Comparison of mixed quantum states

TL;DR

This paper investigates the fundamental limits and conditions for unambiguous comparison of mixed quantum states, revealing that universal comparison is impossible and providing specific criteria for state comparison within a set.

Contribution

It establishes necessary and sufficient conditions for unambiguous comparison of mixed quantum states, advancing understanding of quantum state discrimination.

Findings

Universal comparison of mixed states is impossible.

Conditions for unambiguous comparison depend on the number of states and set size.

A unified condition for simultaneous existence of comparison measurements is derived.

Abstract

In this article, we study the problem of comparing mixed quantum states: given $n$ unknown mixed quantum states, can one determine whether they are identical or not with an unambiguous quantum measurement? We first study universal comparison of mixed quantum states, and prove that this task is generally impossible to accomplish. Then, we focus on unambiguous comparison of $n$ mixed quantum states arbitrarily chosen from a set of $k$ mixed quantum states. The condition for the existence of an unambiguous measurement operator which can produce a conclusive result when the unknown states are actually the same and the condition for the existence of an unambiguous measurement operator when the unknown states are actually different are studied independently. We derive a necessary and sufficient condition for the existence of the first measurement operator, and a necessary condition and two sufficient conditions for the second. Furthermore, we find that the sufficiency of the necessary condition for the second measurement operator has simple and interesting dependence on $n$ and $k$. At the end, a unified condition is obtained for the simultaneous existence of these two unambiguous measurement operators.