Local Unitary Equivalent Classes of Symmetric N-Qubit Mixed States

Majorana Representation (MR) of symmetric $N$-qubit pure states has been used successfully in entanglement classification. Generalization of this has been a long standing open problem due to the difficulties faced in the construction of a Majorana like geometric representation for symmetric mixed state. We have overcome this problem by developing a method of classifying local unitary (LU) equivalent classes of symmetric $N$-qubit mixed states based on the geometrical Multiaxial Representation (MAR) of the density matrix. In addition to the two parameters defined for the entanglement classification of the symmetric pure states based on MR, namely, diversity degree and degeneracy configuration, we show that another parameter called rank needs to be introduced for symmetric mixed state classification. Our scheme of classification is more general as it can be applied to both pure and mixed states. To bring out the similarities/ differences between the MR and MAR, $N$-qubit GHZ state is taken up for a detailed study. We conclude that pure state classification based on MR is not a special case of our classification scheme based on MAR. We also give a recipe to identify the most general symmetric $N$-qubit pure separable states. The power of our method is demonstrated using several well known examples of symmetric two qubit pure and mixed states as well as three qubit pure states. Classification of uniaxial, Biaxial and triaxial symmetric two qubit mixed states which can be produced in the laboratory is studied in detail

A classification of entanglement in three-qubit systems

We present a classification of three-qubit states based in their three-qubit and reduced two-qubit entanglements. For pure states these criteria can be easily implemented, and the different types can be related with sets of equivalence classes under Local Unitary operations. For mixed states characterization of full tripartite entanglement is not yet solved in general; some partial results will be presented here.Comment: Shortened version. Accepted in EPJ

Topology of the three-qubit space of entanglement types

The three-qubit space of entanglement types is the orbit space of the local unitary action on the space of three-qubit pure states, and hence describes the types of entanglement that a system of three qubits can achieve. We show that this orbit space is homeomorphic to a certain subspace of R^6, which we describe completely. We give a topologically based classification of three-qubit entanglement types, and we argue that the nontrivial topology of the three-qubit space of entanglement types forbids the existence of standard states with the convenient properties of two-qubit standard states.Comment: 9 pages, 3 figures, v2 adds a referenc

Speed of qubit states during thermalisation

Classifying quantum states usually demands to observe properties such as the amount of correlation at one point in time. Further insight may be gained by inspecting the dynamics in a given evolution scheme. Here we attempt such a classification looking at single-qubit and two-qubit states at the start of thermalisation with a heat bath. The speed with which the evolution starts is influenced by quantum aspects of the state, however, such signatures do not allow for a systematic classification

Local unitary equivalence and entanglement of multipartite pure states

The necessary and sufficient conditions for the equivalence of arbitrary n-qubit pure quantum states under Local Unitary (LU) operations derived in [B. Kraus Phys. Rev. Lett. 104, 020504 (2010)] are used to determine the different LU-equivalence classes of up to five-qubit states. Due to this classification new parameters characterizing multipartite entanglement are found and their physical interpretation is given. Moreover, the method is used to derive examples of two n-qubit states (with n>2 arbitrary) which have the properties that all the entropies of any subsystem coincide, however, the states are neither LU-equivalent nor can be mapped into each other by general local operations and classical communication

On Locality of Schmidt-Correlated States

We review some results on the equivalence of quantum states under local unitary transformations (LUT). In particular, the classification of two-qubit Schmidt correlated (SC) states under LUT is investigated. By presenting the standard form of quantum states under LUT, the sufficient and necessary conditions of whether two different SC states are local unitary equivalent are provided. The correlations of SC states are also discussed.Comment: 14 page

The inductive entanglement classification yields ten rather than eight classes of four-qubit entangled states

Lamata et al. use an inductive approach to classify the entangled pure states of four qubits under stochastic local operations and classical communication (SLOCC) [PRA 75(2), 022318 (2007)]. The inductive method yields a priori ten different entanglement superclasses, of which they discard three as empty. One of the remaining superclasses is split in two, resulting in eight superclasses of genuine four-qubit entanglement. Here, we show that two of the three discarded superclasses are in fact non-empty and should have been retained. We give explicit expressions for the canonical states for those superclasses, up to SLOCC and qubit permutations. Furthermore, we confirm that the third discarded superclass is indeed empty, yielding a total of ten superclasses of genuine four-qubit entanglement under the inductive classification scheme.Comment: 9 page

Multilinear singular value decomposition for two qubits

Schmidt decomposition has been used in the local unitary (LU) classification of bipartite quantum states for some time. In order to generalize the LU classification of bipartite quantum states into multipartite quantum states, higher order singular value decomposition (HOSVD) is introduced but specific examples have not been explicitly worked out. In this pedagogical paper, we would like to work out such details in two qubits since the LU classification of two qubits is well known. We first demonstrate the method of HOSVD in two-qubit systems and discuss its properties. In terms of the LU classification of two-qubit states, some subtle differences in the stabilizer groups of entanglement classes are noticed when Schmidt decomposition is substituted by HOSVD. To reconcile the differences between the two, further studies are needed