Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0909.1087 · Physical Review A

Entropic characterization of separability in Gaussian states

Sudha, A. R. Usha Devi, A. K. Rajagopal

Published Feb 24, 2010• 9 citations
arXivHTMLPDF

Abstract

We explore separability of bipartite divisions of mixed Gaussian states based on the positivity of the Abe-Rajagopal (AR) $q$-conditional entropy. The AR $q$-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 ($q=1$ 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.

DOI
10.1103/physreva.81.024303

Related papers

Entropic analysis of non-separability in noisy Dicke states

Mohamed Nawareg

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 $$k = N/2$$ k = N / 2 . Additionally, for systems with $$N > 30$$ N > 30 and $$k=1$$ k = 1 , the separability thresholds derived from both criteria converge within $$10^{-8}$$ 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.

Quantum Information Processing•Dec 2, 2025•1 citation

Entanglement Criteria for All Bipartite Gaussian States

G. Giedke, Barbara Kraus, Maciej Lewenstein +1 more

We provide a necessary and sufficient condition for separability of Gaussian states of bipartite systems of arbitrarily many modes. The condition provides an operational criterion since it can be checked by simple computation. Moreover, it allows us to find a pure product-state decomposition of any given separable Gaussian state. We also show that all bipartite Gaussian states with nonpositive partial transpose are distillable.

arXiv:quant-ph/0104050•Physical Review Letters•Oct 1, 2001•190 citations

Generalized entropic criterion for separability

R. Rossignoli, N. Canosa

We discuss the entropic criterion for separability of compound quantum systems for general nonadditive entropic forms based on arbitrary concave functions f. For any separable state, the generalized entropy of the whole system is shown to be not smaller than that of the subsystems, for any choice of f, providing thus a necessary criterion for separability. Nevertheless, the criterion is not sufficient and examples of entangled states with the same property are provided. This entails, in particular, that the conjecture about the positivity of the conditional Tsallis entropy for all q, a more stringent requirement than the positivity of the conditional von Neumann entropy, is actually a necessary but not sufficient condition for separability in general. The direct relation between the entropic criterion and the largest eigenvalues of the full and reduced density operators of the system is also discussed.

arXiv:1505.03608•Physical Review A•Oct 11, 2002•62 citations

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

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.

arXiv:0706.3038•Physical Review A•Oct 31, 2007•10 citations

Distillability criterion for all bipartite Gaussian states

G. Giedke, L-M Duan, J. I. Cirac +1 more

We prove that all inseparable Gaussian states of two modes can be distilled into maximally entangled pure states by local operations. Using this result we show that a bipartite Gaussian state of arbitrarily many modes can be distilled if and only if its partial transpose is not positive.

Quantum Information and Computation•Oct 1, 2001•26 citations

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

A. K. Rajagopal, Sudha, Anantha S. Nayak +1 more

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.

arXiv:1309.6944•Physical Review A•Jan 27, 2014•24 citations
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→\ensuremath∞q\ensuremath{\rightarrow}\ensuremath{\infty}q\ensuremath→\ensuremath∞) 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 discuss the entropic criterion for separability of compound quantum systems for general nonadditive entropic forms based on arbitrary concave functions f. For any separable state, the generalized entropy of the whole system is shown to be not smaller than that of the subsystems, for any choice of f, providing thus a necessary criterion for separability. Nevertheless, the criterion is not sufficient and examples of entangled states with the same property are provided. This entails, in particular, that the conjecture about the positivity of the conditional Tsallis entropy for all q, a more stringent requirement than the positivity of the conditional von Neumann entropy, is actually a necessary but not sufficient condition for separability in general. The direct relation between the entropic criterion and the largest eigenvalues of the full and reduced density operators of the system is also discussed.
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.
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.