Classification of General<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>n</mml:mi></mml:math>-Qubit States under Stochastic Local Operations and Classical Communication in Terms of the Rank of Coefficient Matrix

Li, Xiangrong; Li, Dafa

doi:10.1103/physrevlett.108.180502

Cited by 54 publications

(38 citation statements)

References 37 publications

(47 reference statements)

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“…with q 1 q 2 = 12, 13,14,15,16,23,24,25,26,34,35,36,45,46, 56, a total of 15 E 2 q 1 q 2 (|3, 6 ). Also, there are C 3 6 2 = 10 the measures E 2 q 1 q 2 q 3 (|3, 6 ), and…”

Section: Examplementioning

confidence: 99%

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A multipartite entanglement measure based on coefficient matrices

Zhao

Yang

Hung

et al. 2015

Quantum Inf Process

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The quantification of quantum entanglement has been extensively studied in past years. However, many existing entanglement measures are difficult to calculate. And lots of them are introduced only for bipartite system or only for the systems constituted by qubits. In this paper, we propose an entanglement measure for multipartite system based on vector lengths and the angles between vectors of the coefficient matrices. Our entanglement measure is simple and feasible, with a remarkable geometric meaning. Furthermore, we prove that our entanglement measure satisfies the three necessary conditions which are required for any entanglement measure: (1) It vanishes if and only if the state is (fully) separable; (2) it remains invariant under local unitary transformations; and (3) it cannot increase under local operation and classical communication. Finally, we apply our entanglement measure on some computational examples. It demonstrates that our entanglement measure is capable of dealing with quantum pure states with arbitrary dimensions and parties. Meanwhile, because it only needs to compute the vector lengths and the angles between vectors of every bipartition coefficient matrix, our entanglement measure is easy to calculate.

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“…with q 1 q 2 = 12, 13,14,15,16,23,24,25,26,34,35,36,45,46, 56, a total of 15 E 2 q 1 q 2 (|3, 6 ). Also, there are C 3 6 2 = 10 the measures E 2 q 1 q 2 q 3 (|3, 6 ), and…”

Section: Examplementioning

confidence: 99%

“…where bits 1 to l and l + 1 to N are referred to as the row bits and column bits, respectively [25,26]. That means we split the N subsystems into two disjoint subsets {1, 2, .…”

Section: An Entanglement Measure Based On Vectors Of the Coefficient mentioning

confidence: 99%

A multipartite entanglement measure based on coefficient matrices

Zhao

Yang

Hung

et al. 2015

Quantum Inf Process

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“…For four qubits, there are infinite SLOCC equivalence classes [3] and the infinite SLOCC classes are partitioned into nine inequivalent families [4][5][6][7]. It is known that a SLOCC classification for n qubits remains unsolved because the difficulty increases rapidly as n does [8][9][10][11][12][13]22].…”

Section: Introductionmentioning

confidence: 99%

Stochastic local operations and classical communication (SLOCC) and local unitary operations (LU) classifications of n qubits via ranks and singular values of the spin-flipping matrices

2018

Quantum Inf Process

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We construct ℓ-spin-flipping matrices from the coefficient matrices of pure states of n qubits and show that the ℓ-spin-flipping matrices are congruent and unitary congruent whenever two pure states of n qubits are SLOCC and LU equivalent, respectively. The congruence implies the invariance of ranks of the ℓ-spin-flipping matrices under SLOCC and then permits a reduction of SLOCC classification of n qubits to calculation of ranks of the ℓ-spin-flipping matrices. The unitary congruence implies the invariance of singular values of the ℓ-spin-flipping matrices under LU and then permits a reduction of LU classification of n qubits to calculation of singular values of the ℓ-spin-flipping matrices. Furthermore, we show that the invariance of singular values of the ℓ-spin-flipping matrices Ω (n) 1 implies the invariance of the concurrence for even n qubits and the invariance of the n-tangle for odd n qubits. Thus, the concurrence and the n-tangle can be used for LU classification and computing the concurrence and the n-tangle only performs additions and multiplications of coefficients of states.

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“…The problem of classifying pure states up to stochastic local operations assisted by classical communication (SLOCC) has been intensely studied during the past decade by many authors [1][2][3][4]. Although the SLOCC classes are known for some particular systems, for example, in the cases of three or four qubits, the general method allowing similar derivations for an arbitrary system of many particles which treats, in a unified way, distinguishable and indistinguishable particles has been missing.…”

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confidence: 99%

“…Local unitary operations are represented by the direct product of L copies of SU(N ), K = SU(N ) ×L , and SLOCC operations by of L copies of SL(N ), G = SL(N ) ×L (see Refs. [1][2][3][4]). The action of an element (A 1 , .…”

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confidence: 99%

Critical sets of the total variance can detect all stochastic local operations and classical communication classes of multiparticle entanglement

2012

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We present a general algorithm for finding all classes of pure multiparticle states equivalent under stochastic local operations and classical communication (SLOCC). We parametrize all SLOCC classes by the critical sets of the total variance function. Our method works for arbitrary systems of distinguishable and indistinguishable particles. We also show how to calculate the Morse indices of critical points which have the interpretation of the number of independent nonlocal perturbations increasing the variance and hence entanglement of a state. We illustrate our method by two examples.

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Classification of Generaln-Qubit States under Stochastic Local Operations and Classical Communication in Terms of the Rank of Coefficient Matrix

Cited by 54 publications

References 37 publications

A multipartite entanglement measure based on coefficient matrices

A multipartite entanglement measure based on coefficient matrices

Stochastic local operations and classical communication (SLOCC) and local unitary operations (LU) classifications of n qubits via ranks and singular values of the spin-flipping matrices

Critical sets of the total variance can detect all stochastic local operations and classical communication classes of multiparticle entanglement

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