Stochastic Local Operations And Classical Communication Research Articles

TESTING EQUIVALENCE OF PURE QUANTUM STATES AND GRAPH STATES UNDER SLOCC

A set of necessary and sufficient conditions are derived for the equivalence of an arbitrary pure state and a graph state on n qubits under stochastic local operations and classical communication (SLOCC), using the stabilizer formalism. Because all stabilizer states are equivalent to graph states by local unitary transformations, these conditions constitute a classical algorithm for the determination of SLOCC-equivalence of pure states and stabilizer states. This algorithm provides a distinct advantage over the direct solution of the SLOCC-equivalence condition |ψ〉 = S|g〉 for an unknown invertible local operator S, as it usually allows for easy detection of states that are not SLOCC-equivalent to graph states.

Bounds on probability of transformations between multipartite pure states

For a tripartite pure state of three qubits, it is well known that there are two inequivalent classes of genuine tripartite entanglement, namely the GHZ-class and the W-class. Any two states within the same class can be transformed into each other with stochastic local operations and classical communication (SLOCC) with a non-zero probability. The optimal conversion probability, however, is only known for special cases. Here, we derive new lower and upper bounds for the optimal probability of transformation from a GHZ-state to other states of the GHZ-class. A key idea in the derivation of the upper bounds is to consider the action of the LOCC protocol on a different input state, namely $1/\sqrt{2} [\ket{000} - \ket{111}]$, and demand that the probability of an outcome remains bounded by 1. We also find an upper bound for more general cases by using the constraints of the so-called interference term and 3-tangle. Moreover, we generalize some of our results to the case where each party holds a higher-dimensional system. In particular, we found that the GHZ state generalized to three qutrits, i.e., $\ket{\mathrm{GHZ}_3} = 1/\sqrt{3} [ \ket{000} + \ket{111} + \ket{222} ] $, shared among three parties can be transformed to {\it any} tripartite 3-qubit pure state with probability 1 via LOCC. Some of our results can also be generalized to the case of a multipartite state shared by more than three parties.

Operational Families of Entanglement Classes for SymmetricN-Qubit States

We solve the entanglement classification under stochastic local operations and classical communication (SLOCC) for all multipartite symmetric states in the general N-qubit case. For this purpose, we introduce 2 parameters playing a crucial role, namely, the diversity degree and the degeneracy configuration of a symmetric state. Those parameters give rise to a simple method of identifying operational families of SLOCC entanglement classes of all symmetric N-qubit states, where the number of families grows as the partition function of the number of qubits.

PartialK-way negativities of pure four-qubit entangled states

It has been shown by Versraete et. al [F. Versraete, J. Dehaene, B. De Moor, and H. Verschelde, Phys. Rev. A65, 052112 (2002)] that by stochastic local operations and classical communication (SLOCC), a pure state of four qubits can be transformed to a state belonging to one of a set of nine families of states. By using selective partial transposition, we construct partial K-way negativities to measure the genuine 4-partite, tripartite, and bi-partite entanglement of single copy states belonging to the nine families of four qubit states. Partial K-way negativities are polynomial functions of local invariants characterizing each family of states as such entanglement monotones.

Special entangled quantum systems and the Freudenthal construction

We consider special quantum systems containing both distinguishable and identical constituents. It is shown that for these systems the Freudenthal construction based on cubic Jordan algebras naturally defines entanglement measures invariant under the group of stochastic local operations and classical communication (SLOCC). For this type of multipartite entanglement the SLOCC classes can be explicitly given. These results enable further explicit constructions of multiqubit entanglement measures for distinguishable constituents by embedding them into identical fermionic ones. We also prove that the Plücker relations for the embedding system provide a sufficient and necessary condition for the separability of the embedded one. We argue that this embedding procedure can be regarded as a convenient representation for quantum systems of particles which are really indistinguishable but for some reason they are not in the same state of some inner degree of freedom.

Macroscopic inequivalent entanglement witness in Heisenberg spin chain

Motivated by the wise idea of entanglement witness (EW), we present an inequivalent entanglement witness (IEEW) that can analogously classify certain eigenstates entangled in inequivalent ways under stochastic local operations and classical communication (SLOCC) in the Heisenberg spin chain. Since the IEEW is the absolute value of magnetization |M| that is a macroscopically measurable quantity, our conclusions provide a macroscopic method to detect inequivalent entanglement between microscopic spins, on the one hand, and clearly show that inequivalent entanglement can yield different macroscopic effects, on the other hand.

Classification of depolarized maps

We classify the completely positive maps acting on two $d$-dimensional systems which commute with all $U\ensuremath{\bigotimes}U$ unitaries, where $U∊\mathrm{SU}(d)$. This set of operations map Werner states to Werner states. We find a simple condition for a map to be implementable by stochastic local operations and classical communication (SLOCC). We show that all positive partial transpose (PPT) preserving maps can be implemented by SLOCC. This can be used to prove that the entanglement of Werner states cannot be stochastically increased, even if we allow PPT entanglement for free, which provides a necessary result for proving the result of Masanes [Phys. Rev. Lett. 96, 150501 (2006)], one of the central results in the theory of entanglement.

Epsilon-measures of entanglement

We associate to every entanglement measure a family of measures which depend on a precision parameter, and which we call ε-measures of entanglement. Their definition aims at addressing a realistic scenario in which we need to estimate the amount of entanglement in a state that is only partially known. We show that many properties of the original measure are inherited by the family, in particular weak monotonicity under transformations applied by means of local operations and classical communication (LOCC). On the other hand, they may increase on average under stochastic LOCC. Remarkably, the ε-version of a convex entanglement measure is continuous even if the original entanglement measure is not, so that the ε-version of an entanglement measure may be actually considered a smoothed version of it.

Discussion of the entanglement classification of a 4-qubit pure state

We classify 4-qubit pure states under the stochastic local operation and classical communication (SLOCC). There exist twenty three essentially different classes of states, giving rise to a four-graded partially ordered structure. We also give the criterion to judge which class an arbitrary 4-qubit state belongs to. We re-classify the 4-qubit pure state into 2×2×4, 4×4 aspects. Finally, we give our analysis of the classification difference of methods for the 3-qubit pure state.

Classification and quantification of qubit and qutrit multipartite pure state entanglement

Classification and quantification of bipartite qubit, tripartite qubit, and bipartite qutrit entanglement are studied in detail based on the entanglement measure of multipartite pure states. The results show that different ways of pure state entanglement can be characterized by extremal values of the corresponding entanglement measure under the stochastic local operation and classical communication(SLOCC). Detailed analysis shows that there are three different kinds of entanglement under the SLOCC in both tripartite qubit and bipartite qutrit pure state systems. Comparison of the measure with the recently proposed entropy-product measure is made and briefly discussed.

Inductive classification of multipartite entanglement under stochastic local operations and classical communication

We propose an inductive procedure to classify N-partite entanglement under stochastic local operations and classical communication (SLOCC) provided such a classification is known for N-1 qubits. The method is based upon the analysis of the coefficient matrix of the state in an arbitrary product basis. We illustrate this approach in detail with the well-known bi- and tripartite systems, obtaining as a by-product a systematic criterion to establish the entanglement class of a given pure state without resourcing to any entanglement measure. The general case is proved by induction, allowing us to find an upper bound for the number of N-partite entanglement classes in terms of the number of entanglement classes for N-1 qubits.

LU Invariants and Canonical Forms and SLOCC Classification of Pure 3-QubitStates

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

Classification of tripartite entanglement with one qubit

We present a method to find the decompositions of tripartite entangled pure states which are smaller than two successive Schmidt decompositions. The method becomes very simple when one of the subsystems is a qubit. In this particular case, we get a classification of states according to their decompositions. Furthermore, we also use this method to classify the entangled states that can be inter-converted through stochastic local operations and classical communication (SLOCC). More general tripartite systems are briefly discussed.

On the geometry of a class ofN-qubit entanglement monotones

A family of N-qubit entanglement monotones invariant under stochastic local operations and classical communication (SLOCC) is defined. This class of entanglement monotones includes the well-known examples of the concurrence, the three-tangle, and some of the four, five and N-qubit SLOCC invariants introduced recently. The construction of these invariants is based on bipartite partitions of the Hilbert space in the form ${\bf C}^{2^N}\simeq{\bf C}^L\otimes{\bf C}^l$ with $L=2^{N-n}\geq l=2^n$. Such partitions can be given a nice geometrical interpretation in terms of Grassmannians Gr(L,l) of l-planes in ${\bf C}^L$ that can be realized as the zero locus of quadratic polinomials in the complex projective space of suitable dimension via the Plucker embedding. The invariants are neatly expressed in terms of the Plucker coordinates of the Grassmannian.

Bound Entanglement Provides Convertibility of Pure Entangled States

I show that two distant parties can transform pure entangled states to arbitrary pure states by stochastic local operations and classical communication (SLOCC) at the single copy level, if they share bound entangled states. This is the effect of bound entanglement since this entanglement processing is impossible by SLOCC alone. A similar effect of bound entanglement exists in three qubits where two incomparable entangled states of GHZ (Greenberger, Horne, and Zeilinger) and W can be interconverted. In general, multipartite settings composed by N distant parties, all N-partite pure entangled states are interconvertible by SLOCC with the assistance of bound entangled states with positive partial transpose.

Measuring polynomial invariants of multiparty quantum states

We present networks for directly estimating the polynomial invariants of multi-party quantum states under local transformations. The structure of these networks is closely related to the structure of the invariants themselves and this lends a physical interpretation to these otherwise abstract mathematical quantities. Specifically, our networks estimate the invariants under local unitary (LU) transformations and under stochastic local operations and classical communication (SLOCC). Our networks can estimate the LU invariants for multi-party states, where each party can have a Hilbert space of arbitrary dimension and the SLOCC invariants for multi-qubit states. We analyze the statistical efficiency of our networks compared to methods based on estimating the state coefficients and calculating the invariants.

MULTIPARTITE ENTANGLEMENT UNDER STOCHASTIC LOCAL OPERATIONS AND CLASSICAL COMMUNICATION

Stochastic local operations and classical communication (SLOCC), also called local filtering operations, are a convenient, useful set of quantum operations in grasping essential properties of entanglement. We give a quick overview of the characteristics of multipartite entanglement in terms of SLOCC, illustrating the 2-qubit and the rest (2×2×n) quantum system. This not only includes celebrated results of 3-qubit pure states, but also has implications to 2-qubit mixed states.