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arXiv:2404.02716 · Physical Review A

Conditions for separability in multiqubit systems with an accelerating qubit using a conditional entropy

Harsha Miriam Reji, Hemant Shreepad Hegde, R. Prabhu

Published Sep 4, 2024• 14 citations
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Abstract

We study the separability in multiqubit pure and mixed Greenberger-Horne-Zeilinger (GHZ) and $W$ states with an accelerating qubit using the Abe-Rajagopal (AR) $q$-conditional entropy. We observe that the pure multiqubit GHZ and $W$ 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 $q$-conditional entropy parameter $q$. 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 $q$. These conditions obtained from the AR $q$-conditional entropy serve as necessary conditions for separability in such multiqubit states with a relativistic qubit.

DOI
10.1103/physreva.110.032403

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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.
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.
Abstract Characterizing quantum entanglement in mixed states is a longstanding challenge. Among the various methods available, conditional entropies serve as a powerful tool. Notably, the AR q-conditional entropy introduced by Abe and Rajagopal in 2002 has demonstrated significant promise as it often surpasses other entropy-based criteria. The wide-ranging applications of conditional entropy in quantum information underscore the importance of studying and analyzing it for a deeper understanding of quantum correlations and their implications. In this paper, we investigate the non-separability of noisy Dicke states using the AR approach of conditional entropy. Our findings reveal that the entropic criterion is equally effective as the PPT criterion in identifying non-separability across a large subset of N -partite noisy Dicke states with even N and excitation number k=N/2k = N/2k=N/2 k = N / 2 . Additionally, for systems with N>30N > 30N>30 N > 30 and k=1k=1k=1 k = 1 , the separability thresholds derived from both criteria converge within 10−810^{-8}10−8 10 - 8 , highlighting their strong agreement in this parameter range. Furthermore, we established a condition based on AR q-conditional entropy for identifying genuine multipartite entanglement (GME) in noisy Dicke states and compared its effectiveness to previous methods. Notably, our condition identifies a broader range of GME, particularly when the number of excitations approaches half the number of qubits (i.e., N /2). In contrast, previous methods perform better when the number of excitations is significantly less than N /2. We believe these results will pave the way for further advancements in entanglement theory and the development of potential quantum-based applications for conditional entropy.
We demonstrate the necessary and sufficient criterion for an NNN qubit Greenberger-Horne-Zeilinger (GHZ) state in white noise when it is divided into N−jN-jN−j parties with N≥2j+1N\geq 2j+1N≥2j+1 for arbitrary NNN and jjj. The criterion covers more than a half of all the separabilities of NNN-qubit Greenberger-Horne-Zeilinger states in white noise. For the rest of multipartite separable problems, we present a method to obtain the sufficient conditions of separability using linear programming.
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 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.