Separability from spectrum – Open Quantum Problems

Cite this problem as Problem 15.

Problem

For a mixed state $\rho$ on a bipartite $(N\times M)$ -dimensional Hilbert space, are there any factorizations of the space into an $N$ -dimensional tensor an $M$ -dimensional space with respect to which the state is entangled? The answer to this question depends only on the spectrum of $\rho$ , and the problem is to characterize the spectra for which the answer is “no”. Another reformulation of the problem is: for which spectra $\lambda$ does it happen that $\rho$ is separable whenever it has spectrum $\lambda$ ?

Background

A finite-dimensional bipartite quantum state $\rho_{AB}$ is called separable if it admits a decomposition of the form

$\displaystyle \rho_{AB} = \sum_i p_i\, \alpha_i^A \otimes \beta_i^B ,$

where $\alpha_i^A , \beta_i^B$ are states of the local systems (without loss of generality, pure), and $p_i$ is a probability distribution. A state that is not separable is called entangled. A necessary condition for a state to be separable is that it has a positive partial transpose (PPT), i.e. $\rho^\Gamma \geq 0$ , where the operation $\Gamma$ is defined by $(X\otimes Y)^\Gamma := X \otimes Y^\intercal$ on simple tensors, and extended by linearity.

The above 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 entangled state be obtained? For sufficiently mixed states, this is not possible. This problem is different from Problem 9, because only the spectrum of $\rho$ and not the spectra of the reductions are to be part of the criterion.

States for which $U\rho U^\dag$ is separable for all unitary operators $U$ are called separable from spectrum or also absolutely separable [1]. As the terminology suggests, whether a state is separable from spectrum depends only on its eigenvalues. We will call a spectrum absolutely separable if any state (and hence all states) with that spectrum are absolutely separable.

Partial Results

In [2] some sufficient conditions for a state to be separable from spectrum are found: if $\mathrm{Tr} \rho^2 \leq \frac{1}{NM-1}$ , where $NM$ is the total dimension, then $\rho$ is separable. This is a tight bound as far as the purity is concerned, but it fails to detect many absolutely separable spectra in general. The paper [2] has further relevant results.

For the case of two qubits, the question is solved in [3]: the absolutely separable spectra $\lambda_1 \geq \lambda_2 \geq \lambda_3 \geq \lambda_4 \geq 0$ (arranged in decreasing order) are precisely those obeying $\lambda_1 - \lambda_3 - 2 \sqrt{\lambda_2 \lambda_4} \leq 0$ [3, Theorem 1].

Instead of absolute separability, one can consider absolute PPT-ness: a state $\rho$ (or a spectrum $\lambda$ ) is absolutely PPT, a.k.a. PPT from spectrum, if $U\rho U^\dag$ has a positive partial transpose for all unitaries $U$ . Absolutely PPT spectra have been completely classified [4]. Another variation is to consider all spectra whose corresponding states are guaranteed to satisfy the reduction criterion [5].

The original problem of absolute separability has been solved also for qubit-qudit systems [6]. In this case, separability from spectrum coincides with PPT-ness from spectrum, leading to the simple eigenvalue condition in $(2\times d)$ -dimensional systems $\lambda_1-\lambda_{2d-1}-\sqrt{\lambda_{2d-2}\lambda_{2d}}\leq0$ , where again $\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_{2d} \geq 0$ is sorted in decreasing order.

Currently research [7] is going in the direction of deciding the question of whether absolute separability is equivalent to absolute PPT-ness, which, as mentioned, has an effective characterization [4].

The absolute separability problem has many analogs in different settings, including continuous-variable systems [8, 9], quantum channel theory [10], bosonic systems [11], and quantum coherence [12].

References

[1] M. Kus and K. Zyczkowski, Geometry of entangled states, Phys. Rev A 63, 032307 (2001) and arXiv:quant-ph/0006068 (2000).

[2] L. Gurvits and H. Barnum, Largest separable balls around the maximally mixed bipartite quantum state, Phys. Rev. A 66, 062311 (2002) and arXiv:quant-ph/0204159 (2002).

[3] 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) arXiv:quant-ph/0011110 (2000).

[4] R. Hildebrand, Positive partial transpose from spectra, Phys. Rev. A 76, 052325 (2007), and arXiv:quant-ph/0502170 (2005).

[5] M. A. Jivulescu, N. Lupa, I. Nechita, and D. Reeb, Positive reduction from spectra, Lin. Alg. Appl. 469, 276-304 (2015) and arXiv:1406.1277 (2014).

[6] N. Johnston, Separability from spectrum for qubit-qudit states, Phys. Rev. A 88, 062330 (2013) and arXiv:1309.2006 (2013).

[7] S. Arunachalam, N. Johnston, and V. Russo, Is absolute separability determined by the partial transpose?, Quant. Inf. Comput. 15, 0694-0720 (2015) and arXiv:1405.5853 (2014).

[8] M. M. Wolf, J. Eisert, and M. B. Plenio, The entangling power of passive optical elements, Phys. Rev. Lett. 90, 047904 (2003) and arXiv:quant-ph/0206171 (2002).

[9] L. Lami, A. Serafini, and G. Adesso, Gaussian entanglement revisited, New J. Phys. 20, 023030 (2018) and arXiv:1612.05215 (2016).

[10] S. N. Filippov, K. Yu. Magadov, and M. A. Jivulescu, Absolutely separating quantum maps and channels, New J. Phys. 19, 083010 (2017) and arXiv:1703.00344 (2017).

[11] G. Champagne, N. Johnston, M. MacDonald, and L. Pipes, Spectral properties of symmetric quantum states and symmetric entanglement witnesses, Linear Algebra Appl. 649, 273-300 (2022) and arXiv:2108.10405 (2021).

[12] N. Johnston, S. Moein, R. Pereira, and S. Plosker, Absolutely $k$ -incoherent quantum states and spectral inequalities for factor width of a matrix, Phys. Rev. A 106, 052417 (2022) and arXiv:2205.05110 (2022).