Three-tangle for rank-three mixed states: Mixture of Greenberger-Horne-Zeilinger,<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>W</mml:mi></mml:math>, and flipped-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>W</mml:mi></mml:math>states

Jung, Eylee; Hwang, Mi-Ra; Park, DaeKil; Son, Jin-Woo

doi:10.1103/physreva.79.024306

Cited by 46 publications

(49 citation statements)

References 21 publications

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“…However, the residual entanglement for several special mixtures was computed in Ref. [31][32][33][34][35]. More recently, the three-tangle for all GHZ-symmetric states [36] was computed analytically [37].…”

Section: Classification Of Entanglementmentioning

confidence: 99%

Entanglement of four-qubit rank-2 mixed states

Jung

Park

2015

Quantum Inf Process

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It is known that there are three maximally entangled states |Φ 1 = (|0000 +|1111 )/ √ 2, |Φ 2 = ( √ 2|1111 +|1000 +|0100 +|0010 +|0001 )/ √ 6, and |Φ 3 = (|1111 + |1100 + |0010 + |0001 )/2 in four-qubit system. It is also known that there are three independent measures F (4) j ( j = 1, 2, 3) for true fourway quantum entanglement in the same system. In this paper, we compute F (4) j and their corresponding linear monotones G (4) j for three rank-two mixed states ρ j = p|Φ j Φ j |+(1 − p)|W 4 W 4 |, where |W 4 = (|0111 + |1011 + |1101 + |1110 )/2. We discuss the possible applications of our results briefly.

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Section: Classification Of Entanglementmentioning

confidence: 99%

Entanglement of four-qubit rank-2 mixed states

Jung

Park

2015

Quantum Inf Process

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“…In practice we deal with mixed states rather than pure states due to decoherence effects and hence it is of great importance to study mixed separable states. There exists many important papers [6][7][8][9][10][11][12][13][14] has presented a complete set of 18 local polynomial invariants of two qubit mixed states and demonstrated the usefulness of these invariants to study entanglement. Also, detection of multipartite entanglement has been studied in depth(see for example [18][19][20]).…”

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confidence: 99%

Geometric multiaxial representation of N -qubit mixed symmetric separable states

Sirsi

Hegde

et al. 2017

Phys. Rev. A

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Study of an N qubit mixed symmetric separable states is a long standing challenging problem as there exist no unique separability criterion. In this regard, we take up the N-qubit mixed symmetric separable states for a detailed study as these states are of experimental importance and offer elegant mathematical analysis since the dimension of the Hilbert space reduces from 2 N to N + 1. Since there exists a one to one correspondence between spin-j system and an N-qubit symmetric state, we employ Fano statistical tensor parameters for the parametrization of spin density matrix. Further, we use geometric multiaxial representation(MAR) of density matrix to characterize the mixed symmetric separable states. Since separability problem is NP hard, we choose to study it in the continuum limit where mixed symmetric separable states are characterized by the P-distribution function λ(θ, φ). We show that the N-qubit mixed symmetric separable state can be visualized as a uniaxial system if the distribution function is independent of θ and φ. We further choose distribution function to be the most general positive function on a sphere and observe that the statistical tensor parameters characterizing the N-qubit symmetric system are the expansion coefficients of the distribution function. As an example for the discrete case, we investigate the MAR of a uniformly weighted two qubit mixed symmetric separable state. We also observe that there exists a correspondence between separability and classicality of states.Study of separable states is the corner stone of entanglement problem. Bell inequalities were first used for the identification of entanglement. The most operationally convenient criterion for the detection of entanglement is given by Peres-Horodecki and is called positive partial transpose (PPT) criterion [1] which is necessary and sufficient for 2 × 2 and 2 × 3 systems only [2]. There exist other criteria in literature for detecting entanglement. One among them is the realignment criterion [3], which exhibits a powerful PPT entanglement detection capability. Entanglement witness[2, 4] and uncertainty relations [5] pose operational difficulty as they depend on the expectation value of some observables for the state in question. In practice we deal with mixed states rather than pure states due to decoherence effects and hence it is of great importance to study mixed separable states. There exists many important papers [6][7][8][9][10][11][12][13][14] has presented a complete set of 18 local polynomial invariants of two qubit mixed states and demonstrated the usefulness of these invariants to study entanglement. Also, detection of multipartite entanglement has been studied in depth(see for example [18][19][20]). Geometric entanglement properties of pure symmetric N qubit states are studied in detail [21]. To this day, no generally * sumarkr@gmail.com accepted theory exists for the classification and quantification of entanglement for mixed states. A general N qubit mixed state resides in the Hilbert space of dimension 2 N...

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“…We consider the rank d of the density matrix d, rather than the dimension of the Hilbert space on which it acts, because d is the parameter that determines the computational difficulty of the convex roof minimization. A number of special cases of computation of the convex roof have been solved for cases of restricted rank [11][12][13].Minimal ensembles have been found analytically for the concurrence of arbitrary two-qubit mixed states [8], and for the three-tangle of rank-two mixtures of generalized GHZ and generalized W states [12,13], as well as on rank-three mixtures of a GHZ state, a W state, and a state obtained by flipping all three bits of a W state [14]. When the minimal ensemble is not known analytically, which is the typical case, one may evaluate an upper bound on E(ρ) using, for example, a steepest descent algorithm [15].…”

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“…Minimal ensembles have been found analytically for the concurrence of arbitrary two-qubit mixed states [8], and for the three-tangle of rank-two mixtures of generalized GHZ and generalized W states [12,13], as well as on rank-three mixtures of a GHZ state, a W state, and a state obtained by flipping all three bits of a W state [14]. When the minimal ensemble is not known analytically, which is the typical case, one may evaluate an upper bound on E(ρ) using, for example, a steepest descent algorithm [15].…”

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confidence: 99%

Bounding polynomial entanglement measures for mixed states

2014

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We generalize the notion of the best separable approximation (BSA) and best W-class approximation (BWA) to arbitrary pure state entanglement measures, defining the best zero-E approximation (BEA). We show that for any polynomial entanglement measure E, any mixed state ρ admits at least one "S-decomposition," i.e., a decomposition in terms of a mixed state on which E is equal to zero, and a single additional pure state with (possibly) non-zero E. We show that the BEA is not in general the optimal S-decomposition from the point of view of bounding the entanglement of ρ, and describe an algorithm to construct the entanglement-minimizing S-decomposition for ρ and place an upper bound on E(ρ). When applied to the three-tangle, the cost of the algorithm is linear in the rank d of the density matrix and has accuracy comparable to a steepest descent algorithm whose cost scales as d 8 log d. We compare the upper bound to a lower bound algorithm given by Eltschka and Siewert for the three-tangle, and find that on random rank-two three-qubit density matrices, the difference between the upper and lower bounds is 0.14 on average. We also find that the three-tangle of random full-rank three qubit density matrices is less than 0.023 on average.Non-classical correlations in quantum states such as entanglement distinguish quantum from classical information theory. The ability to calculate entanglement of mixed quantum states is relevant for the analysis of tomography data for systems of multiple qubits in several implementations [1][2][3]. Multipartite systems can contain multiple inequivalent types of entanglement that cannot be converted into one another by local operations and classical communication [4].One approach to characterizing pure-state entanglement in a system of qubits associates a polynomial function that is invariant under determinant 1 local operations with each type of entanglement [5][6][7]. Examples of such polynomial invariants include the concurrence for two qubits [8] and the three-tangle, which quantifies the amount of entanglement in a three-qubit system that cannot be accounted for by entanglement between pairs of the qubits [9].A polynomial invariant E is extended to mixed states by way of the convex roof, given for a rank-d density matrix ρ by:where E = {p i , |ψ i } is a pure-state ensemble for ρ and Υ ρ is the set of all such ensembles. Caratheodory's theorem allows us to restrict the optimization to ensembles containing no more than d 2 elements [10]. An ensemble that minimizes eq. (1) is said to be minimal. We consider the rank d of the density matrix d, rather than the dimension of the Hilbert space on which it acts, because d is the parameter that determines the computational difficulty of the convex roof minimization. A number of special cases of computation of the convex roof have been solved for cases of restricted rank [11][12][13].Minimal ensembles have been found analytically for the concurrence of arbitrary two-qubit mixed states [8], and for the three-tangle of rank-two mixtures of gener...

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Three-tangle for rank-three mixed states: Mixture of Greenberger-Horne-Zeilinger,W, and flipped-Wstates

Cited by 46 publications

References 21 publications

Entanglement of four-qubit rank-2 mixed states

Entanglement of four-qubit rank-2 mixed states

Geometric multiaxial representation of N -qubit mixed symmetric separable states

Bounding polynomial entanglement measures for mixed states

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