Excluding false negative error in certification of quantum channels

Aleksandra Krawiec,Łukasz Pawela,Zbigniew Puchała

doi:10.1038/s41598-021-00444-x

Abstract

Certification of quantum channels is based on quantum hypothesis testing and involves also preparation of an input state and choosing the final measurement. This work primarily focuses on the scenario when the false negative error cannot occur, even if it leads to the growth of the probability of false positive error. We establish a condition when it is possible to exclude false negative error after a finite number of queries to the quantum channel in parallel, and we provide an upper bound on the number of queries. On top of that, we found a class of channels which allow for excluding false negative error after a finite number of queries in parallel, but cannot be distinguished unambiguously. Moreover, it will be proved that parallel certification scheme is always sufficient, however the number of steps may be decreased by the use of adaptive scheme. Finally, we consider examples of certification of various classes of quantum channels and measurements.

Highlights

Certification of quantum channels is based on quantum hypothesis testing and involves preparation of an input state and choosing the final measurement
For a fixed upper bound on the probability of false positive error, ǫ, we introduce the minimal number of steps needed for adaptive certification, Nǫ, as the minimal number of steps after which p2A,N (|ψ, 0) = 0 and p1A,N (|ψ, 0) ≤ ǫ for some input state |ψ and measurement effect 0
As certification of quantum channels is in the Noisy Intermediate-Scale Quantum (NISQ) era a task of significant importance, the main aim of this work was to give an insight into this problem from theoretical perspective

Summary

Introduction

Certification of quantum channels is based on quantum hypothesis testing and involves preparation of an input state and choosing the final measurement. We found a class of channels which allow for excluding false negative error after a finite number of queries in parallel, but cannot be distinguished unambiguously. There are three major theoretical approaches towards verification of quantum channels called minimum error discrimination, unambiguous discrimination and certification. All these three approaches can be generalized to the multiple-shot case, that is when the given channel can be used multiple times in various configurations. The most straightforward possibility is the parallel scheme and the most sophisticated is the adaptive scheme (where we are allowed to use any processing between the uses of the given channel)

Objectives

Conclusion