Using partial transpose and realignment to generate local unitary invariants

Bhosale, Udaysinh T.; Shuddhodan, K. V.; Lakshminarayan, Arul

doi:10.1103/physreva.87.052311

Cited by 5 publications

(13 citation statements)

References 29 publications

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“…Here, ρ i k im is a bipartite state obtained by tracing out all other subsystems except i k and i m . It was also shown in [27] that the characteristic polynomial of P is real and hence these real coefficients are also LU invariants. Note that the "link transformation" [27,28] is a combination of both PT and realignment, executed in that order.…”

Section: Introductionmentioning

confidence: 92%

“…It is clear that the states in Eq. (27) gives the maximum 3-tangle for the given value of C 12 . Also as a consequence of this maximization it is found that C 13 = C 23 = 0 for these states, which is a reflection of the monogamy of entanglement.…”

Section: Lower Boundary Are From Maximally 3-tangled Statesmentioning

confidence: 99%

“…In the same work [27], it was shown that for the case of bipartite pure state of two qubits the quantity det[R(ρ…”

Section: Introductionmentioning

confidence: 99%

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Simple permutation-based measure of quantum correlations and maximally-3-tangled states

Bhosale¹,

Lakshminarayan²

2016

Phys. Rev. A

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Quantities invariant under local unitary transformations are of natural interest in the study of entanglement. This paper deduces and studies a particularly simple quantity that is constructed from a combination of two standard permutations of the density matrix, namely realignment and partial transpose. This bipartite quantity, denoted here as R12, vanishes on large classes of separable states including classical-quantum correlated states, while being maximum for only maximally entangled states. It is shown to be naturally related to the 3-tangle in three qubit states via their two-qubit reduced density matrices. Upper and lower bounds on concurrence and negativity of twoqubit density matrices for all ranks are given in terms of R12. Ansatz states satisfying these bounds are given and verified using various numerical methods. In rank-2 case it is shown that the states satisfying the lower bound on R12 vs concurrence define a class of three qubit states that maximizes the tripartite entanglement (the 3-tangle) given an amount of entanglement between a pair of them. The measure R12 is conjectured, via numerical sampling, to be always larger than the concurrence and negativity. In particular this is shown to be true for the physically interesting case of X states.

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Section: Introductionmentioning

confidence: 92%

Section: Lower Boundary Are From Maximally 3-tangled Statesmentioning

confidence: 99%

See 1 more Smart Citation

Simple permutation-based measure of quantum correlations and maximally-3-tangled states

Bhosale¹,

Lakshminarayan²

2016

Phys. Rev. A

Self Cite

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“…with ρ PT = PT(N −r : r). Unitarity trivially comes from the fact that indices of the Pauli matrices and the Kronecker deltas in (33) are all distinct, so that the identity (19) can be applied to each pair of matrices. To show (34), we first write ρ PT in the computational basis with the help of the tensor representation.…”

Section: General R Matricesmentioning

confidence: 99%

Partial transpose criteria for symmetric states

2016

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We express the positive partial transpose (PPT) separability criterion for symmetric states of multi-qubit systems in terms of matrix inequalities based on the recently introduced tensor representation for spin states. We construct a matrix from the tensor representation of the state and show that it is similar to the partial transpose of the density matrix written in the computational basis. Furthermore, the positivity of this matrix is equivalent to the positivity of a correlation matrix constructed from tensor products of Pauli operators. This allows for a more transparent experimental interpretation of the PPT criteria for an arbitrary spin-j state. The unitary matrices connecting our matrix to the partial transpose of the state generalize the so-called magic basis that plays a central role in Wootters' explicit formula for the concurrence of a 2-qubit system and the Bell bases used for the teleportation of a one or two-qubit state.Comment: 8 page

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“…In this paper, we study the local unitary equivalence problem in terms of matrix realignment [23,24] and partial transposition [25,26], which are the techniques used in dealing with the separability problem of quantum states and also in generating local unitary invariants [27]. We present a necessary and sufficient criterion for the local unitary equivalence of multipartite states, together with explicit forms of the local unitary operators.…”

mentioning

confidence: 99%

Criterion of local unitary equivalence for multipartite states

Zhang

Zhao

et al. 2013

Phys. Rev. A

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We study the local unitary equivalence of arbitrary dimensional multipartite quantum mixed states. We present a necessary and sufficient criterion of the local unitary equivalence for general multipartite states based on matrix realignment. The criterion is shown to be operational, even for particularly degenerated states, by detailed examples. In addition, explicit expressions of the local unitary operators are constructed for locally equivalent states. Complementary to the criterion, an alternative approach based on the partial transposition of matrices is also given, which makes the criterion more effective in dealing with generally degenerated mixed states.Quantum entanglement is one of the most extraordinary features of quantum physics. Multipartite entanglement plays a vital role in quantum information processing [1,2] and interferometry [3]. One fact is that the degree of entanglement of a quantum state remains invariant under local unitary transformations, while two quantum states with the same degree of entanglement, e.g., entanglement of formation [4,5] or concurrence [6,7], may not be equivalent under local unitary transformations. Another fact is that two entangled states are said to be equivalent in implementing quantum information tasks if they can be mutually exchanged under local operations and classical communication (LOCC). LOCC equivalent states are interconvertible also by local unitary transformations [8]. Therefore, it is important to classify and characterize quantum states in terms of local unitary transformations.To deal with this problem, one approach is to construct invariants of local unitary transformations. The method developed in [9,10], in principle, allows one to compute all the invariants of local unitary transformations for bipartite states, though it is not easy to do this operationally. In [11], a complete set of 18 polynomial invariants is presented for the local unitary equivalence of two-qubit mixed states. Partial results have been obtained for three-qubit states [12,13], some generic mixed states [14][15][16], and tripartite pure and mixed states [17]. Reference [18] gives explicit index-free formulas for all of the degree 6 algebraically independent local unitary invariant polynomials for finite-dimensional k-partite pure and mixed quantum states. The local unitary equivalence problem for multipartite pure qubit states has been solved in [19]. By exploiting the high-order singular-value decomposition technique and local symmetries of the states, Ref.[20] presents a practical scheme of classification under local unitary transformations for general multipartite pure states with arbitrary dimensions, which extends the results of n-qubit pure states [19] to those of n-qudit pure states. For mixed states, Ref.[21] solves the local unitary equivalence problem of arbitrary dimensional bipartite nondegenerated quantum systems by presenting a complete set of invariants, such that two density matrices are local unitary equivalent if and only if all of these invariants have equal ...

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Using partial transpose and realignment to generate local unitary invariants

Cited by 5 publications

References 29 publications

Simple permutation-based measure of quantum correlations and maximally-3-tangled states

Simple permutation-based measure of quantum correlations and maximally-3-tangled states

Partial transpose criteria for symmetric states

Criterion of local unitary equivalence for multipartite states

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