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:1307.2323 · Physical Review A

Bounding polynomial entanglement measures for mixed states

Samuel G. Rodriques, Nilanjana Datta, Peter J. Love

Published Jul 30, 2014• 21 citations
arXivHTMLPDF

Abstract

We generalize the notion of the best separable approximation (BSA) and best $W$-class approximation (BWA) to arbitrary pure-state entanglement measures, defining the best zero-$E$ approximation (BEA). We show that for any polynomial entanglement measure $E$, any mixed state $\ensuremath{\rho}$ admits at least one ``$S$ decomposition,'' i.e., a decomposition in terms of a mixed state on which $E$ is equal to zero, and a single additional pure state with (possibly) nonzero $E$. We show that the BEA is not, in general, the optimal $S$ decomposition from the point of view of bounding the entanglement of $\ensuremath{\rho}$ and describe an algorithm to construct the entanglement-minimizing $S$ decomposition for $\ensuremath{\rho}$ and place an upper bound on $E(\ensuremath{\rho})$. When applied to the three-tangle, the cost of the algorithm is linear in the rank $d$ of the density matrix and has an accuracy comparable to a steepest-descent algorithm whose cost scales as ${d}^{8}logd$. We compare the upper bound to a lower-bound algorithm given by C. Eltschka and J. Siewert [Phys. Rev. Lett. 108, 020502 (2012)] for the three-tangle and find that on random rank-2 three-qubit density matrices, the difference between the upper and lower bounds is $0.14$ on average. We also find that the three-tangle of random full-rank three-qubit density matrices is less than $0.023$ on average.

DOI
10.1103/physreva.90.012340

Related papers

Upper bound for SL-invariant entanglement measures of mixed states

Andreas Osterloh

An algorithm is proposed that serves to handle full-rank density matrices when coming from a lower-rank method to compute the convex roof. This is in order to calculate an upper bound for any polynomial SL-invariant multipartite entanglement measure $E$. This study exemplifies how this algorithm works based on a method for calculating convex roofs of rank-2 density matrices. It iteratively considers the decompositions of the density matrix into two states each, exploiting the knowledge for the rank-2 case. The algorithm is therefore quasiexact as far as the rank-2 case is concerned, and it also hints where it should include more states in the decomposition of the density matrix. Focusing on the measure of three-way entanglement of qubits (called three-tangle), I show the results the algorithm gives for two states, one of which is the Greenberger-Horne-Zeilinger--Werner (GHZ-$W$) state, for which the exact convex roof is known. It overestimates the three-tangle in the state, thereby giving insight into the optimal decomposition the GHZ-$W$ state has. As a proof of principle, I have run the algorithm for the three-tangle on the transverse quantum Ising model. I give qualitative and quantitative arguments why the convex roof should be close to the upper bound found here.

arXiv:1512.05987•Physical Review A•May 18, 2016•6 citations

Entanglement Quantification Made Easy: Polynomial Measures Invariant under Convex Decomposition

Bartosz Regula, Gerardo Adesso

Quantifying entanglement in composite systems is a fundamental challenge, yet exact results are available in only a few special cases. This is because hard optimization problems are routinely involved, such as finding the convex decomposition of a mixed state with the minimal average pure-state entanglement, the so-called convex roof. We show that under certain conditions such a problem becomes trivial. Precisely, we prove by a geometric argument that polynomial entanglement measures of degree 2 are independent of the choice of pure-state decomposition of a mixed state, when the latter has only one pure unentangled state in its range. This allows for the analytical evaluation of convex roof extended entanglement measures in classes of rank-2 states obeying such a condition. We give explicit examples for the square root of the three-tangle in three-qubit states, and we show that several representative classes of four-qubit pure states have marginals that enjoy this property.

arXiv:1512.03326•Physical Review Letters•Feb 19, 2016•25 citations

Separable approximations of density matrices of composite quantum systems

Karnas, S, Maciej Lewenstein

We investigate optimal separable approximations (decompositions) of states rho of bipartite quantum systems A and B of arbitrary dimensions MxN following the lines of Ref. [M. Lewenstein and A. Sanpera, Phys. Rev. Lett. 80, 2261 (1998)]. Such approximations allow to represent in an optimal way any density operator as a sum of a separable state and an entangled state of a certain form. For two qubit systems (M=N=2) the best separable approximation has a form of a mixture of a separable state and a projector onto a pure entangled state. We formulate a necessary condition that the pure state in the best separable approximation is not maximally entangled. We demonstrate that the weight of the entangled state in the best separable approximation in arbitrary dimensions provides a good entanglement measure. We prove in general for arbitrary M and N that the best separable approximation corresponds to a mixture of a separable and an entangled state which are both unique. We develop also a theory of optimal separable approximations for states with positive partial transpose (PPT states). Such approximations allow to decompose any density operator with positive partial transpose as a sum of a separable state and an entangled PPT state. We discuss procedures of constructing such decompositions.

•Nov 15, 2000•38 citations

Experimental Examination of Entanglement Estimates

Songbo Xie, Yuanyuan Zhao, Chao Zhang +4 more

Recently, a proper genuine multipartite entanglement measure has been found for three-qubit pure states [see Xie and Eberly, Phys. Rev. Lett. 127, 040403 (2021)PRLTAO0031-900710.1103/PhysRevLett.127.040403], but capturing useful entanglement measures for mixed states has remained an open challenge. So far, it requires not only a full tomography in experiments, but also huge calculational labor. A leading proposal was made by Gühne, Reimpell, and Werner [Phys. Rev. Lett. 98, 110502 (2007)PRLTAO0031-900710.1103/PhysRevLett.98.110502], who used expectation values of entanglement witnesses to describe a lower bound estimation of entanglement. We provide here an extension that also gives genuine upper bounds of entanglement. This advance requires only the expectation value of any Hermitian operator. Moreover, we identify a class of operators A_{1} that not only give good estimates, but also require a remarkably small number of experimental measurements. In this Letter, we define our approach and illustrate it by estimating entanglement measures for a number of pure and mixed states prepared in our recent experiments.

arXiv:2207.07584•Physical Review Letters•Apr 12, 2023•5 citations

Radial Convex Geometry of Quantum States and Its Relation to Best Separable Approximation

Haonan Qiang

We study bipartite entanglement from the convex geometry of quantum states. Taking the maximally mixed state as a reference point, we define a geometric entangled-space quantity $G(ρ)=[1-L(ρ)]/L(ρ)$ from the relative positions of the separable and quantum-state boundaries, and introduce a relative entanglement degree $Q(ρ)=[(1-p)/p]/G(ρ)$ by comparing it with the robustness relative to the maximally mixed state. We apply this geometric construction to the Best Separable Approximation (BSA). For the optimal decomposition, we derive the general bound $(1-p)L_B/[p(1-L_B)]\leq p_0\leq Q(ρ)$ and show that the entangled component of the BSA has an entangled-space size no smaller than that of the original mixed state, namely $[1-L_R]/L_R\leq[1-L_B]/L_B$. For two-qubit states, the BSA entangled component is a pure entangled state. Using the PPT criterion, its geometric parameter can be evaluated explicitly, giving the bound $(1-p)/(2p)\leq p_0\leq Q(ρ)$. These results provide a simple geometric description of the relation between BSA, robustness, and the entangled region of the quantum-state space.

quant-pharXiv:2608.20050•Aug 21, 2026•0 citations

Best separable approximation of N-qubit symmetric states

Ruben Quesada, Anna Sanpera

The structural study of entanglement in multipartite systems is hindered by the lack of necessary and sufficient operational criteria able to discriminate among the various entanglement properties of a given mixed state. Here, we pursue a different route to the study of multipartite entanglement based on the closeness of a multipartite state to the set of separable ones. In particular, we analyze multipartite diagonal symmetric N qubit states and provide the analytical expression for their Best Separable Approximation (BSA [Phys. Rev. Lett. 80, 2261 (1998)]), that is, their unique convex decomposition into a separable part and an entangled one with maximal weight of the separable one.

•Jun 26, 2013•0 citations
We generalize the notion of the best separable approximation (BSA) and best WWW-class approximation (BWA) to arbitrary pure-state entanglement measures, defining the best zero-EEE approximation (BEA). We show that for any polynomial entanglement measure EEE, any mixed state \ensuremathρ\ensuremath{\rho}\ensuremathρ admits at least one ``SSS decomposition,'' i.e., a decomposition in terms of a mixed state on which EEE is equal to zero, and a single additional pure state with (possibly) nonzero EEE. We show that the BEA is not, in general, the optimal SSS decomposition from the point of view of bounding the entanglement of \ensuremathρ\ensuremath{\rho}\ensuremathρ and describe an algorithm to construct the entanglement-minimizing SSS decomposition for \ensuremathρ\ensuremath{\rho}\ensuremathρ and place an upper bound on E(\ensuremathρ)E(\ensuremath{\rho})E(\ensuremathρ). When applied to the three-tangle, the cost of the algorithm is linear in the rank ddd of the density matrix and has an accuracy comparable to a steepest-descent algorithm whose cost scales as d8logd{d}^{8}logdd8logd. We compare the upper bound to a lower-bound algorithm given by C. Eltschka and J. Siewert [Phys. Rev. Lett. 108, 020502 (2012)] for the three-tangle and find that on random rank-2 three-qubit density matrices, the difference between the upper and lower bounds is 0.140.140.14 on average. We also find that the three-tangle of random full-rank three-qubit density matrices is less than 0.0230.0230.023 on average.
An algorithm is proposed that serves to handle full-rank density matrices when coming from a lower-rank method to compute the convex roof. This is in order to calculate an upper bound for any polynomial SL-invariant multipartite entanglement measure EEE. This study exemplifies how this algorithm works based on a method for calculating convex roofs of rank-2 density matrices. It iteratively considers the decompositions of the density matrix into two states each, exploiting the knowledge for the rank-2 case. The algorithm is therefore quasiexact as far as the rank-2 case is concerned, and it also hints where it should include more states in the decomposition of the density matrix. Focusing on the measure of three-way entanglement of qubits (called three-tangle), I show the results the algorithm gives for two states, one of which is the Greenberger-Horne-Zeilinger--Werner (GHZ-WWW) state, for which the exact convex roof is known. It overestimates the three-tangle in the state, thereby giving insight into the optimal decomposition the GHZ-WWW state has. As a proof of principle, I have run the algorithm for the three-tangle on the transverse quantum Ising model. I give qualitative and quantitative arguments why the convex roof should be close to the upper bound found here.
Quantifying entanglement in composite systems is a fundamental challenge, yet exact results are available in only a few special cases. This is because hard optimization problems are routinely involved, such as finding the convex decomposition of a mixed state with the minimal average pure-state entanglement, the so-called convex roof. We show that under certain conditions such a problem becomes trivial. Precisely, we prove by a geometric argument that polynomial entanglement measures of degree 2 are independent of the choice of pure-state decomposition of a mixed state, when the latter has only one pure unentangled state in its range. This allows for the analytical evaluation of convex roof extended entanglement measures in classes of rank-2 states obeying such a condition. We give explicit examples for the square root of the three-tangle in three-qubit states, and we show that several representative classes of four-qubit pure states have marginals that enjoy this property.
We investigate optimal separable approximations (decompositions) of states rho of bipartite quantum systems A and B of arbitrary dimensions MxN following the lines of Ref. [M. Lewenstein and A. Sanpera, Phys. Rev. Lett. 80, 2261 (1998)]. Such approximations allow to represent in an optimal way any density operator as a sum of a separable state and an entangled state of a certain form. For two qubit systems (M=N=2) the best separable approximation has a form of a mixture of a separable state and a projector onto a pure entangled state. We formulate a necessary condition that the pure state in the best separable approximation is not maximally entangled. We demonstrate that the weight of the entangled state in the best separable approximation in arbitrary dimensions provides a good entanglement measure. We prove in general for arbitrary M and N that the best separable approximation corresponds to a mixture of a separable and an entangled state which are both unique. We develop also a theory of optimal separable approximations for states with positive partial transpose (PPT states). Such approximations allow to decompose any density operator with positive partial transpose as a sum of a separable state and an entangled PPT state. We discuss procedures of constructing such decompositions.
Recently, a proper genuine multipartite entanglement measure has been found for three-qubit pure states [see Xie and Eberly, Phys. Rev. Lett. 127, 040403 (2021)PRLTAO0031-900710.1103/PhysRevLett.127.040403], but capturing useful entanglement measures for mixed states has remained an open challenge. So far, it requires not only a full tomography in experiments, but also huge calculational labor. A leading proposal was made by Gühne, Reimpell, and Werner [Phys. Rev. Lett. 98, 110502 (2007)PRLTAO0031-900710.1103/PhysRevLett.98.110502], who used expectation values of entanglement witnesses to describe a lower bound estimation of entanglement. We provide here an extension that also gives genuine upper bounds of entanglement. This advance requires only the expectation value of any Hermitian operator. Moreover, we identify a class of operators A_{1} that not only give good estimates, but also require a remarkably small number of experimental measurements. In this Letter, we define our approach and illustrate it by estimating entanglement measures for a number of pure and mixed states prepared in our recent experiments.
We study bipartite entanglement from the convex geometry of quantum states. Taking the maximally mixed state as a reference point, we define a geometric entangled-space quantity G(ρ)=[1−L(ρ)]/L(ρ)G(ρ)=[1-L(ρ)]/L(ρ)G(ρ)=[1−L(ρ)]/L(ρ) from the relative positions of the separable and quantum-state boundaries, and introduce a relative entanglement degree Q(ρ)=[(1−p)/p]/G(ρ)Q(ρ)=[(1-p)/p]/G(ρ)Q(ρ)=[(1−p)/p]/G(ρ) by comparing it with the robustness relative to the maximally mixed state. We apply this geometric construction to the Best Separable Approximation (BSA). For the optimal decomposition, we derive the general bound (1−p)LB/[p(1−LB)]≤p0≤Q(ρ)(1-p)L_B/[p(1-L_B)]\leq p_0\leq Q(ρ)(1−p)LB​/[p(1−LB​)]≤p0​≤Q(ρ) and show that the entangled component of the BSA has an entangled-space size no smaller than that of the original mixed state, namely [1−LR]/LR≤[1−LB]/LB[1-L_R]/L_R\leq[1-L_B]/L_B[1−LR​]/LR​≤[1−LB​]/LB​. For two-qubit states, the BSA entangled component is a pure entangled state. Using the PPT criterion, its geometric parameter can be evaluated explicitly, giving the bound (1−p)/(2p)≤p0≤Q(ρ)(1-p)/(2p)\leq p_0\leq Q(ρ)(1−p)/(2p)≤p0​≤Q(ρ). These results provide a simple geometric description of the relation between BSA, robustness, and the entangled region of the quantum-state space.
The structural study of entanglement in multipartite systems is hindered by the lack of necessary and sufficient operational criteria able to discriminate among the various entanglement properties of a given mixed state. Here, we pursue a different route to the study of multipartite entanglement based on the closeness of a multipartite state to the set of separable ones. In particular, we analyze multipartite diagonal symmetric N qubit states and provide the analytical expression for their Best Separable Approximation (BSA [Phys. Rev. Lett. 80, 2261 (1998)]), that is, their unique convex decomposition into a separable part and an entangled one with maximal weight of the separable one.