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.
The absolute separability 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 investigate whether or not it is the case that
ρ is absolutely separable if and only if
UρU† has positive partial transpose for all unitary matrices
U. 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.
We investigate the geometric structure of the set of spectra of bipartite absolute separable states (
ASEPm,n) and absolute positive partial transpose states (
APPTm,n), \emph{i.e.}, bipartite quantum states that remain separable or PPT, respectively, under all global unitary transformations. First, we establish general geometric properties of absolute convex sets of matrices, their spectra and extreme points. Regarding absolute separability, we present a permutation-symmetric reformulation of the absolute PPT criterion and use it to demonstrate that
APPTm,n is a spectrahedron for all
m≤n: in particular, all its faces are exposed. While
ASEP2,n is a spectrahedron, too, for
ASEPm,n we can in general only show that it is a semialgebraic set for all
m,n≥3. Furthermore, we provide a complete characterization of the faces and extreme points of
APPTm,n and demonstrate that the dimension of a face is determined by the rank of a certain matrix, with maximal proper faces having dimension
mn−m−1.
In the quantitative setting, we provide a rigorous lower bound on the maximal attainable purity of
APPTm,n via an inscribed polytope
Pm,n and conjecture that the maximal purity of
APPTm,n (along with its spectra) coincides with the polytope for arbitrary dimensions except when
m=n=2. Additionally, we also provide a rigorous upper bound on the minimal von Neumann entropy of
APPTm,n and demonstrate numerically that the minimum entropy eventually coincides with the polytope
Pm,n as the local system dimension
n increases. Finally, we show that the relative spectral volume of
APPTm,n decays exponentially in
n by a constant multiplicative factor of the relative volume of the inscribed polytope
Pm,n.
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
U. This problem has been solved when the local dimensions
m and
n satisfy
m=2 and $n\ensuremath{\le}3$. We solve all remaining qubit-qudit cases (i.e., when
m=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
U. 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.
The absolute separability 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 investigate whether or not it is the case that
ρ is absolutely separable if and only if
UρU† has positive partial transpose for all unitary matrices
U. 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.
States of sufficiently low purity are separable and cannot be entangled by unital (purity-non-generating) operations. Since high-purity states are experimentally demanding, it is natural to ask how much purity a state must possess to enable entanglement generation. Absolutely separable states remain separable under all deterministic unital channels, and so cannot deterministically generate entanglement in this setting. We show, however, that some absolutely separable states can generate entanglement via probabilistic protocols that do not produce purity. This motivates the study of states that fail to generate entanglement with any non-zero probability, which we call completely absolutely separable; we give a full characterization of this class. Along the way, we derive a novel sufficient condition for separability that depends only on the largest and smallest eigenvalues, along with the smallest local dimension and is independent of all previously known spectral separability criteria.