantages of the use of adaptive discrimination scheme in the asymptotic regime were studied in 21 .…”

On the optimal certification of von Neumann measurements

“…In the case of discrimination of unitary channels and von Neumann measurements, the positive answer was obtained 24,40 . Nevertheless, in some special cases of quantum channels or quantum measurements by using an adaptive scenario we are able to improve the probability of correct discrimination of such objects 20,21 . In the case of certification of quantum objects, the optimality of parallel scenario was showed for unitary channels 27 .…”

Multiple-shot and unambiguous discrimination of von Neumann measurements

“…It is worth mentioning here that the optimality of the parallel scheme is no longer true when studying the discrimination of general (non-projective) POVMs. In the case of discrimination of POVMs with rank-one effects one may need to use adaptive discrimination scheme to obtain perfect distinguishability [23]. While considering the discrimination of von Neumann measurements, one could expect that we can improve the discrimination by performing some processing based on the obtained measurement's labels.…”

“…In the work [19] the authors formulated necessary and sufficient conditions under which quantum channels can be perfectly discriminated. Further works investigated the adaptive [20][21][22][23] and parallel [24] schemes for discrimination of channels. Finally, some asymptotic results on discrimination of typical quantum channels in large dimensions were obtained in [25].…”