Eigenvalue Determination for Mixed Quantum States using Overlap Statistics

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.

Eigenstate extraction with neural-network tomography

“…Note that there exist proposals to retrieve spectral information of unknown density matrices ρ by applying additional, general random unitaries to ρ [19][20][21] (see also [22]). In contrast, we here leverage knowledge of the dominant eigenstate in order to extract information about its corresponding eigenvalue.…”