Eigenvalue Determination for Mixed Quantum States using Overlap Statistics

TL;DR

This paper investigates how overlap statistics between a mixed quantum state and its transformed images can be used to estimate the state's eigenvalues, with explicit results for qutrits and considerations for restricted transformations.

Contribution

It provides explicit formulas for overlap distributions in qutrits and analyzes the impact of restricted transformation sets like SO(3) on eigenvalue estimation.

Findings

Eigenvalues can be estimated from overlap bounds under full unitary transformations.

Overlap distributions depend only on eigenvalues for Haar-random unitaries.

Restricted transformations introduce systematic uncertainties in eigenvalue estimation.

Abstract

We consider the statistics of overlaps between a mixed state and its image under random unitary transformations. Choosing the transformations from the unitary group with its invariant (Haar) measure, the distribution of overlaps depends only on the eigenvalues of the mixed state. This allows one to estimate these eigenvalues from the overlap statistics. In the first part of this work, we present explicit results for qutrits, including a discussion of the expected uncertainties in the eigenvalue estimation. In the second part, we assume that the set of available unitary transformations is restricted to $SO(3)$, realized as Wigner $D$-matrices. In that case, the overlap statistics does not depend only on the eigenvalues, but also on the eigenstates of the mixed state under scrutiny. The overlap distribution then shows a complicated pattern, which may be considered as a fingerprint of the mixed state. When using random transformations from the unitary group, the eigenvalues can be determined quite simply from the lower and the upper limit of the overlap statistics. This may still be possible in the $SO(3)$ case, but only at the expense of a finite systematic uncertainty.