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Is Separability from Spectrum Determined by the Partial Transpose?

Srinivasan Arunachalam, Nathaniel Johnston, Vincent Russo

Published Jan 1, 2014• 1 citation
OpenAlex

Abstract

The separability from spectrum problem asks for a characterization of the quantum states ρ ∈ Mm ⊗Mn with the property that UρU † is separable for all unitary matrices U. We provide evidence that ρ is separable from spectrum if and only if UρU † has positive partial transpose for all unitary matrices U. In particular, we show that many well-known separability criteria are unable to detect entanglement in any such state, including the range criterion, the realignment criterion, the Choi map and its generalizations, and the Breuer–Hall map. We also show that these two properties coincide for the families of isotropic and Werner states, and several eigenvalue results for entanglement witnesses are proved along the way that are of independent interest. 1

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The separability from spectrum problem asks for a characterization of the quantum states ρ ∈ Mm ⊗Mn with the property that UρU † is separable for all unitary matrices U. We provide evidence that ρ is separable from spectrum if and only if UρU † has positive partial transpose for all unitary matrices U. In particular, we show that many well-known separability criteria are unable to detect entanglement in any such state, including the range criterion, the realignment criterion, the Choi map and its generalizations, and the Breuer–Hall map. We also show that these two properties coincide for the families of isotropic and Werner states, and several eigenvalue results for entanglement witnesses are proved along the way that are of independent interest. 1
The absolute separability problem asks for a characterization of the quantum states ρ∈Mm⊗Mnρ\in M_m\otimes M_nρ∈Mm​⊗Mn​ with the property that UρU†UρU^\daggerUρU† is separable for all unitary matrices UUU. We investigate whether or not it is the case that ρρρ is absolutely separable if and only if UρU†UρU^\daggerUρU† has positive partial transpose for all unitary matrices UUU. In particular, we develop an easy-to-use method for showing that an entanglement witness or positive map is unable to detect entanglement in any such state, and we apply our method to many well-known separability criteria, including the range criterion, the realignment criterion, the Choi map and its generalizations, and the Breuer-Hall map. We also show that these two properties coincide for the family of isotropic states, and several eigenvalue results for entanglement witnesses are proved along the way that are of independent interest.
The absolute separability problem asks for a characterization of the quantum states ρ∈Mm⊗Mn\rho \in M_m\otimes M_nρ∈Mm​⊗Mn​ with the property that UρU†U\rho U^\daggerUρU† is separable for all unitary matrices UUU. We investigate whether or not it is the case that ρ\rhoρ is absolutely separable if and only if UρU†U\rho U^\daggerUρU† has positive partial transpose for all unitary matrices UUU. In particular, we develop an easy-to-use method for showing that an entanglement witness or positive map is unable to detect entanglement in any such state, and we apply our method to many well-known separability criteria, including the range criterion, the realignment criterion, the Choi map and its generalizations, and the Breuer--Hall map. We also show that these two properties coincide for the family of isotropic states, and several eigenvalue results for entanglement witnesses are proved along the way that are of independent interest.
The separability from spectrum problem asks for a characterization of the eigenvalues of the bipartite mixed states $\ensuremath{\rho}$ with the property that ${U}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\rho}U$ is separable for all unitary matrices UUU. This problem has been solved when the local dimensions mmm and nnn satisfy m=2m=2m=2 and $n\ensuremath{\le}3$. We solve all remaining qubit-qudit cases (i.e., when m=2m=2m=2 and $n\ensuremath{\ge}4$ is arbitrary). In all of these cases we show that a state is separable from spectrum if and only if ${U}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\rho}U$ has positive partial transpose for all unitary matrices UUU. This equivalence is in stark contrast with the usual separability problem, where a state having positive partial transpose is a strictly weaker property than it being separable.
We present a framework for deciding whether a quantum state is separable or entangled using covariance matrices of locally measurable observables. This leads to the covariance matrix criterion as a general separability criterion. We demonstrate that this criterion allows us to detect many states where the familiar criterion of the positivity of the partial transpose fails. It turns out that a large number of criteria that have been proposed to complement the positive partial transpose criterion---the computable cross norm or realignment criterion, the criterion based on local uncertainty relations, criteria derived from extensions of the realignment map, and others---are in fact a corollary of the covariance matrix criterion.
Recently, a powerful separability criterion was introduced by O. Rudolf in [5] and by K. Chen et al. in [6] — basing on realignment of elements of density matrix. Composing the main idea behind the above criterion and the necessary and sufficient condition in terms of positive maps, we provide a characterization of separable states by means of linear contractions. The latter need not be positive maps. We extend the idea to multipartite systems, and find that, somewhat suprisingly, partial realigment (unlike partial transposition) can detect genuinely tri-parite entanglement. We generalize it by introducing a family of so called permutation separability criteria for multipartite states. Namely, any permutation of indices of density matrix written in product basis leads to a separability criterion. Partial transpose and realignment criterion are special cases of permutation criteria.
Separability from the spectrum is an important and ongoing research topic in quantum entanglement. In this study, we investigate properties related to absolute separability from the spectrum in qudit-qudit states in the bipartite state space Hmn=Hm⊗Hn. First, we propose the necessary and sufficient conditions for absolute positive partial transpose states in the Hilbert space H4n. These conditions are equivalent to the positive semidefiniteness of twelve matrices resulting from the symmetric matricizations of eigenvalues. Furthermore, we demonstrate that these sufficient conditions can be extended to the general Hmn case, improving existing conclusions in the literature. These sufficient conditions depend only on the first few leading and last few leading eigenvalues, significantly reducing the complexity of determining absolute separable states. On the other hand, we also introduce additional sufficient conditions for determining that states in Hmn are not absolutely separable. These conditions only depend on 2m − 1 eigenvalues of the mixed states. Our sufficient conditions are not only simple and easy to implement, but also effective in practice. As applications, we derive distance bounds for eigenvalues and purity bounds for general absolutely separable states.