Permutation tests for quantum state identity

TL;DR

This paper analyzes the quantum state identity problem, developing optimal and approximate tests using semidefinite programming, representation theory, and classical permutation strategies.

Contribution

It provides the first optimal test for the general two-sided error regime and introduces a general G-test based on subgroup symmetries.

Findings

Derived an exact optimal test via SDP for any input distribution.

Proposed a G-test framework using arbitrary subgroups of S_n.

Presented an approximation of the permutation test with classical permutation and Swap tests.

Abstract

The quantum analogue of the equality function, known as the quantum state identity problem, is the task of deciding whether $n$ unknown quantum states are equal or unequal, given the promise that all states are either pairwise orthogonal or identical. Under the one-sided error requirement, it is known that the permutation test is optimal for this task, and for two input states this coincides with the well-known Swap test. Until now, the optimal measurement in the general two-sided error regime was unknown. Under more specific promises, the problem can be solved approximately or even optimally with simpler tests, such as the circle test. This work attempts to capture the underlying structure of the quantum state identity problem. Using tools from semidefinite programming and representation theory, we (i) give an optimal test for any input distribution without the one-sided error requirement by writing the problem as an SDP, giving the exact solutions to the primal and dual programs and showing that the two values coincide; (ii) propose a general $G$-test which uses an arbitrary subgroup $G$ of $\text{S}_n$, giving an analytic expression of the performance of the specific test, and (iii) give an approximation of the permutation test using only a classical permutation and $n-1$ Swap tests.