Nonexistence of maximally entangled mixed states for a fixed spectrum

• URL: http://arxiv.org/abs/2511.08285v1
• Date: Wed, 12 Nov 2025 01:50:51 GMT
• Title: Nonexistence of maximally entangled mixed states for a fixed spectrum
• Authors: Gonzalo Camacho, Julio I. de Vicente,
• Abstract summary: We consider whether a notion of maximal entanglement is possible among all states with the same spectrum.<n>Despite positive evidence in the past, it was recently shown in [Phys. Rev. Lett. 133, 050202 (2024) that the answer to the above question is negative.<n>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.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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