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:quant-ph/0207129 · Journal of Physics A Mathematical and General

Conditionalq-entropies and quantum separability: a numerical exploration

Josep Amengual i Batle, Angel Ricardo Plastino, M. Casas, Angel Ricardo Plastino

Published Nov 19, 2002• 51 citations
arXivHTMLPDF

Abstract

We revisit the relationship between quantum separability and the sign of the relative q -entropies of composite quantum systems. The q -entropies depend on the density matrix eigenvalues p i through the quantity ω q = ∑ i p q i . Rényi's and Tsallis' measures constitute particular instances of these entropies. We perform a systematic numerical survey of the space of mixed states of two-qubit systems in order to determine, as a function of the degree of mixture, and for different values of the entropic parameter q , the volume in state space occupied by those states characterized by positive values of the relative entropy. Similar calculations are performed for qubit–qutrit systems and for composite systems described by Hilbert spaces of larger dimensionality. We pay particular attention to the limit case q → ∞. Our numerical results indicate that, as the dimensionalities of both subsystems increase, typical mixed states of composite quantum systems tend, as far as their relative q -entropies are concerned, to behave in a classical way.

DOI
10.1088/0305-4470/35/48/307

Related papers

Separability and correlations in composite states based on entropy methods

A. K. Rajagopal, R. W. Rendell

This work is an enquiry into the circumstances under which entropy methods can give an answer to the questions of both quantum separability and classical correlations of a composite state. Several entropy functionals are employed to examine the entanglement and correlation properties guided by the corresponding calculations of concurrence. It is shown that the entropy difference between that of the composite and its marginal density matrices may be of arbitrary sign except under special circumstances when conditional probability can be defined appropriately. This ambiguity is a consequence of the fact that the overlap matrix elements of the eigenstates of the composite density matrix with those of its marginal density matrices also play important roles in the definitions of probabilities and the associated entropies, along with their respective eigenvalues. The general results are illustrated using pure- and mixed-state density matrices of two-qubit systems. Two classes of density matrices are found for which the conditional probability can be defined: (1) density matrices with commuting decompositions and (2) those that are decohered in the representation where the density matrices of the marginals are diagonal. The first class of states encompass those whose separability is currently understood as due to particular symmetries of the states. The second are a class of states that are expected to be useful for understanding separability. Examples of entropy functionals of these decohered states including the crucial isospectral case are discussed.

arXiv:quant-ph/0203124•Physical Review A•Aug 6, 2002•67 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

Entropic uncertainty relations and entanglement

Otfried Gühne, Maciej Lewenstein

We discuss the relationship between entropic uncertainty relations and entanglement. We present two methods for deriving separability criteria in terms of entropic uncertainty relations. In particular, we show how any entropic uncertainty relation on one part of the system results in a separability condition on the composite system. We investigate the resulting criteria using the Tsallis entropy for two and three qubits.

arXiv:quant-ph/0403219•Physical Review A•Aug 24, 2004•195 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

Volume of the set of separable states

Karol Życzkowski, Paweł Horodecki, Anna Sanpera +1 more

The question of how many entangled or, respectively, separable states there are in the set of all quantum states is considered. We propose a natural measure in the space of density matrices $\ensuremath{\varrho}$ describing $N$-dimensional quantum systems. We prove that, under this measure, the set of separable states possesses a nonzero volume. Analytical lower and upper bounds of this volume are also derived for $N=2\ifmmode\times\else\texttimes\fi{}2$ and $N=2\ifmmode\times\else\texttimes\fi{}3$ cases. Finally, numerical Monte Carlo calculations allow us to estimate the volume of separable states, providing numerical evidence that it decreases exponentially with the dimension of the composite system. We have also analyzed a conditional measure of separability under the condition of fixed purity. Our results display a clear dualism between purity and separability: entanglement is typical of pure states, while separability is connected with quantum mixtures. In particular, states of sufficiently low purity are necessarily separable.

arXiv:quant-ph/9804024•Physical Review A•Aug 1, 1998•1,409 citations

Nonlocality and entanglement in qubit systems

Josep Amengual i Batle, M. Casas

Nonlocality and quantum entanglement constitute two special aspects of the quantum correlations existing in quantum systems, which are of paramount importance in quantum-information theory. Traditionally, they have been regarded as identical (equivalent, in fact, for pure two qubit states, that is, Gisin's Theorem ), yet they constitute different resources. Describing nonlocality by means of the violation of several Bell inequalities, we obtain by direct optimization those states of two qubits that maximally violate a Bell inequality, in terms of their degree of mixture as measured by either their participation ratio R = 1/Tr(ρ 2 ) or their maximum eigenvalue λ max . This optimum value is obtained as well, which coincides with previous results. Comparison with entanglement is performed too. An example of an application is given in the XY model. In this novel approximation, we also concentrate on the nonlocality for linear combinations of pure states of two qubits, providing a closed form for their maximal nonlocality measure. The case of Bell diagonal mixed states of two qubits is also extensively studied. Special attention concerning the connection between nonlocality and entanglement for mixed states of two qubits is paid to the so-called maximally entangled mixed states. Additional aspects for the case of two qubits are also described in detail. Since we deal with qubit systems, we will perform an analogous study for three qubits, employing similar tools. Relation between distillability and nonlocality is explored quantitatively for the whole space of states of three qubits. We finally extend our analysis to four-qubit systems, where nonlocality for generalized Greenberger–Horne–Zeilinger states of arbitrary number of parties is computed.

arXiv:1102.4653•Journal of Physics A Mathematical and Theoretical•Oct 14, 2011•61 citations
We revisit the relationship between quantum separability and the sign of the relative q -entropies of composite quantum systems. The q -entropies depend on the density matrix eigenvalues p i through the quantity ω q = ∑ i p q i . Rényi's and Tsallis' measures constitute particular instances of these entropies. We perform a systematic numerical survey of the space of mixed states of two-qubit systems in order to determine, as a function of the degree of mixture, and for different values of the entropic parameter q , the volume in state space occupied by those states characterized by positive values of the relative entropy. Similar calculations are performed for qubit–qutrit systems and for composite systems described by Hilbert spaces of larger dimensionality. We pay particular attention to the limit case q → ∞. Our numerical results indicate that, as the dimensionalities of both subsystems increase, typical mixed states of composite quantum systems tend, as far as their relative q -entropies are concerned, to behave in a classical way.
This work is an enquiry into the circumstances under which entropy methods can give an answer to the questions of both quantum separability and classical correlations of a composite state. Several entropy functionals are employed to examine the entanglement and correlation properties guided by the corresponding calculations of concurrence. It is shown that the entropy difference between that of the composite and its marginal density matrices may be of arbitrary sign except under special circumstances when conditional probability can be defined appropriately. This ambiguity is a consequence of the fact that the overlap matrix elements of the eigenstates of the composite density matrix with those of its marginal density matrices also play important roles in the definitions of probabilities and the associated entropies, along with their respective eigenvalues. The general results are illustrated using pure- and mixed-state density matrices of two-qubit systems. Two classes of density matrices are found for which the conditional probability can be defined: (1) density matrices with commuting decompositions and (2) those that are decohered in the representation where the density matrices of the marginals are diagonal. The first class of states encompass those whose separability is currently understood as due to particular symmetries of the states. The second are a class of states that are expected to be useful for understanding separability. Examples of entropy functionals of these decohered states including the crucial isospectral case are discussed.
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 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.
The question of how many entangled or, respectively, separable states there are in the set of all quantum states is considered. We propose a natural measure in the space of density matrices $\ensuremath{\varrho}$ describing NNN-dimensional quantum systems. We prove that, under this measure, the set of separable states possesses a nonzero volume. Analytical lower and upper bounds of this volume are also derived for $N=2\ifmmode\times\else\texttimes\fi{}2$ and $N=2\ifmmode\times\else\texttimes\fi{}3$ cases. Finally, numerical Monte Carlo calculations allow us to estimate the volume of separable states, providing numerical evidence that it decreases exponentially with the dimension of the composite system. We have also analyzed a conditional measure of separability under the condition of fixed purity. Our results display a clear dualism between purity and separability: entanglement is typical of pure states, while separability is connected with quantum mixtures. In particular, states of sufficiently low purity are necessarily separable.
Nonlocality and quantum entanglement constitute two special aspects of the quantum correlations existing in quantum systems, which are of paramount importance in quantum-information theory. Traditionally, they have been regarded as identical (equivalent, in fact, for pure two qubit states, that is, Gisin's Theorem ), yet they constitute different resources. Describing nonlocality by means of the violation of several Bell inequalities, we obtain by direct optimization those states of two qubits that maximally violate a Bell inequality, in terms of their degree of mixture as measured by either their participation ratio R = 1/Tr(ρ 2 ) or their maximum eigenvalue λ max . This optimum value is obtained as well, which coincides with previous results. Comparison with entanglement is performed too. An example of an application is given in the XY model. In this novel approximation, we also concentrate on the nonlocality for linear combinations of pure states of two qubits, providing a closed form for their maximal nonlocality measure. The case of Bell diagonal mixed states of two qubits is also extensively studied. Special attention concerning the connection between nonlocality and entanglement for mixed states of two qubits is paid to the so-called maximally entangled mixed states. Additional aspects for the case of two qubits are also described in detail. Since we deal with qubit systems, we will perform an analogous study for three qubits, employing similar tools. Relation between distillability and nonlocality is explored quantitatively for the whole space of states of three qubits. We finally extend our analysis to four-qubit systems, where nonlocality for generalized Greenberger–Horne–Zeilinger states of arbitrary number of parties is computed.