Quantum state certification

TL;DR

This paper introduces efficient algorithms for quantum state certification that significantly reduce the number of copies needed compared to full tomography, with optimal copy complexities for testing state fidelity and trace distance.

Contribution

The paper presents two robust quantum state certification algorithms with optimal copy complexities, improving efficiency over previous methods.

Findings

Fidelity-based certification uses O(d/ε) copies.

Trace distance certification uses O(d/ε^2) copies.

Algorithms are optimal up to constant factors.

Abstract

We consider the problem of quantum state certification, where one is given $n$ copies of an unknown $d$-dimensional quantum mixed state $\rho$, and one wants to test whether $\rho$ is equal to some known mixed state $\sigma$ or else is $\epsilon$-far from $\sigma$. The goal is to use notably fewer copies than the $\Omega(d^2)$ needed for full tomography on $\rho$ (i.e., density estimation). We give two robust state certification algorithms: one with respect to fidelity using $n = O(d/\epsilon)$ copies, and one with respect to trace distance using $n = O(d/\epsilon^2)$ copies. The latter algorithm also applies when $\sigma$ is unknown as well. These copy complexities are optimal up to constant factors.