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arXiv:1309.6944 · Physical Review A

From the quantum relative Tsallis entropy to its conditional form: Separability criterion beyond local and global spectra

A. K. Rajagopal, Sudha, Anantha S. Nayak, A. R. Usha Devi

Published Jan 27, 2014• 24 citations
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Abstract

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 $W$ 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 $W$ states, in one of its partitions. The criterion is shown to be nonspectral through its usefulness in identifying entanglement in isospectral density matrices.

DOI
10.1103/physreva.89.012331

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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.
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.
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 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.
General physical background of Peres-Horodecki positive partial transpose (ppt-) separability criterion is revealed. Especially, the physical sense of partial transpose operation is shown to be equivalent to the local causality reversal (LCR-) procedure for all separable quantum systems or to the uncertainty in a global time arrow direction in all entangled cases. Using these universal causal considerations the heuristic causal separability criterion has been proposed for arbitrary $ D^{N} \times D^{N}$ density matrices acting in $ \mathcal{H}_{D}^{\otimes N} $ Hilbert spaces which describe the ensembles of $ N $ quantum systems of $ D $ eigenstates each. Resulting general formulas have been then analyzed for the widest special type of one-parametric density matrices of arbitrary dimensionality, which model equivalent quantum subsystems being equally connected (EC-) with each other by means of a single entnaglement parameter $ p $. In particular, for the family of such EC-density matrices it has been found that there exists a number of $ N $- and $ D $-dependent separability (or entanglement) thresholds $ p_{th}(N,D) $ which in the case of a qubit-pair density matrix in $ \mathcal{H}_{2} \otimes \mathcal{H}_{2} $ Hilbert space are shown to reduce to well-known results obtained earlier by Peres [5] and Horodecki [6]. As the result, a number of remarkable features of the entanglement thresholds for EC-density matrices has been described for the first time. All novel results being obtained for the family of arbitrary EC-density matrices are shown to be applicable for a wide range of both interacting and non-interacting multi-partite quantum systems, such as arrays of qubits, spin chains, ensembles of quantum oscillators, strongly correlated quantum many-body systems with the possibility of many-body localization, etc.
General physical background of famous Peres-Horodecki positive partial transpose (PH- or PPT-) separability criterion is revealed. Especially, the physical sense of partial transpose operation is shown to be equivalent to what one could call as the "local causality reversal" (LCR-) procedure for all separable quantum systems or to the uncertainty in a global time arrow direction in all entangled cases. Using these universal causal considerations brand new general relations for the heuristic causal separability criterion have been proposed for arbitrary [Formula: see text] density matrices acting in [Formula: see text] Hilbert spaces which describe the ensembles of N quantum systems of D eigenstates each. Resulting general formulas have been then analyzed for the widest special type of one-parametric density matrices of arbitrary dimensionality, which model a number of equivalent quantum subsystems being equally connected (EC-) with each other to arbitrary degree by means of a single entanglement parameter p. In particular, for the family of such EC-density matrices it has been found that there exists a number of N- and D-dependent separability (or entanglement) thresholds [Formula: see text] for the values of the corresponded entanglement parameter p, which in the simplest case of a qubit-pair density matrix in [Formula: see text] Hilbert space are shown to reduce to well-known results obtained earlier independently by Peres (Phys Rev Lett 77:1413-1415, 1996) and Horodecki (Phys Lett A 223(1-2):1-8, 1996). As the result, a number of remarkable features of the entanglement thresholds for EC-density matrices has been described for the first time. All novel results being obtained for the family of arbitrary EC-density matrices are shown to be applicable to a wide range of both interacting and non-interacting (at the moment of measurement) multi-partite quantum systems, such as arrays of qubits, spin chains, ensembles of quantum oscillators, strongly correlated quantum many-body systems with the possibility of many-body localization, etc.