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JNNCE Journal of Engineering and Management

Separability in one parameter families of 3, 4 qubit W and GHZ states using Conditional Sandwiched Rényi Relative Entropy

Anantha S. Nayak

Published Jul 31, 2022• 0 citations
Publisher

Abstract

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.

DOI
10.37314/jjem.2023.070103

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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.
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.
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 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 $q\ensuremath{\rightarrow}\ensuremath{\infty}$, 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 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.
A comprehensive investigation of the entanglement characteristics is carried out on tripartite spin-1/2 systems, examining prototypical tripartite states, the thermal Heisenberg model, and the transverse field Ising model. The entanglement is computed using the Rényi relative entropy. In the traditional Rényi relative entropy, the generalization parameter α can take values only in the range [Formula: see text] due to the requirements of joint convexity of the measure. To use the Rényi relative entropy over a wider range of α, we use the sandwiched form which is jointly convex in the regime [Formula: see text]. In prototypical tripartite states, we find that GHZ states are monogamous, but surprisingly so are W states. On the other hand, star states exhibit polygamy, due to the higher level of purity of the bipartite subsystems. For spin models, we study the dependence of entanglement on various parameters such as temperature, spin-spin interaction, and anisotropy, and identify regions where entanglement is the largest. The Rényi parameter α scales the amount of entanglement in the system. The entanglement measure based on the traditional and the sandwiched Rényi relative entropies obey the Araki-Lieb-Thirring inequality. In the Heisenberg models, namely the XYZ, XXZ, and XY models, the system is always monogamous. However, in the transverse field Ising model, the state is initially polygamous and becomes monogamous with temperature and coupling.