Multipartite Mixed States Research Articles

On Classes of Local Unitary Transformations

We give a one-to-one correspondence between classes of density matrices under local unitary invariance and the double cosets of unitary groups. We show that the interrelationship among classes of local unitary equivalent multi-partite mixed states is independent from the actual values of the eigenvalues and only depends on the multiplicities of the eigenvalues. The interpretation in terms of homogeneous spaces of unitary groups is also discussed.

Beyond pure state entanglement for atomic ensembles

We analyze multipartite entanglement between atomic ensembles within quantum matter–light interfaces. In our proposal, a polarized light beam crosses sequentially several polarized atomic ensembles impinging on each of them at a given angle αi. These angles are crucial parameters for shaping the entanglement since they are directly connected to the appropriate combinations of the collective atomic spins that are squeezed. We exploit such a scheme to go beyond the pure state paradigm proposing realistic experimental settings to address multipartite mixed state entanglement in continuous variables.

Simple algorithm for computing the geometric measure of entanglement

We present an easy implementable algorithm for approximating the geometric measure of entanglement from above. The algorithm can be applied to any multipartite mixed state. It involves only the solution of an eigenproblem and finding a singular value decomposition, no further numerical techniques are needed. To provide examples, the algorithm was applied to the isotropic states of 3 qubits and the 3-qubit XX model with external magnetic field.

Quantification of Macroscopic Quantum Superpositions within Phase Space

Based on phase-space structures of quantum states, we propose a novel measure to quantify macroscopic quantum superpositions. Our measure simultaneously quantifies two different kinds of essential information for a given quantum state in a harmonious manner: the degree of quantum coherence and the effective size of the physical system that involves the superposition. It enjoys remarkably good analytical and algebraic properties. It turns out to be the most general and inclusive measure ever proposed that it can be applied to any types of multipartite states and mixed states represented in phase space.

Linking a distance measure of entanglement to its convex roof

An important problem in quantum information theory is the quantification of entanglement in multipartite mixed quantum states. In this work, a connection between the geometric measure of entanglement and a distance measure of entanglement is established. We present a new expression for the geometric measure of entanglement in terms of the maximal fidelity with a separable state. A direct application of this result provides a closed expression for the Bures measure of entanglementof two qubits. We also prove that the number of elements in an optimal decomposition w.r.t. the geometric measure of entanglement is bounded from above by the Caratheodory bound, and we find necessary conditions for the structure of an optimal decomposition.

ENTANGLEMENT PROPERTY AND MONOGAMY RELATION OF GENERALIZED MIXED W STATES

We introduce a new class of multipartite entangled mixed states with pure state decompositions of generalized W states, similar to Schmidt-correlated states having generalized GHZ states in the pure state decomposition. The entanglement and separability properties are studied according to PPT operations. Monogamy relations linked to these states are also investigated.

A lower bound of concurrence for multipartite quantum states

We study the concurrence of arbitrary multipartite mixed quantum states. An explicit lower bound of the concurrence is derived, which detects quantum entanglement of some states better than some separability criteria, and gives sufficient conditions for distilling GHZ states from tripartite states. An interesting relation between the lower bound of the concurrence for bipartite states and for tripartite states has been presented.

Multipartite entanglement criterion from uncertainty relations

We formulate an entanglement criterion using Peres-Horodecki positive partial transpose operations combined with the Schrodinger-Robertson uncertainty relation. We show that any pure entangled bipartite and tripartite state can be detected by experimentally measuring mean values and variances of specific observables. Those observables must satisfy a specific condition in order to be used, and we show their general form in the 2×2 (two qubits) dimension case. The criterion is applied on a variety of physical systems including bipartite and multipartite mixed states and reveals itself to be stronger than the Bell inequalities and other criteria. The criterion also work on continuous variable cat states and angular momentum states of the radiation field.

Mixed-state localizable entanglement for continuous variables

We investigate localization of entanglement of multipartite mixed Gaussian states into a pair of modes by local Gaussian measurements on the remaining modes and classical communication. We provide a detailed proof that for arbitrary symmetric Gaussian state maximum entanglement can be localized by homodyne detection of either amplitude or phase quadrature on each mode. We then consider arbitrary mixed three-mode Gaussian states and show that the optimal Gaussian measurement on one mode yielding maximum entanglement among the other two modes can be determined by calculating roots of a high-order polynomial. Finally, we discuss localization of entanglement with single-photon detection.

Estimation of concurrence for multipartite mixed states

We investigate the lower bound of concurrence for multipartite quantum mixed states. Analytical lower bounds are derived for some multipartite systems, by establishing functional relations between concurrence and the generalized partial transpositions.

Multipartite entangled states in particle mixing

In the physics of flavor mixing, the flavor states are given by superpositions of mass eigenstates. By using the occupation number to define a multiqubit space, the flavor states can be interpreted as multipartite mode-entangled states. By exploiting a suitable global measure of entanglement, based on the entropies related to all possible bipartitions of the system, we analyze the correlation properties of such states in the instances of three- and four-flavor mixing. Depending on the mixing parameters, and, in particular, on the values taken by the free phases, responsible for the CP-violation, entanglement concentrates in certain bipartitions. We quantify in detail the amount and the distribution of entanglement in the physically relevant cases of flavor mixing in quark and neutrino systems. By using the wave packet description for localized particles, we use the global measure of entanglement, suitably adapted for the instance of multipartite mixed states, to analyze the decoherence, induced by the free evolution dynamics, on the quantum correlations of stationary neutrino beams. We define a decoherence length as the distance associated with the vanishing of the coherent interference effects among massive neutrino states. We investigate the role of the CP-violating phase in the decoherence process.

On the degree conjecture for separability of multipartite quantum states

We settle the so-called degree conjecture for the separability of multipartite quantum states, which are normalized graph Laplacians, first given by Braunstein et al. [Phys. Rev. A 73, 012320 (2006)]. The conjecture states that a multipartite quantum state is separable if and only if the degree matrix of the graph associated with the state is equal to the degree matrix of the partial transpose of this graph. We call this statement to be the strong form of the conjecture. In its weak version, the conjecture requires only the necessity, that is, if the state is separable, the corresponding degree matrices match. We prove the strong form of the conjecture for pure multipartite quantum states using the modified tensor product of graphs defined by Hassan and Joag [J. Phys. A 40, 10251 (2007)], as both necessary and sufficient condition for separability. Based on this proof, we give a polynomial-time algorithm for completely factorizing any pure multipartite quantum state. By polynomial-time algorithm, we mean that the execution time of this algorithm increases as a polynomial in m, where m is the number of parts of the quantum system. We give a counterexample to show that the conjecture fails, in general, even in its weak form, for multipartite mixed states. Finally, we prove this conjecture, in its weak form, for a class of multipartite mixed states, giving only a necessary condition for separability.

A complete set of local invariants for a family of multipartite mixed states

We study the equivalence of quantum states under local unitary transformations by using the singular value decomposition. A complete set of invariants under local unitary transformations is presented for several classes of tripartite mixed states in composite systems. Two density matrices in the same class are equivalent under local unitary transformations if and only if all these invariants have equal values for these density matrices.

Analytically solvable two-qubit entanglement monotone

A so-called quasientanglement measure was obtained [J. Zhang et al., Phys. Rev. A 73, 022319 (2006)] by extending Jaeger's Minkowskian norm entanglement measure to general multipartite mixed states. The obtained quasientanglement measure has good properties, but it is still left open whether it is nonincreasing under general local operations and classical communications (LOCC). We show that the obtained quasientanglement measure is indeed nonincreasing on average under LOCC at least for two-qubit states. Thus, it is a good candidate for the two-qubit entanglement monotone. Further, it is shown that this entanglement measure is less than the concurrence for a large class of mixed entangled states.

Teleportations of Mixed States and Multipartite QuantumStates

In this paper, we propose a protocol to deterministically teleportan unknown mixed state of qubit by utilizing a maximally bipartiteentangled state of qubits as quantum channel. If a non-maximallyentangled bipartite pure state is employed as quantum channel, theunknown mixed quantum state of qubit can be teleported with1−(1−C2)1/2 probability, where C is the concurrence of thequantum channel. The protocol can also be generalized to teleport amixed state of qudit or a multipartite mixed state. More importantpurpose is that, on the basis of the protocol, the teleportation ofan arbitrary multipartite (pure or mixed) quantum state can bedecomposed into the teleportation of each subsystem by employingseparate entangled states as quantum channels. In the case ofdeterministic teleportation, Bob only needs to perform unitarytransformations on his single particles in order to recover theinitial teleported multipartite quantum state.

Entangled Three-Qubit States without Concurrence and Three-Tangle

We provide a complete analysis of mixed three-qubit states composed of a Greenberger-Horne-Zeilinger state and a W state orthogonal to the former. We present optimal decompositions and convex roofs for the three-tangle. Further, we provide an analytical method to decide whether or not an arbitrary rank-2 state of three qubits has vanishing three-tangle. These results highlight intriguing differences compared to the properties of two-qubit mixed states, and may serve as a quantitative reference for future studies of entanglement in multipartite mixed states. By studying the Coffman-Kundu-Wootters inequality we find that, while the amounts of inequivalent entanglement types strictly add up for pure states, this "monogamy" can be lifted for mixed states by virtue of vanishing tangle measures.

Matrix Tensor Product Approach to the Equivalence of Multipartite States underLocal Unitary Transformations

The equivalence of multipartite quantum mixed states underlocal unitary transformations is studied. A criterion for theequivalence of non-degenerate mixed multipartite quantum states underlocal unitary transformations is presented.

Quasimultipartite entanglement measure based on quadratic functions

We develop a new entanglement measure by extending Jaeger's Minkowskian norm entanglement measure. This measure can be applied to a much wider class of multipartite mixed states, although still "quasi" in the sense that it is still incapable of dividing precisely the sets of all separable and entangled states. As a quadratic scalar function of the system density matrix, the quasi measure can be easily expressed in terms of the so-called coherence vector of the system density matrix, by which we show the basic properties of the quasi measure including (1) zero-entanglement for all separable states, (2) invariance under local unitary operations, and (3) non-increasing under local POVM (positive operator-valued measure) measurements. These results open up perspectives in further studies of dynamical problems in open systems, especially the dynamic evolution of entanglement, and the entanglement preservation against the environment-induced decoherence effects.

Mixed-state entanglement of assistance and the generalized concurrence

We consider the maximum bipartite entanglement that can be distilled from a single copy of a multipartite mixed entangled state, where we focus mostly on $d\times d\times n$-dimensional tripartite mixed states. We show that this {\em assisted entanglement}, when measured in terms of the generalized concurrence (named G-concurrence) is (tightly) bounded by an entanglement monotone, which we call the G-concurrence of assistance. The G-concurrence is one of the possible generalizations of the concurrence to higher dimensions, and for pure bipartite states it measures the {\em geometric mean} of the Schmidt numbers. For a large (non-trivial) class of $d\times d$-dimensional mixed states, we are able to generalize Wootters formula for the concurrence into lower and upper bounds on the G-concurrence. Moreover, we have found an explicit formula for the G-concurrence of assistance that generalizes the expression for the concurrence of assistance for a large class of $d\times d\times n$ dimensional tripartite pure states.

CRITERIA OF PARTIAL SEPARABILITY OF MULTIPARTITE QUBIT MIXED STATES

We discuss the partial separability and its criteria problems of multipartite qubit mixed states. First, we strictly define what is the partial separability of a multipartite qubit system. Next, we provide a way of reduction from N-partite qubit density matrixes to bipartite qubit density matrixes, and prove a necessary condition that an N-partite qubit mixed state to be partially separable is its reduction to satisfy the PPT condition.