Nonexistence of maximally entangled mixed states for a fixed spectrum

TL;DR

This paper proves that for two-qubit states with fixed spectra, there are no maximally entangled mixed states beyond pure states, extending previous results to higher ranks and broader eigenvalue distributions.

Contribution

It generalizes the nonexistence of maximally entangled states for fixed spectra to all rank-2, rank-3 two-qubit states, and many rank-4 cases, broadening the scope of prior findings.

Findings

No maximally entangled states for fixed spectra in rank-2 and rank-3 two-qubit states.

Extended nonexistence results to certain rank-4 eigenvalue distributions.

Confirmed that pure states are the unique maximally entangled states for a fixed spectrum.

Abstract

The existence of a maximally entangled pure state is a cornerstone result of entanglement theory that has paramount consequences in quantum information theory. A natural generalization of this property is to consider whether a notion of maximal entanglement is possible among all states with the same spectrum (where the aforementioned case of pure states corresponds to the particular choice in which the spectrum is a delta distribution, i.e., rank-1 states). Despite positive evidence in the past that such a notion might exist at least in the case of two-qubit states, it was recently shown in [Phys. Rev. Lett. 133, 050202 (2024)] that the answer to the above question is negative. This reference proved this for particular choices of the spectrum in the case of rank-2 two-qubit density matrices. While this settles the problem in general, it still leaves open whether there are other choices of the spectrum outside the case of pure states where a maximally entangled state for a fixed spectrum might exist. In this work we extend this impossibility result to all rank-2 and rank-3 two-qubit states as well as for a large class of eigenvalue distributions in the case where the rank equals four.