Strategies for optimal single-shot discrimination of quantum measurements

Abstract

In this work we study the problem of single-shot discrimination of von Neumann measurements, which we associate with measure-and-prepare channels. There are two possible approaches to this problem. The first one is simple and does not utilize entanglement. We focus only on the discrimination of classical probability distributions, 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 of entanglement. We quantify the distance between 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 between the measurements and hence provides an upper bound on the probability of their correct distinction. As a by-product, the program finds a strategy (input state) which achieves this bound. Finally, we provide a full description for the cases of Fourier matrices and mirror isometries.

Mitigation of readout noise in near-term quantum devices by classical post-processing based on detector tomography

“…TV-distance is related to the operational distance [30][31][32] between arbitrary POVMs M and N via the equality…”

“…where the maximization is over all subsets of indices enumerating effects (i.e., all possible sets of outcomes) and ||.|| ∞ denotes operator norm [33]. Operational distance between POVMs [34] has an interesting operational interpretation through the formula D op (M, N) = 2p disc (M, N) − 1, where p disc (M, N) is the optimal probability of distinguishing between measurements M and N (without using entanglement) [31].…”

On the optimal certification of von Neumann measurements

“…Now we recall some technical tools which will be used to prove the main result of this work. It was shown in 23 (Theorem 1) that the diamond norm distance between von Neumann measurements P U and P 1 is given by where DU d is the subgroup of diagonal unitary matrices of dimension d. As we can see, the problem of discrimination of von Neumann measurements reduces to the problem of discrimination of unitary channels. From 12 we know that the diamond norm distance between two unitary channels U and 1 is expressed as where ν(U) = min{|x| : x ∈ W(U)}.…”

“…Thanks to that, we will show the lower bound for p II . In the second part we will use some technical lemmas presented in Online Appendix C in the Supplementary Materials and we will utilize the results from 23 to show the upper bound for p II .…”

Discrimination of POVMs with rank-one effects

“…However, it may happen that it suffices to use the parallel one. For example, in the case of unitary channels [8,9] and von Neumann measurements it was shown in [11][12][13] that the parallel scheme is always optimal. It is also known that asymptotically the use of adaptive strategy does not give an advantage over the parallel one for the discrimination of classical [14] and classical-quantum channels [15].…”