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.
Physical transformations are described by linear maps that are completely positive and trace preserving (CPTP). However, maps that are positive (P) but not completely positive (CP) are instrumental to derive separability/entanglement criteria. Moreover, the properties of such maps can be linked to entanglement properties of the states they detect. Here, we extend the results presented in [34], where sufficient separability criteria for bipartite systems were derived. In particular, we analyze the entanglement depth of an [Formula: see text]-qubit system by proposing linear maps that, when applied to any state, result in a biseparable state for the [Formula: see text] partitions, i.e., [Formula: see text]-entanglement depth. Furthermore, we derive criteria to detect arbitrary [Formula: see text]-entanglement depth tailored to states in close vicinity of the completely depolarized state (the normalized identity matrix). We also provide separability (or [Formula: see text]-entanglement depth) conditions in the symmetric sector, including the diagonal states. Finally, we suggest how similar map techniques can be used to derive sufficient conditions for a set of expectation values to be compatible with separable states or local-hidden-variable theories. We dedicate this paper to the memory of the late Andrzej Kossakowski, our spiritual and intellectual mentor in the field of linear maps.
Absolutely separable states ϱ remain separable under arbitrary unitary transformations . By example of a three qubit system we show that in a multipartite scenario neither full separability implies bipartite absolute separability nor the reverse statement holds. The main goal of the paper is to analyze quantum maps resulting in absolutely separable output states. Such absolutely separating maps affect the states in a way, when no Hamiltonian dynamics can make them entangled afterwards. We study the general properties of absolutely separating maps and channels with respect to bipartitions and multipartitions and show that absolutely separating maps are not necessarily entanglement breaking. We examine the stability of absolutely separating maps under a tensor product and show that is absolutely separating for any N if and only if Φ is the tracing map. Particular results are obtained for families of local unital multiqubit channels, global generalized Pauli channels, and combination of identity, transposition, and tracing maps acting on states of arbitrary dimension. We also study the interplay between local and global noise components in absolutely separating bipartite depolarizing maps and discuss the input states with high resistance to absolute separability.
A quantum state is called absolutely separable (resp. absolutely positive partial transpose (APPT)) if it remains separable (resp. positive partial transpose (PPT)) under any global unitary transformation. It is known that the set of all absolutely separable states is a subset of the set of all APPT states. Moreover, these two sets are identical for qubit-qudit systems. In this note, we give an upper bound for the purity of APPT bipartite states (and therefore for the purity of absolutely separable bipartite states). For qubit-qudit systems, this upper bound becomes the maximum purity of APPT (and absolutely separable) states.
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.
We study families of positive and completely positive maps acting on a bipartite system ${\mathbb{C}}^{M}\ensuremath{\bigotimes}{\mathbb{C}}^{N}$ (with $M\ensuremath{\le}N$). The maps have a property that, when applied to any state (of a given entanglement class), result in a separable state or, more generally, a state of another certain entanglement class (e.g., Schmidt number $\ensuremath{\le}\phantom{\rule{0.16em}{0ex}}k$). This allows us to derive useful families of sufficient separability criteria. Explicit examples of such criteria have been constructed for arbitrary
M,N, with a special emphasis on
M=2. Our results can be viewed as generalizations of the known facts that in the sufficiently close vicinity of the completely depolarized state (the normalized identity matrix), all states are separable (belong to ``weakly'' entangled classes). Alternatively, some of our results can be viewed as an entanglement classification for a certain family of states, corresponding to mixtures of the completely polarized state with pure states, partial transposes, and/or local transformations thereof.