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

Frontier between separability and quantum entanglement in a many spin system

Alcaraz, F C, Constantino Tsallis

Published Oct 10, 2001• 10 citations
OpenAlex

Abstract

We discuss the critical point $x_c$ separating the quantum entangled and separable states in two series of N spins S in the simple mixed state characterized by the matrix operator $\\rho=x|\\tilde{\\phi}><\\tilde{\\phi}| + \\frac{1-x}{D^N} I_{D^N}$ where $x \\in [0,1]$, $D =2S+1$, ${\\bf I}_{D^N}$ is the $D^N \\times D^N$ unity matrix and $|\\tilde {\\phi}>$ is a special entangled state. The cases x=0 and x=1 correspond respectively to fully random spins and to a fully entangled state. In the first of these series we consider special states $|\\tilde{\\phi}>$ invariant under charge conjugation, that generalizes the N=2 spin S=1/2 Einstein-Podolsky-Rosen state, and in the second one we consider generalizations of the Weber density matrices. The evaluation of the critical point $x_c$ was done through bounds coming from the partial transposition method of Peres and the conditional nonextensive entropy criterion. Our results suggest the conjecture that whenever the bounds coming from both methods coincide the result of $x_c$ is the exact one. The results we present are relevant for the discussion of quantum computing, teleportation and cryptography.

Related papers

Peres criterion for separability through nonextensive entropy

Constantino Tsallis, Seth Lloyd, Michel Baranger

A bipartite spin-1/2 system having the probabilities $(1+3x)/4$ of being in the Einstein-Podolsky-Rosen (EPR) entangled state $|{\ensuremath{\Psi}}^{\ensuremath{-}}〉\ensuremath{\equiv}(1/\sqrt{2})(|\ensuremath{\uparrow}{〉}_{A}|\ensuremath{\downarrow}{〉}_{B}\ensuremath{-}|\ensuremath{\downarrow}{〉}_{A}|\ensuremath{\uparrow}{〉}_{B})$ and $3(1\ensuremath{-}x)/4$ of being orthogonal is known to admit a local realistic description if and only if $x<1/3$ (Peres criterion). We consider here a more general case where the probabilities of being in the entangled states $|{\ensuremath{\Phi}}^{\ifmmode\pm\else\textpm\fi{}}〉\ensuremath{\equiv}(1/\sqrt{2})(|\ensuremath{\uparrow}{〉}_{A}|\ensuremath{\uparrow}{〉}_{B}\ifmmode\pm\else\textpm\fi{}|\ensuremath{\downarrow}{〉}_{A}|\ensuremath{\downarrow}{〉}_{B})$ and $|{\ensuremath{\Psi}}^{\ifmmode\pm\else\textpm\fi{}}〉\ensuremath{\equiv}(1/\sqrt{2})(|\ensuremath{\uparrow}{〉}_{A}|\ensuremath{\downarrow}{〉}_{B}\ifmmode\pm\else\textpm\fi{}|\ensuremath{\downarrow}{〉}_{A}|\ensuremath{\uparrow}{〉}_{B})$ (Bell basis) are given, respectively, by $(1\ensuremath{-}x)/4,$ $(1\ensuremath{-}y)/4,$ $(1\ensuremath{-}z)/4,$ and $(1+x+y+z)/4.$ Following Abe and Rajagopal, we use the nonextensive entropic form ${S}_{q}\ensuremath{\equiv}(1\ensuremath{-}\mathrm{Tr}{\ensuremath{\rho}}^{q})/(q\ensuremath{-}1)$$(q\ensuremath{\in}\mathcal{R}{;S}_{1}=\ensuremath{-}\mathrm{Tr}\ensuremath{\rho}\mathrm{ln}\ensuremath{\rho})$ which has enabled a current generalization of Boltzmann-Gibbs statistical mechanics, and determine the entire region in the $(x,y,z)$ space where the system is separable. For instance, in the vicinity of the EPR state, separability occurs if and only if $x+y+z<1,$ which recovers Peres' criterion when $x=y=z.$ In the vicinity of the other three states of the Bell basis, the situation is identical. These results illustrate the computational power of this nonextensive-quantum-information procedure. In addition to this, a critical-phenomenon-like scenario emerges which enrichens the discussion.

Physical Review A•Mar 15, 2001•106 citations

Volume of the set of separable states. II

Karol Życzkowski

The problem of how many entangled or, respectively, separable states there are in the set of all quantum states is investigated. We study to what extent the choice of a measure in the space of density matrices $\ensuremath{\varrho}$ describing N-dimensional quantum systems affects the results obtained. We demonstrate that the link between the purity of the mixed states and the probability of entanglement is not sensitive to the measure chosen. Since the criterion of partial transposition is not sufficient to distinguish all separable states for $N>~8$, we develop an efficient algorithm to calculate numerically the entanglement of formation of a given mixed quantum state, which allows us to compute the volume of separable states for $N=8$ and to estimate the volume of the bound entangled states in this case.

arXiv:quant-ph/9902050•Physical Review A•Nov 1, 1999•261 citations

Entanglement in SU(2)-invariant quantum spin systems

John Schliemann

We analyze the entanglement of SU(2)-invariant density matrices of two spins ${S}_{1},$ ${S}_{2}$ using the Peres-Horodecki criterion. Such density matrices arise from thermal equilibrium states of isotropic-spin systems. The partial transpose of such a state has the same multiplet structure and degeneracies as the original matrix with the eigenvalue of largest multiplicity being non-negative. The case ${S}_{1}=S,$ ${S}_{2}=1/2$ can be solved completely and is discussed in detail with respect to isotropic Heisenberg spin models. Moreover, in this case the Peres-Horodecki criterion turns out to be a sufficient condition for nonseparability. We also characterize SU(2)-invariant states of two spins of length 1.

arXiv:quant-ph/0212114•Physical Review A•Jul 11, 2003•66 citations

Separability in 2×N composite quantum systems

Barbara Kraus, J. I. Cirac, Siniša Karnas +1 more

We analyze the separability properties of density operators supported on ${C}^{2}\ensuremath{\bigotimes}{C}^{N}$ whose partial transposes are positive operators. We show that if the rank of $\ensuremath{\rho}$ equals N then it is separable, and that bound entangled states have ranks larger than N. We also give a separability criterion for a generic density operator such that the sum of its rank and the one of its partial transpose does not exceed $3N.$ If it exceeds this number, we show that one can subtract product vectors until it decreases to $3N,$ while keeping the positivity of $\ensuremath{\rho}$ and its partial transpose. This automatically gives us a sufficient criterion for separability for general density operators. We also prove that all density operators that remain invariant after partial transposition with respect to the first system are separable.

arXiv:quant-ph/9912010•Physical Review A•May 8, 2000•159 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

Entanglement in SU(2)-invariant quantum systems: The positive partial transpose criterion and others

John Schliemann

We study entanglement in mixed bipartite quantum states which are invariant under simultaneous SU(2) transformations in both subsystems. Previous results on the behavior of such states under partial transposition are substantially extended. The spectrum of the partial transpose of a given SU(2)-invariant density matrix $\ensuremath{\rho}$ is entirely determined by the diagonal elements of $\ensuremath{\rho}$ in a basis of tensor-product states of both spins with respect to a common quantization axis. We construct a set of operators which act as entanglement witnesses on SU(2)-invariant states. A sufficient criterion for $\ensuremath{\rho}$ having a negative partial transpose is derived in terms of a simple spin correlator. The same condition is a necessary criterion for the partial transpose to have the maximum number of negative eigenvalues. Moreover, we derive a series of sum rules which uniquely determine the eigenvalues of the partial transpose in terms of a system of linear equations. Finally we compare our findings with other entanglement criteria including the reduction criterion, the majorization criterion, and the recently proposed local uncertainty relations.

arXiv:quant-ph/0503123•Physical Review A•Jul 11, 2005•36 citations
We discuss the critical point xcx_cxc​ separating the quantum entangled and separable states in two series of N spins S in the simple mixed state characterized by the matrix operator rho=x∣tildephi><tildephi∣+frac1−xDNIDN\\rho=x|\\tilde{\\phi}><\\tilde{\\phi}| + \\frac{1-x}{D^N} I_{D^N}rho=x∣tildephi><tildephi∣+frac1−xDNIDN​ where xin[0,1]x \\in [0,1]xin[0,1], D=2S+1D =2S+1D=2S+1, bfIDN{\\bf I}_{D^N}bfIDN​ is the DNtimesDND^N \\times D^NDNtimesDN unity matrix and ∣tildephi>|\\tilde {\\phi}>∣tildephi> is a special entangled state. The cases x=0 and x=1 correspond respectively to fully random spins and to a fully entangled state. In the first of these series we consider special states ∣tildephi>|\\tilde{\\phi}>∣tildephi> invariant under charge conjugation, that generalizes the N=2 spin S=1/2 Einstein-Podolsky-Rosen state, and in the second one we consider generalizations of the Weber density matrices. The evaluation of the critical point xcx_cxc​ was done through bounds coming from the partial transposition method of Peres and the conditional nonextensive entropy criterion. Our results suggest the conjecture that whenever the bounds coming from both methods coincide the result of xcx_cxc​ is the exact one. The results we present are relevant for the discussion of quantum computing, teleportation and cryptography.
A bipartite spin-1/2 system having the probabilities (1+3x)/4(1+3x)/4(1+3x)/4 of being in the Einstein-Podolsky-Rosen (EPR) entangled state $|{\ensuremath{\Psi}}^{\ensuremath{-}}〉\ensuremath{\equiv}(1/\sqrt{2})(|\ensuremath{\uparrow}{〉}_{A}|\ensuremath{\downarrow}{〉}_{B}\ensuremath{-}|\ensuremath{\downarrow}{〉}_{A}|\ensuremath{\uparrow}{〉}_{B})$ and $3(1\ensuremath{-}x)/4$ of being orthogonal is known to admit a local realistic description if and only if x<1/3x<1/3x<1/3 (Peres criterion). We consider here a more general case where the probabilities of being in the entangled states $|{\ensuremath{\Phi}}^{\ifmmode\pm\else\textpm\fi{}}〉\ensuremath{\equiv}(1/\sqrt{2})(|\ensuremath{\uparrow}{〉}_{A}|\ensuremath{\uparrow}{〉}_{B}\ifmmode\pm\else\textpm\fi{}|\ensuremath{\downarrow}{〉}_{A}|\ensuremath{\downarrow}{〉}_{B})$ and $|{\ensuremath{\Psi}}^{\ifmmode\pm\else\textpm\fi{}}〉\ensuremath{\equiv}(1/\sqrt{2})(|\ensuremath{\uparrow}{〉}_{A}|\ensuremath{\downarrow}{〉}_{B}\ifmmode\pm\else\textpm\fi{}|\ensuremath{\downarrow}{〉}_{A}|\ensuremath{\uparrow}{〉}_{B})$ (Bell basis) are given, respectively, by $(1\ensuremath{-}x)/4,$ $(1\ensuremath{-}y)/4,$ $(1\ensuremath{-}z)/4,$ and (1+x+y+z)/4.(1+x+y+z)/4.(1+x+y+z)/4. Following Abe and Rajagopal, we use the nonextensive entropic form ${S}_{q}\ensuremath{\equiv}(1\ensuremath{-}\mathrm{Tr}{\ensuremath{\rho}}^{q})/(q\ensuremath{-}1)$$(q\ensuremath{\in}\mathcal{R}{;S}_{1}=\ensuremath{-}\mathrm{Tr}\ensuremath{\rho}\mathrm{ln}\ensuremath{\rho})$ which has enabled a current generalization of Boltzmann-Gibbs statistical mechanics, and determine the entire region in the (x,y,z)(x,y,z)(x,y,z) space where the system is separable. For instance, in the vicinity of the EPR state, separability occurs if and only if x+y+z<1,x+y+z<1,x+y+z<1, which recovers Peres' criterion when x=y=z.x=y=z.x=y=z. In the vicinity of the other three states of the Bell basis, the situation is identical. These results illustrate the computational power of this nonextensive-quantum-information procedure. In addition to this, a critical-phenomenon-like scenario emerges which enrichens the discussion.
The problem of how many entangled or, respectively, separable states there are in the set of all quantum states is investigated. We study to what extent the choice of a measure in the space of density matrices $\ensuremath{\varrho}$ describing N-dimensional quantum systems affects the results obtained. We demonstrate that the link between the purity of the mixed states and the probability of entanglement is not sensitive to the measure chosen. Since the criterion of partial transposition is not sufficient to distinguish all separable states for N> 8N>~8N> 8, we develop an efficient algorithm to calculate numerically the entanglement of formation of a given mixed quantum state, which allows us to compute the volume of separable states for N=8N=8N=8 and to estimate the volume of the bound entangled states in this case.
We analyze the entanglement of SU(2)-invariant density matrices of two spins S1,{S}_{1},S1​, S2{S}_{2}S2​ using the Peres-Horodecki criterion. Such density matrices arise from thermal equilibrium states of isotropic-spin systems. The partial transpose of such a state has the same multiplet structure and degeneracies as the original matrix with the eigenvalue of largest multiplicity being non-negative. The case S1=S,{S}_{1}=S,S1​=S, S2=1/2{S}_{2}=1/2S2​=1/2 can be solved completely and is discussed in detail with respect to isotropic Heisenberg spin models. Moreover, in this case the Peres-Horodecki criterion turns out to be a sufficient condition for nonseparability. We also characterize SU(2)-invariant states of two spins of length 1.
We analyze the separability properties of density operators supported on ${C}^{2}\ensuremath{\bigotimes}{C}^{N}$ whose partial transposes are positive operators. We show that if the rank of $\ensuremath{\rho}$ equals N then it is separable, and that bound entangled states have ranks larger than N. We also give a separability criterion for a generic density operator such that the sum of its rank and the one of its partial transpose does not exceed 3N.3N.3N. If it exceeds this number, we show that one can subtract product vectors until it decreases to 3N,3N,3N, while keeping the positivity of $\ensuremath{\rho}$ and its partial transpose. This automatically gives us a sufficient criterion for separability for general density operators. We also prove that all density operators that remain invariant after partial transposition with respect to the first system are separable.
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 study entanglement in mixed bipartite quantum states which are invariant under simultaneous SU(2) transformations in both subsystems. Previous results on the behavior of such states under partial transposition are substantially extended. The spectrum of the partial transpose of a given SU(2)-invariant density matrix $\ensuremath{\rho}$ is entirely determined by the diagonal elements of $\ensuremath{\rho}$ in a basis of tensor-product states of both spins with respect to a common quantization axis. We construct a set of operators which act as entanglement witnesses on SU(2)-invariant states. A sufficient criterion for $\ensuremath{\rho}$ having a negative partial transpose is derived in terms of a simple spin correlator. The same condition is a necessary criterion for the partial transpose to have the maximum number of negative eigenvalues. Moreover, we derive a series of sum rules which uniquely determine the eigenvalues of the partial transpose in terms of a system of linear equations. Finally we compare our findings with other entanglement criteria including the reduction criterion, the majorization criterion, and the recently proposed local uncertainty relations.