Set Of Separable States Research Articles

Thermodynamic precision in the nonequilibrium exchange scenario.

We discuss exchange scenario thermodynamic uncertainty relations for the work done on a two-qubit entangled nonequilibrium steady state obtained by coupling the two qubits and putting each of them in weak contact with a thermal bath. In this way we investigate the use of entangled nonequilibrium steady states as end points of thermodynamic cycles. In this framework we prove analytically that for a paradigmatic unitary it is possible to construct an exchange scenario thermodynamic uncertainty relation. However, despite holding in many cases, we also show that such a relation ceases to be valid when considering other suitable unitary quenches. Furthermore, this paradigmatic example allows us to shed light on the role of the entanglement between the two qubits for precise work absorption. By considering the projection of the entangled steady state onto the set of separable states, we provide examples where such projection implies an increase of the relative uncertainty, showing the usefulness of entanglement.

Attainability and Lower Semi-continuity of the Relative Entropy of Entanglement and Variations on the Theme

The relative entropy of entanglement E_R is defined as the distance of a multipartite quantum state from the set of separable states as measured by the quantum relative entropy. We show that this optimisation is always achieved, i.e. any state admits a closest separable state, even in infinite dimensions; also, E_R is everywhere lower semi-continuous. We use this to derive a dual variational expression for E_R in terms of an external supremum instead of infimum. These results, which seem to have gone unnoticed so far, hold not only for the relative entropy of entanglement and its multipartite generalisations, but also for many other similar resource quantifiers, such as the relative entropy of non-Gaussianity, of non-classicality, of Wigner negativity—more generally, all relative entropy distances from the sets of states with non-negative lambda -quasi-probability distribution. The crucial hypothesis underpinning all these applications is the weak*-closedness of the cone generated by free states, and for this reason, the techniques we develop involve a bouquet of classical results from functional analysis. We complement our analysis by giving explicit and asymptotically tight continuity estimates for E_R and closely related quantities in the presence of an energy constraint.

Entangling power of symmetric two-qubit quantum gates and three-level operations

The capacity of a quantum gate to produce entangled states on a bipartite system is quantified in terms of the entangling power. This quantity is defined as the average of the linear entropy of entanglement of the states produced after applying a quantum gate over the whole set of separable states. Here we focus on symmetric two-qubit quantum gates, acting on the symmetric two-qubit space, and calculate the entangling power in terms of the appropriate local-invariant. A geometric description of the local equivalence classes of gates is given in terms of the $\mathfrak{su}(3)$ Lie algebra root vectors. These vectors define a primitive cell with hexagonal symmetry on a plane, and through the Weyl group the minimum area on the plane containing the whole set of locally equivalent quantum gates is identified. We give conditions to determine when a given quantum gate produces maximally entangled states from separable ones (perfect entanglers). We found that these gates correspond to one fourth of the whole set of locally-distinct quantum gates. The theory developed here is applicable to three-level systems in general, where the non-locality of a quantum gate is related to its capacity to perform non-rigid transformations on the Majorana constellation of a state. The results are illustrated by an anisotropic Heisenberg model, the Lipkin-Meshkov-Glick model, and two coupled quantized oscillators with cross-Kerr interaction.

The Set of Separable States has no Finite Semidefinite Representation Except in Dimension 3times 2

Given integers n ge m, let text {Sep}(n,m) be the set of separable states on the Hilbert space mathbb {C}^n otimes mathbb {C}^m. It is well-known that for (n,m)=(3,2) the set of separable states has a simple description using semidefinite programming: it is given by the set of states that have a positive partial transpose. In this paper we show that for larger values of n and m the set text {Sep}(n,m) has no semidefinite programming description of finite size. As text {Sep}(n,m) is a semialgebraic set this provides a new counterexample to the Helton–Nie conjecture, which was recently disproved by Scheiderer in a breakthrough result. Compared to Scheiderer’s approach, our proof is elementary and relies only on basic results about semialgebraic sets and functions.

Hilbert-Schmidt distance and entanglement witnessing

Gilbert proposed an algorithm for bounding the distance between a given point and a convex set. In this article we apply the Gilbert's algorithm to get an upper bound on the Hilbert-Schmidt distance between a given state and the set of separable states. While Hilbert Schmidt Distance does not form a proper entanglement measure, it can nevertheless be useful for witnessing entanglement. We provide here a few methods based on the Gilbert's algorithm that can reliably qualify a given state as strongly entangled or practically separable, while being computationally efficient. The method also outputs successively improved approximations to the Closest Separable State for the given state. We demonstrate the efficacy of the method with examples.

Trivariate analysis of two qubit symmetric separable state

One of the main problems of quantum information theory is developing the separability criterion which is both necessary and sufficient in nature. Positive partial transposition test (PPT) is one such criterion which is both necessary and sufficient for $$2\times 2$$ and $$2\times 3$$ systems but not otherwise. We express this strong PPT criterion for a system of 2-qubit symmetric states in terms of the well-known Fano statistical tensor parameters and prove that a large set of separable symmetric states are characterised by real statistical tensor parameters only. The physical importance of these states are brought out by employing trivariate representation of density matrix wherein the components of $${\mathbf{J}}$$ , namely $$J_{x}$$ , $$J_{y}$$ , $$J_{z}$$ are considered to be the three variates. We prove that this set of separable states is characterised by the vanishing average expectation value of $$J_{y}$$ and its covariance with $$J_{x}$$ and $$J_{z}$$ . This allows us to identify a symmetric separable state easily. We illustrate our criterion using 2-qubit system as an example.

Equivalence of qubit-environment entanglement and discord generation via pure dephasing interactions and the resulting consequences

We find that when a qubit initialized in a pure state experiences pure dephasing due to interaction with an environment, separable qubit-environment states generated during the evolution also have zero quantum discord with respect to the environment. What follows is that the set of separable states which can be reached during the evolution has zero volume and hence, such effects as sudden death of qubit-environment entanglement are very unlikely. In case of the discord with respect to the qubit, a vast majority of separable states qubit-environment is discordant, but in specific situations zero-discord states are possible. This is conceptually important since there is a connection between the discordance with respect to a given subsystem and the possibility of describing the evolution of this subsystem using completely positive maps. Finally, we use the formalism to find an exemplary evolution of an entangled state of two qubits that is completely positive, occurs solely due to interaction of only one of the qubits with its environment (so one could guess that it corresponds to a local operation, since it is local in a physical sense), but which nevertheless causes the enhancement of entanglement between the qubits. While this simply means that the considered evolution is completely positive, but does not belong to LOCC, it shows how much caution has to be exercised when identifying evolution channels that belong to that class.

Oblique discord

Discord and entanglement characterize two kinds of quantum correlations, and discord captures more correlation than entanglement in the sense that even separable states may have nonzero discord. In this paper, we propose a new kind of quantum correlation that we call as oblique discord. A zero-discord state corresponds to an orthonormal basis, while a zero-oblique-discord state corresponds to a basis which is not necessarily orthogonal. Under this definition, the set of zero-discord states is properly contained inside the set of zero-oblique-discord states, and the set of zero-oblique-discord states is properly contained inside the set of separable states. We give a characterization of zero-oblique-discord states via quantum mapping, provide a geometric measure for oblique discord, and raise a conjecture, which if it holds, then we can define an information-theoretic measure for oblique discord. Also, we point out that the definition of oblique discord can be properly extended to some different versions just as the case of quantum discord.

Grover’s algorithm and the secant varieties

In this paper we investigate the entanglement nature of quantum states generated by Grover's search algorithm by means of algebraic geometry. More precisely we establish a link between entanglement of states generated by the algorithm and auxiliary algebraic varieties built from the set of separable states. This new perspective enables us to propose qualitative interpretations of earlier numerical results obtained by M. Rossi et al. We also illustrate our purpose with a couple of examples investigated in details.

Geometric descriptions of entangled states by auxiliary varieties

The aim of the paper is to propose geometric descriptions of multipartite entangled states using algebraic geometry. In the context of this paper, geometric means each stratum of the Hilbert space, corresponding to an entangled state, is an open subset of an algebraic variety built by classical geometric constructions (tangent lines, secant lines) from the set of separable states. In this setting, we describe well-known classifications of multipartite entanglement such as 2 × 2 × (n + 1), for n ⩾ 1, quantum systems and a new description with the 2 × 3 × 3 quantum system. Our results complete the approach of Miyake and make stronger connections with recent work of algebraic geometers. Moreover, for the quantum systems detailed in this paper, we propose an algorithm, based on the classical theory of invariants, to decide to which subvariety of the Hilbert space a given state belongs.

Optimal multi-setting witness operators for two qudits

Quantum entanglement in bipartite systems is approached with the help of pseudo Bell operators, a relaxed form of Bell operators. Their connection to optimal witness operators is established. A method to construct pseudo Bell operators for pairs of qudits, based on multiple two-dimensional local measuring settings, is presented. Each of them provide two optimal (multi-setting) witness operators. They are rigorously proved to fulfill the conditions in the definition. We explain how to approximate the set of separable states and the set of entangled states. A class of Werner-like entangled states is identified. The spectral resolution of all multi-setting pseudo Bell operators is carried out in full.

The geometry of multi-qubit entanglement

This paper, a continuation of a previous one (Iwai 2007 J. Phys. A: Math. Theor. 40 1361), studies the geometry of multi-qubit entanglement with respect to bipartite partitions. An n-qubit system (C2)⊗n is isomorphic with , the linear space of 2ℓ × 2m complex matrices, where ℓ + m = n. According to the isomorphism, the local transformation group U(2ℓ) × U(2m) acts on the space of normalized states in . Let M and G denote the space of normalized states and the local transformation group acting on M, respectively. According to the orbit types, M is stratified into strata, among which a principal stratum and the sets of separable states and maximally entangled states will be identified. For , a function F(C) = det(I − CC*) proves to serve as a measure of entanglement, where I denotes the 2ℓ × 2ℓ identity matrix. The F(C) attains the minimal and the maximal values, respectively, on the sets of separable states and maximally entangled states, and further takes no extremal values on the principal stratum. The F(C) projects to a function on the factor space G\\M. A naturally defined metric on M also projects to that on G\\M, which serves to measure the distance between the separable states, F−1(0), and the states, F−1(k), of prescribed value k of measure. To be precise, one has to restrict M to the principal stratum, when projecting the metric. Three- and four-qubit systems will be studied intensively, and then multi-qubit systems discussed.

Nonlinear entanglement witnesses, covariance matrices and the geometry of separable states

Entanglement witnesses provide a standard tool for the analysis of entanglement in experiments. We investigate possible nonlinear entanglement witnesses from several perspectives. First, we demonstrate that they can be used to show that the set of separable states has no facets. Second, we give a new derivation of nonlinear witnesses based on covariance matrices. Finally, we investigate extensions to the multipartite case.

Multipartite entanglement and hyperdeterminants

We classify multipartite entanglement in a unified manner, focusing on a duality between the set of separable states and that of entangled states. Hyperdeterminants, derived from the duality, are natural generalizations of entanglement measures, the concurrence, 3-tangle for 2, 3 qubits respectively. Our approach reveals how inequivalent multipartite entangled classes of pure states constitute a partially ordered structure under local actions, significantly different from a totally ordered one in the bipartite case. Moreover, the generic entangled class of the maximal dimension, given by the nonzero hyperdeterminant, does not include the maximally entangled states in Bell's inequalities in general (e.g., in the \(n \!\geq\! 4\) qubits), contrary to the widely known bipartite or 3-qubit cases. It suggests that not only are they never locally interconvertible with the majority of multipartite entangled states, but they would have no grounds for the canonical \(n\)-partite entangled states. Our classification is also useful for that of mixed states.

Thermodynamics: Energy of closed and open systems

The definition of energy, in thermodynamics, is deepened by stating operative definitions of the basic concepts of physics on which it rests, such as those of isolated system, ambient of a system, separable system and set of separable states. Then the definition of energy is rigorously extended to open systems. The extension gives a clear physical meaning to the concept of energy difference between two states with arbitrarily different compositions.