# Separability from spectrum

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# Problem

For a mixed state $\rho$ on an N M-dimensional Hilbert space: Are there any factorizations into an N tensor an M dimensional space with respect to which the state is not seperable? This depends only on the spectrum of $\rho$ and the problem is to characterize the spectra for which the answer is "no".

# Background

The question arises in the context where we are given a highly mixed state on two quantum systems and the ability to apply any unitary operator. Can an inseperable state be obtained? For sufficiently mixed states, this is not possible.

This problem is different from No. 9, because only the spectrum of $\rho$ and not the spectra of the reductions are to be part of the criterion.

# Partial Results

See the generic bounds on how close a state has to be to the completely mixed state to be guaranteed not to have entanglement. The paper of Leonid Gurvits and Howard Barnum [1] has further relevant results.

For the case of two qubits, the question is solved in [2]: Exactly the states with eigenvalues $x_1, x_2, x_3, x_4$ (arranged in decreasing order) obeying $x_1 - x_3 - 2 \sqrt{x_2 x_4} \leq 0$ cannot be transformed into a state with non-zero entanglement of formation by applying any unitary operator (Theorem 1).

# Literature

1. L. Gurvits and H. Barnum, »Size of the Separable Neighborhood of the Maximally Mixed Bipartite Quantum State«, quant-ph/0204159 (2002)
2. F. Verstraete, K. Audenaert, and B. De Moor, »Maximally entangled mixed states of two qubits«, Phys. Rev. A 64, 012316 (2001) and (together with T. De Bie) quant-ph/0011110 (2000)