Majorization uncertainty relations are generalized for an arbitrary mixed quantum state *ρ* of a finite size *N*. In particular, a lower bound for the sum of two entropies characterizing probability distributions corresponding to measurements with respect to arbitrary two orthogonal bases is derived in terms of the spectrum of *ρ* and the entries of a unitary matrix *U* relating both bases. The obtained results can also be formulated for two measurements performed on a single subsystem of a bipartite system described by a pure state, and consequently expressed as uncertainty relation for the sum of conditional entropies.

We present an in-depth study of the problem of discrimination of von Neumann measurements in finite-dimensional Hilbert spaces. Specifically, we consider two scenarios: unambiguous and multiple-shot discrimination. In the first scenario we give the general expressions for the optimal discrimination probabilities with and without the assistance of entanglement. In the case of multiple-shot discrimination, we focus on discrimination of measurements with the assistance of entanglement. Interestingly, we prove that in this setting all pairs of distinct von Neumann measurements can be distinguished perfectly (i.e. with the unit success probability) using only a finite number of queries. We also show that in this scenario queering the measurements \emph{in parallel} gives the optimal strategy and hence any possible adaptive methods do not offer any advantage over the parallel scheme. Finally, we show that typical pairs of Haar-random von Neumann measurements can be perfectly distinguished with only two queries.

}, url = {https://arxiv.org/abs/1810.05122}, author = {Zbigniew Pucha{\l}a and {\L}ukasz Pawela and Aleksandra Krawiec and Ryszard Kukulski and Micha{\l} Oszmaniec} } @article {2683, title = {Strategies for optimal single-shot discrimination of quantum measurements}, journal = {Physical Review A}, volume = {98}, year = {2018}, chapter = {042103}, abstract = {In this work we study the problem of single-shot discrimination of von Neumann measurements. We associate each measurement with a measure-and-prepare channel. There are two possible approaches to this problem. The first one, which is simple, does not utilize entanglement. We focus only on discrimination of classical probability distribution, which are outputs of the channels. We find necessary and sufficient criterion for perfect discrimination in this case. A more advanced approach requires the usage and entanglement. We quantify the distance of the two measurements in terms of the diamond norm (called sometimes the completely bounded trace norm). We provide an exact expression for the optimal probability of correct distinction and relate it to the discrimination of unitary channels. We also state a necessary and sufficient condition for perfect discrimination and a semidefinite program which checks this condition. Our main result, however, is a cone program which calculates the distance of these measurements and hence provides an upper bound on the probability of their correct distinction. As a by-product the program also finds a strategy (input state) which achieves this bound. Finally, we provide a full description for the cases of Fourier matrices and mirror isometries.

}, doi = {10.1103/PhysRevA.98.042103}, url = {https://arxiv.org/abs/1804.05856}, author = {Zbigniew Pucha{\l}a and {\L}ukasz Pawela and Aleksandra Krawiec and Ryszard Kukulski} } @article {2618, title = {Vertices cannot be hidden from quantum spatial search for almost all random graphs}, journal = {Quantum Information Processing}, volume = {17}, year = {2018}, pages = {81}, doi = {10.1007/s11128-018-1844-7}, url = {https://doi.org/10.1007/s11128-018-1844-7}, author = {Adam Glos and Aleksandra Krawiec and Ryszard Kukulski and Zbigniew Pucha{\l}a} }