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

The equivalence of multipartite quantum mixed states under local unitary transformations is studied. A criterion for the equivalence of non-degenerate mixed multipartite quantum states under local unitary transformations is presented.     

Local Unitary Invariants for Multipartite Quantum Systems

We present an approach of constructing invariants under local unitary transformations for multipartite quantum systems. The invariants constructed in this way can be complement to that in [Science 340 (2013) 1205-1208]. Detailed examples are given to compute such invariant in detail. It is shown that these invariants can be used to detect the local unitary equivalence of degenerated quantum states.     

Invariants for a Class of Nongeneric Three-Qubit States

We investigate the equivalence of quantum states under local unitary transformations. A complete set of invariants under local unitary transformations are presented for a class of non-generic three-qubit mixed states. It is shown that two such states in this class are locally equivalent if and only if all these invariants have equal values for them.     

Local Invariants for a Class of Tripartite Quantum Mixed States

We study the equivalence of tripartite mixed states under local unitary transformations. The nonlocal properties for a class of tripartite quantum states in ${\mathbb C}^K \otimes {\mathbb C}^M \otimes {\mathbb C}^N$ composite systems are investigated and a complete set of invariants under local unitary transformations for these states is presented. It is shown that two of these states are locally equivalent if and only if all these invariants have the same values.     

A New Multipartite Entanglement Measure for Arbitrary <Emphasis Type="Italic">n</Emphasis>-qudit Pure States

In this paper, we propose a multipartite entanglement measure for arbitrary pure states, which is presented based on reduced density matrices of multi-qudit pure states. We review some multipartite entanglement measures based on density matrices. This is helpful for us to introduce a new good entanglement measure, which is vanishing if and only if a state is separable, invariant under local unitary transformations and non-increasing under local operations assisted by classical communication. We apply our entanglement measure for some explicit examples. It demonstrates that our entanglement measure is practical and convenient for computation. It can also distinguish the relatively high entanglement and the maximal entanglement. In short, our entanglement measure is good at characterizing multipartite entanglement.     

Local unitary classification of arbitrary dimensional multipartite pure states

We propose a practical entanglement classification scheme for general multipartite pure states in arbitrary dimensions under local unitary equivalence by exploiting the high order singular value decomposition technique and local symmetries of the states. By virtue of this scheme, the method of determining the local unitary equivalence of n-qubit states proposed by Kraus is extended to the case for arbitrary dimensional multipartite states.     

Randomized Entangled Mixed States from Phase States

We construct randomized entangled mixed states by using the formalism of phase states for d-dimensional systems (qudits). The randomized entangled mixed states are a special kind of mixed states that exhibit genuine multipartite correlation. Such states are obtained by the application of randomized entangling operators to an arbitrary pair of qudits of a multiqudit system. The study of the entanglement of randomized mixed states is of great importance in quantum computation since any experimental implementation of entangled states in a realistic environment can be made by imperfect entangling gates. We give a brief review of some necessary background about unitary phase operators and phase states of a multi-qudit system. Evolved density matrices arise when qudits of the multi-qudit system interact via a Hamiltonian of Heisenberg type. The randomized entangled states associated with evolved density matrices are derived via the action of an entangling operator on a pair of two qudits {i, j} of the multi-qudit system with some probability p. The randomized entangled mixed states for bipartite, tripartite and multipartite systems are explicitly expressed and their Kraus decomposition properties are discussed.

     

A note on local unitary equivalence of isotropic-like states

We consider the local unitary equivalence of a class of quantum states in a bipartite case and a multipartite case. The necessary and sufficient condition is presented. As special cases, the local unitary equivalent classes of isotropic state and Werner state are provided. Then we study the local unitary similar equivalence of this class of quantum states and analyze the necessary and sufficient condition.     

On the Non-k-Separability of Dicke Class of States and N-Qudit W States

In this paper, we present the separability criteria to identify non-k-separability and genuine multipartite entanglement in mixed multipartite states using elements of density matrices. Our criteria can detect the non-k-separability of Dicke class of states, anti W states and mixtures thereof and higher dimensional W class of states. We then investigate the performance of our criteria by considering N-qubit Dicke states with arbitrary excitations added with white noise and mixture of N-qudit W state with white noise. We also study the robustness of our criteria against white noise. Further, we demonstrate that our criteria are experimentally implementable by means of local observables such as Pauli matrices and generalized Gell-Mann matrices.     

Completely mixed state is a critical point for three-qubit entanglement

Pure three-qubit states have five algebraically independent and one algebraically dependent polynomial invariants under local unitary transformations and an arbitrary entanglement measure is a function of these six invariants. It is shown that if the reduced density operator of a some qubit is a multiple of the unit operator, than the geometric entanglement measure of the pure three-qubit state is absolutely independent of the polynomial invariants and is a constant for such tripartite states. Hence a one-particle completely mixed state is a critical point for the geometric measure of entanglement.     

Entanglement in four qubit states: Polynomial invariant of degree 2, genuine multipartite concurrence and one-tangle

Characterization of the multipartite mixed state entanglement is still a challenging problem. This is due to the fact that the entanglement for the mixed states, in general, is defined by a convex-roof extension. That is the entanglement measure of a mixed state ρ of a quantum system can be defined as the minimum average entanglement of an ensemble of pure states. In this paper, we show that polynomial entanglement measures of degree 2 of even-N qubits X states is in the full agreement with the genuine multipartite (GM) concurrence. Then, we plot the hierarchy of entanglement classification for four qubit pure states and then using new invariants, we classify the four qubit pure states. We focus on the convex combination of the classes whose at most the one of the invariants is non-zero and find the relationship between entanglement measures consist of non-zero-invariant, GM concurrence and one-tangle. We show that in many entanglement classes of four qubit states, GM concurrence is equal to the square root of one-tangle.     

Teleportations of Mixed States and Multipartite Quantum States

In this paper, we propose a protocol to deterministically teleport an unknown mixed state of qubit by utilizing a maximally bipartite entangled state of qubits as quantum channel. Ira non-maximally entangled bipartite pure state is employed as quantum channel, the unknown mixed quantum state of qubit can be teleported with 1 -√ 1- C^2 probability, where C is the concurrence of the quantum channel. The protocol can also be generalized to teleport a mixed state of qudit or a multipartite mixed state. More important purpose is that, on the basis of the protocol, the teleportation of an arbitrary multipartite （pure or mixed） quantum state can be decomposed into the teleportation of each subsystem by employing separate entangled states as quantum channels. In the case of deterministic teleportation, Bob only needs to perform unitary transformations on his single particles in order to recover the initial teleported multipartite quantum state.     

Entanglement evolution of three-qubit mixed states in multipartite cavity—reservoir systems

We analyze the multipartite entanglement evolution of three-qubit mixed states composed of a GHZ state and a W state.For a composite system consisting of three cavities interacting with independent reservoirs,it is shown that the entanglement evolution is restricted by a set of monogamy relations.Furthermore,as quantified by the negativity,the entanglement dynamical property of the mixed entangled state of cavity photons is investigated.It is found that the three cavity photons can exhibit the phenomenon of entanglement sudden death(ESD).However,compared with the evolution of a generalized three-qubit GHZ state which has the equal initial entanglement,the ESD time of mixed states is later than that of the pure state.Finally,we discuss the entanglement distribution in the multipartite system,and point out the intrinsic relation between the ESD of cavity photons and the entanglement sudden birth of reservoirs.     

A Note on Normal Forms of Quantum States and Separability

We study the normal form of multipartite density matrices. It is shown that the correlation matrix (CM) separability criterion can be improved from the normal form we obtained under filtering transformations. Based on CM criterion the entanglement witness is further constructed in terms of local orthogonal observables for both bipartite and multipartite systems.     

LU Invariants and Canonical Forms and SLOCC Classification of Pure 3-Qubit States

In this paper the entanglement of pure 3-qubit states is discussed. The local unitary (LU) polynomial invariants that are closely related to the canonical forms are constructed and the relations of the coefficients of the canonical forms are given. Then the stochastic local operations and classical communication (SLOCC) classification of the states are discussed on the basis of the canonical forms, and the symmetric canonical form of the states without 3-tangle is discussed. Finally, we give the relation between the LU polynomial invariants and SLOCC classification.     

Rescaling multipartite entanglement measures for mixed states

A relevant problem regarding entanglement measures is the following: Given an arbitrary mixed state, how does a measure for multipartite entanglement change if general local operations are applied to the state? This question is nontrivial as the normalization of the states has to be taken into account. Here we answer it for pure-state entanglement measures which are invariant under determinant-one local operations and homogeneous in the state coefficients, and their convex-roof extension which quantifies mixed-state entanglement. Our analysis allows us to enlarge the set of mixed states for which these important measures can be calculated exactly. In particular, our results hint at a distinguished role of entanglement measures which have homogeneous degree 2 in the state coefficients.     

多粒子纠缠的保护方案

量子纠缠是量子信息的重要物理资源. 然而当量子系统与环境相互作用时, 会不可避免地产生消相干导致纠缠下降, 因此保护纠缠不受环境的影响具有重要意义. 振幅衰减是一种典型的衰减机制. 如果探测环境保证没有激发从系统中流出, 即视为对系统的一种弱测量. 本文基于局域脉冲序列和弱测量, 提出了一种可以保护多粒子纠缠不受振幅衰减影响的有效物理方案, 保护的对象是在量子通信和量子计算中发挥重要作用的Cluster态和Maximal slice态.     

Bell-Klyshko inequalities to characterize maximally entangled states of n qubits

This Letter presents the first rigorous proof of the conjecture raised by Gisin and Bechmann-Pasquinucci [Phys. Lett. A 246, 1 (1998)]], that the Greenberger-Horne-Zeilinger states of n qubits and the states obtained from them by local unitary transformations are the unique states that maximally violate the Bell-Klyshko inequalities. The proof is obtained by using the certain algebraic properties that Pauli's matrices satisfy and some subtle mathematical techniques. Since all states obtained by local unitary transformations of a maximally entangled state are equally valid entangled states, we thus give a characterization of maximally entangled states of n qubits in terms of the Bell-type inequality.     

Distribution of the local density of states, reflection coefficient, and Wigner delay time in absorbing ergodic systems at the point of chiral symmetry

Employing the chiral Gaussian unitary ensemble of random matrices, we calculate the probability distribution of the local density of states for zero-dimensional ("quantum chaotic") two-sublattice systems at the point of chiral symmetry E=0 and in the presence of uniform absorption. The obtained result can be used to find the distributions of the reflection coefficient and of the Wigner time delay for such systems.     

Quantification and scaling of multipartite entanglement in continuous variable systems

We present a theoretical method to determine the multipartite entanglement between different partitions of multimode, fully or partially symmetric Gaussian states of continuous variable systems. For such states, we determine the exact expression of the logarithmic negativity and show that it coincides with that of equivalent two-mode Gaussian states. Exploiting this reduction, we demonstrate the scaling of the multipartite entanglement with the number of modes and its reliable experimental estimate by direct measurements of the global and local purities.