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Nonexistence of maximally entangled mixed states for a fixed spectrum

Gonzalo Camacho, Julio I. de Vicente

TL;DR

The paper tackles whether a universal maximally entangled state exists within the fixed-spectrum class $S(\lambda_1,\lambda_2,\lambda_3,\lambda_4)$ for two-qubit systems. It combines SEP-impossibility proofs with NE-convertibility analyses and a linear-programming framework to test whether the MEMS $\rho_{\vec{\lambda}}$ can be transformed to any isospectral target; it proves nonexistence for rank-2 and rank-3 spectra and for large classes of rank-4 spectra, while providing explicit NE-transformations in a spectral region. The results indicate that a universally maximally entangled two-qubit state for a fixed spectrum does not exist outside the pure-state case, reinforcing the idea that Nielsen-type maximality does not extend to mixed states in general. The work also supplies a practical LP-based methodology to study LOCC convertibility between mixed states and highlights the nuanced landscape of entanglement resource theories beyond the pure-state regime.

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.

Nonexistence of maximally entangled mixed states for a fixed spectrum

TL;DR

The paper tackles whether a universal maximally entangled state exists within the fixed-spectrum class for two-qubit systems. It combines SEP-impossibility proofs with NE-convertibility analyses and a linear-programming framework to test whether the MEMS can be transformed to any isospectral target; it proves nonexistence for rank-2 and rank-3 spectra and for large classes of rank-4 spectra, while providing explicit NE-transformations in a spectral region. The results indicate that a universally maximally entangled two-qubit state for a fixed spectrum does not exist outside the pure-state case, reinforcing the idea that Nielsen-type maximality does not extend to mixed states in general. The work also supplies a practical LP-based methodology to study LOCC convertibility between mixed states and highlights the nuanced landscape of entanglement resource theories beyond the pure-state regime.

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.

Paper Structure

This paper contains 10 sections, 8 theorems, 64 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Impossibility results via SEP transformations
  4. Impossibility results via NE transformations
  5. Eigenvalue distributions for which a NE transformation to Bell-diagonal states is possible
  6. Eigenvalue distributions for which a NE transformation to Bell-diagonal states is impossible
  7. Rank-2 case
  8. Rank-3 case
  9. Rank-4 case
  10. Conclusions

Key Result

Lemma 1

If $V$ is a 2-dimensional subspace of $\mathbb{C}^{2}\otimes\mathbb{C}^2$ and $A,B\in\mathbb{C}^{2\times2}$ are invertible, then $(A\otimes B)V$ is a 2-dimensional subspace that has the property of having all states separable or just one or just two iff $V$ has the same property.

Figures (1)

  • Figure 1: Feasibility of the LP for values of $\vec{\lambda}$ constrained to $\lambda_1>1/2$ for the choices $\lambda_4=0,0.01,0.03,0.06$ (notice that $\lambda_1>\lambda_3+2\sqrt{\lambda_2\lambda_4}$ is always satisfied in these cases). Note that $\lambda_4=0$ corresponds to the rank-3 case. Feasible regions of the LP are in blue and green and infeasible regions are in orange and black. The green region corresponds to the points where we analytically know that the LP is feasible due to Theorem \ref{['thsufNE']}, while the black region corresponds to the points of infeasibility due to Theorems \ref{['th:thne1']}, \ref{['th:thne2']} and \ref{['th:thne4']}.

Theorems & Definitions (16)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • ...and 6 more