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arXiv:0706.3038 · Physical Review A

Separability of a family of one-parameter W and Greenberger-Horne-Zeilinger multiqubit states using the Abe-Rajagopal q -conditional-entropy approach

R. Prabhu, A. R. Usha Devi, G. Padmanabha

Published Oct 31, 2007• 10 citations
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Abstract

We employ conditional Tsallis $q$ entropies to study the separability of symmetric one parameter $W$ and GHZ multiqubit mixed states. The strongest limitation on separability is realized in the limit $q\ensuremath{\rightarrow}\ensuremath{\infty}$, and is found to be much superior to the condition obtained using the von Neumann conditional entropy ($q=1$ case). Except for the example of two qubit and three qubit symmetric states of GHZ family, the $q$-conditional entropy method leads to sufficient---but not necessary---conditions on separability.

DOI
10.1103/physreva.76.042337

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We employ conditional Tsallis qqq entropies to study the separability of symmetric one parameter WWW and GHZ multiqubit mixed states. The strongest limitation on separability is realized in the limit q\ensuremath→\ensuremath∞q\ensuremath{\rightarrow}\ensuremath{\infty}q\ensuremath→\ensuremath∞, and is found to be much superior to the condition obtained using the von Neumann conditional entropy (q=1q=1q=1 case). Except for the example of two qubit and three qubit symmetric states of GHZ family, the qqq-conditional entropy method leads to sufficient---but not necessary---conditions on separability.
We study the separability in multiqubit pure and mixed Greenberger-Horne-Zeilinger (GHZ) and WWW states with an accelerating qubit using the Abe-Rajagopal (AR) qqq-conditional entropy. We observe that the pure multiqubit GHZ and WWW states in the inertial-noninertial bipartition with one of their qubits accelerated will remain nonseparable irrespective of the qubit's acceleration. In these systems, we capture the variation of their nonseparability with respect to the acceleration of the qubit and the AR qqq-conditional entropy parameter qqq. However, in the corresponding multiqubit mixed states obtained by introducing a global noise to the above pure states, we could get stronger conditions on their separability in the inertial-noninertial bipartition, in terms of the acceleration of the qubit, the noise parameter, and the number of qubits in the system, in the asymptotic limit of the parameter qqq. These conditions obtained from the AR qqq-conditional entropy serve as necessary conditions for separability in such multiqubit states with a relativistic qubit.
An increase in activity has been seen recently in the study of entanglement in bipartite quantum systems.The entropic characterization of separability of composite quantum systems has attracted a lot of interest among diverse techniques.In this research article, a new form of conditional Rényi relative entropy (known as Conditional Sandwiched Rényi Relative Entropy (CSRRE) form) is used to determine the bipartite separability limit in single parameter families of mixed W-and GHZ-states.For N=3, 4 qubits, the non-entanglement limit in the 1: N-1 partition of these states determined using CSRRE criterion is more restrictive than the separability range discovered by Abe-Rajagopal's form of Tsallis entropy.It's established that1: N-1separabilty range determined using CSRRE approach precisely with the range determined by positivity of partial transpose criterion.
We study the noisy GHZ-W mixture (where GHZ denotes Greenberger-Horne-Zeilinger). We demonstrate some necessary but not sufficient criteria for different classes of separability of these states. It turns out that the partial transposition criterion of Peres [Phys. Rev. Lett. 77, 1413 (1996)] and the criteria of G\"uhne and Seevinck [New J. Phys. 12, 053002 (2010)] dealing with matrix elements are the strongest ones for different separability classes of this two-parameter state. As a result, we determine a set of entangled states of positive partial transpose.
The quantum relative R\'enyi entropy of two noncommuting density matrices was recently defined from which its conditional entropy is deduced. This framework is here extended to the corresponding Tsallis relative entropy and to its conditional form. This expression of Tsallis conditional entropy is shown to witness entanglement beyond the method based on global and local spectra of composite quantum states. When the reduced density matrix happens to be a maximally mixed state, this conditional entropy coincides with the expression in terms of Tsallis entropies derived earlier by Abe and Rajagopal [Physica A 289, 157 (2001)]. The separability range in a noisy one-parameter family of WWW and Greenberger-Horne-Zeilinger states with three and four qubits is explored here and it is shown that the results inferred from negative Tsallis conditional entropy matches that obtained through Peres's partial transpose criterion for one-parameter family of WWW states, in one of its partitions. The criterion is shown to be nonspectral through its usefulness in identifying entanglement in isospectral density matrices.
We explore separability of bipartite divisions of mixed Gaussian states based on the positivity of the Abe-Rajagopal (AR) qqq-conditional entropy. The AR qqq-conditional entropic characterization provide more stringent restrictions on separability (in the limit $q\ensuremath{\rightarrow}\ensuremath{\infty}$) than that obtained from the corresponding von Neumann conditional entropy (q=1q=1q=1 case)---similar to the situation in finite dimensional states. Effectiveness of this approach, in relation to the results obtained by partial transpose criterion, is explicitly analyzed in three illustrative examples of two-mode Gaussian states of physical significance.
The conditional version of sandwiched Tsallis relative entropy (CSTRE) is employed to study the bipartite separability of one parameter family of N-qudit Werner-Popescu states in their 1:N-1 partition. For all N, the strongest limitation on bipartite separability is realized in the limit and is found to match exactly with the separability range obtained using an algebraic method which is both necessary and sufficient. The theoretical superiority of using CSTRE criterion to find the bipartite separability range over the one using Abe-Rajagopal (AR) q-conditional entropy is illustrated by comparing the convergence of the parameter x with respect to q, in the implicit plots of AR q-conditional entropy and CSTRE.