On the optimal certification of von Neumann measurements

Paulina Lewandowska,Łukasz Pawela,Zbigniew Puchała,Ryszard Kukulski,Aleksandra Krawiec

doi:10.1038/s41598-021-81325-1

Abstract

In this report we study certification of quantum measurements, which can be viewed as the extension of quantum hypotheses testing. This extension involves also the study of the input state and the measurement procedure. Here, we will be interested in two-point (binary) certification scheme in which the null and alternative hypotheses are single element sets. Our goal is to minimize the probability of the type II error given some fixed statistical significance. In this report, we begin with studying the two-point certification of pure quantum states and unitary channels to later use them to prove our main result, which is the certification of von Neumann measurements in single-shot and parallel scenarios. From our main result follow the conditions when two pure states, unitary operations and von Neumann measurements cannot be distinguished perfectly but still can be certified with a given statistical significance. Moreover, we show the connection between the certification of quantum channels or von Neumann measurements and the notion of q-numerical range.

Highlights

The validation of sources producing quantum states and measurement devices, which are involved in quantum computation workflows, is a necessary step of quantum technology[1,2,3]
One of our results presented in this work is a geometric interpretation of the formula for minimized probability of the type II error in the problem of certification of unitary channels, which is strictly connected with the notion of q-numerical range
In this work we studied the two-point certification of quantum states, unitary channels and von Neumann measurements

Summary

Numericaπlirange

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 PU and P1 is given by. Our goal is to find an optimal input state and a measurement for which the probability of the type II error is saturated, while the statistical significance δ is assumed. In the scheme of certification of von Neumann measurements the optimized probability of type II error can be expressed as. Utilizing data processing inequality in Lemma 1 in Online Appendix C, along with the certification scheme of unitary channels in Theorem 2, the optimized probability of the type II error is lower-bounded by. I=1 where the existence of ρ0 together with its properties are described in Lemma 2 and Corollary 1 in Online

Numerical ranges

Findings

Conclusions