Separability from spectrum

Cite this problem as Problem 15.


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”.


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.

States, for which U\rho U^* is separable for all unitary operators U are also called absolutely separable [3]. This terminology actually predates the current problem statement. 

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).

For the case of a qubit-qudit system, the question is solved in [4]: Separability from spectrum coincides with positivity under partial transpose from spectrum, leading to the simple eigenvalue condition in 2xd dimensional systems  x_1-x_{2d-1}-\sqrt{x_{2d-2}x_{2d}}\leq0.

Currently research [5] is going in the direction of deciding the question whether absolute separability is equivalent to the “absolute” version of the ppt condition, which has an effective characterization [6].



[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).

[3] M. Kus and K. Zyczkowski, »Geometry of entangled states«, arXiv/quant-ph/0006068

[4] N. Johnston, »Separability from Spectrum for Qubit-Qudit States«, Phys. Rev. A 88, 062330 (2013)

[5] S. Arunachalam, N. Johnston, and V. Russo, »Is absolute separability determined by the partial transpose?«, arXiv:1405.5853

[6]  R. Hildebrand, “Positive partial transpose from spectra,” Phys. Rev. A, vol. 76, p. 052325, 2007, and quant-ph/0502170.