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arXiv:2211.02871 · Open Systems & Information Dynamics

Linear Maps as Sufficient Criteria for Entanglement Depth and Compatibility in Many-Body Systems

Maciej Lewenstein, Guillem Müller-Rigat, Jordi Tura, Anna Sanpera

Published Sep 1, 2022• 4 citations

Abstract

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.

DOI: 10.1142/s1230161222500111

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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.

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.

Abstract Quantum states that remain separable (i.e. not entangled) under any global unitary transformation are known as absolutely separable and form a convex set. Despite extensive efforts, the complete characterization of this set remains largely unknown. In this work, we employ linear maps and their inverses to derive new sufficient analytical conditions for absolute separability in arbitrary dimensions, providing extremal points of this set and improving its characterization. Additionally, we employ convex geometry optimization to refine the characterization of the set when multiple non-comparable criteria for absolute separability are available. We also address the closely related problem of characterizing the absolute PPT (positive partial transposition) set, which consists of quantum states that remain positive under partial transposition across all unitary transformations. Finally, we extend our results to multipartite states. We are proud to dedicate our work to Professor Ryszard Horodecki, whose pioneering contributions to the field of quantum entanglement continue to inspire us all. With deep gratitude and respect.

Let

H

and

K

be (finite or infinite dimensional) complex Hilbert spaces. A characterization of positive completely bounded normal linear maps from

{\mathcal B}(H)

into

{\mathcal B}(K)

is given, which particularly gives a characterization of positive elementary operators including all positive linear maps between matrix algebras. This characterization is then applied give a representation of quantum channels (operations) between infinite-dimensional systems. A necessary and sufficient criterion of separability is give which shows that a state

\rho

on

H\otimes K

is separable if and only if

(\Phi\otimes I)\rho\geq 0

for all positive finite rank elementary operators

\Phi

. Examples of NCP and indecomposable positive linear maps are given and are used to recognize some entangled states that cannot be recognized by the PPT criterion and the realignment criterion.

A family of separability criteria based on correlation matrix (tensor) is provided. Interestingly, it unifies several criteria known before like, e.g., computable cross-norm or realignment criterion (CNNR), de Vicente criterion, and derived recently separability criterion based on symmetric informationally complete positive operator valued measures (SIC POVMs). It should be stressed that, unlike the well-known correlation matrix criterion or criterion based on local uncertainty relations, our criteria are linear in the density operator and hence one may find unexplored classes of entanglement witnesses and positive maps. Interestingly, there is a natural generalization to multipartite scenario using multipartite correlation matrix. We illustrate the detection power of the above criteria on several well-known examples of quantum states.

Abstract We analyze positivity of a tensor product of two linear qubit maps, Φ 1 ⊗ Φ 2 . Positivity of maps Φ 1 and Φ 2 is a necessary but not a sufficient condition for positivity of Φ 1 ⊗ Φ 2 . We find a non-trivial sufficient condition for positivity of the tensor product map beyond the cases when both Φ 1 and Φ 2 are completely positive or completely co-positive. We find necessary and (separately) sufficient conditions for n -tensor-stable positive qubit maps, i.e. such qubit maps Φ that Φ ⊗ n is positive. Particular cases of 2- and 3-tensor-stable positive qubit maps are fully characterized, and the decomposability of 2-tensor-stable positive qubit maps is discussed. The case of non-unital maps is reduced to the case of appropriate unital maps. Finally, n -tensor-stable positive maps are used in characterization of multipartite entanglement, namely, in the entanglement depth detection.