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arXiv:1305.2385

Extremal entanglement witnesses

Leif Ove Hansen, Andreas Hauge, Jan Myrheim, Per Øyvind Sollid

Published May 10, 2013• 3 citations
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Abstract

We study extremal entanglement witnesses on a bipartite quantum system. We define the cone of witnesses as the dual of the set of separable density matrices, thus $\textrm{Tr}\,Ωρ\geq 0$ when $Ω$ is a witness and $ρ$ a pure product state, $ρ=ψψ^{\dagger}$ with $ψ=ϕ\otimesχ$. The set of witnesses of unit trace is a compact convex set, defined by its extremal points. The expectation value $f(ϕ,χ)=\mathrm{Tr}\,Ωρ$ as a function of $ϕ$ and $χ$ is a nonnegative biquadratic form. Every zero of $f(ϕ,χ)$ imposes real-linear constraints on $f$ and $Ω$. The Hessian matrix at the zero must be nonnegative. Its eigenvectors with zero eigenvalue, if any, we call Hessian zeros. A zero of $f(ϕ,χ)$ is quadratic if it has no Hessian zeros, otherwise it is quartic. We call a witness quadratic if it has only quadratic zeros, and quartic otherwise. We prove that a witness is extremal if and only if no other witness has the same, or a larger, set of zeros and Hessian zeros. A quadratic extremal witness has a minimum number of isolated zeros depending on dimensions. If a witness is not extremal, the constraints defined by its zeros and Hessian zeros determine all directions in which to search for witnesses having more zeros or Hessian zeros. A finite number of iterated searches in random directions lead to an extremal witness which is usually quadratic with the minimum number of zeros. We discuss some topics related to extremal witnesses, in particular the relation between the facial structures of the dual sets of witnesses and separable states. We discuss the relation between extremality and optimality of witnesses, and a conjecture of separability of the structural physical approximation (SPA) of an optimal witness. We discuss how to treat the entanglement witnesses on a complex Hilbert space as witnesses on a real Hilbert space.

DOI
10.48550/arxiv.1305.2385

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We study extremal entanglement witnesses on a bipartite quantum system. We define the cone of witnesses as the dual of the set of separable density matrices, thus Tr Ωρ≥0\textrm{Tr}\,Ωρ\geq 0TrΩρ≥0 when ΩΩΩ is a witness and ρρρ a pure product state, ρ=ψψ†ρ=ψψ^{\dagger}ρ=ψψ† with ψ=ϕ⊗χψ=ϕ\otimesχψ=ϕ⊗χ. The set of witnesses of unit trace is a compact convex set, defined by its extremal points. The expectation value f(ϕ,χ)=Tr Ωρf(ϕ,χ)=\mathrm{Tr}\,Ωρf(ϕ,χ)=TrΩρ as a function of ϕϕϕ and χχχ is a nonnegative biquadratic form. Every zero of f(ϕ,χ)f(ϕ,χ)f(ϕ,χ) imposes real-linear constraints on fff and ΩΩΩ. The Hessian matrix at the zero must be nonnegative. Its eigenvectors with zero eigenvalue, if any, we call Hessian zeros. A zero of f(ϕ,χ)f(ϕ,χ)f(ϕ,χ) is quadratic if it has no Hessian zeros, otherwise it is quartic. We call a witness quadratic if it has only quadratic zeros, and quartic otherwise. We prove that a witness is extremal if and only if no other witness has the same, or a larger, set of zeros and Hessian zeros. A quadratic extremal witness has a minimum number of isolated zeros depending on dimensions. If a witness is not extremal, the constraints defined by its zeros and Hessian zeros determine all directions in which to search for witnesses having more zeros or Hessian zeros. A finite number of iterated searches in random directions lead to an extremal witness which is usually quadratic with the minimum number of zeros. We discuss some topics related to extremal witnesses, in particular the relation between the facial structures of the dual sets of witnesses and separable states. We discuss the relation between extremality and optimality of witnesses, and a conjecture of separability of the structural physical approximation (SPA) of an optimal witness. We discuss how to treat the entanglement witnesses on a complex Hilbert space as witnesses on a real Hilbert space.
We present a study of extremal entanglement witnesses on a bipartite composite quantum system. We define the cone of witnesses as the dual of the set of separable density matrices, thus [Formula: see text] when [Formula: see text] is a witness and [Formula: see text] is a pure product state, [Formula: see text] with [Formula: see text]. The set of witnesses of unit trace is a compact convex set, uniquely defined by its extremal points. The expectation value [Formula: see text] as a function of vectors [Formula: see text] and [Formula: see text] is a positive semidefinite biquadratic form. Every zero of [Formula: see text] imposes strong real-linear constraints on f and [Formula: see text]. The real and symmetric Hessian matrix at the zero must be positive semidefinite. Its eigenvectors with zero eigenvalue, if such exist, we call Hessian zeros. A zero of [Formula: see text] is quadratic if it has no Hessian zeros, otherwise it is quartic. We call a witness quadratic if it has only quadratic zeros, and quartic if it has at least one quartic zero. A main result we prove is that a witness is extremal if and only if no other witness has the same, or a larger, set of zeros and Hessian zeros. A quadratic extremal witness has a minimum number of isolated zeros depending on dimensions. If a witness is not extremal, then the constraints defined by its zeros and Hessian zeros determine all directions in which we may search for witnesses having more zeros or Hessian zeros. A finite number of iterated searches in random directions, by numerical methods, leads to an extremal witness which is nearly always quadratic and has the minimum number of zeros. We discuss briefly some topics related to extremal witnesses, in particular the relation between the facial structures of the dual sets of witnesses and separable states. We discuss the relation between extremality and optimality of witnesses, and a conjecture of separability of the so-called structural physical approximation (SPA) of an optimal witness. Finally, we discuss how to treat the entanglement witnesses on a complex Hilbert space as a subset of the witnesses on a real Hilbert space.
We define an entanglement witness in a composite quantum system as an observable having a nonnegative expectation value in every separable state. Then a state is entangled if and only if it has a negative expectation value of some entanglement witness. Equivalent representations of entanglement witnesses are nonnegative biquadratic forms or positive linear maps of Hermitian matrices. As reported elsewhere, we have studied extremal entanglement witnesses in dimension $3\ifmmode\times\else\texttimes\fi{}3$ by constructing numerical examples of generic extremal nonnegative forms. These are so complicated that we do not know how to handle them other than by numerical methods. However, the corresponding extremal positive maps can be presented graphically, as we attempt to do in the present paper. We understand that a positive map is extremal when the image of D\mathcal{D}D, the set of density matrices, fills out D\mathcal{D}D maximally, in a certain sense. For the graphical presentation of a map we transform it to a standard form where it is unital and trace preserving. We present an iterative algorithm for the transformation, which converges rapidly in all our numerical examples and presumably works for any positive map. This standard form of an entanglement witness is unique up to unitary product transformations.
Let ${H}^{[N]}{=H}^{[{d}_{1}]}\ensuremath{\bigotimes}\ensuremath{\cdots}\ensuremath{\bigotimes}{H}^{[{d}_{n}]}$ be a tensor product of Hilbert spaces and let ${\ensuremath{\tau}}_{0}$ be the closest separable state in the Hilbert-Schmidt norm to an entangled state ${\ensuremath{\rho}}_{0}.$ Let ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}}_{0}$ denote the closest separable state to ${\ensuremath{\rho}}_{0}$ along the line segment from I/NI/NI/N to ${\ensuremath{\rho}}_{0}$ where I is the identity matrix. Following A. O. Pittenger and M. H. Rubin [Linear Algebr. Appl. 346, 75 (2002)] a witness W0{W}_{0}W0​ detecting the entanglement of ${\ensuremath{\rho}}_{0}$ can be constructed in terms of I, ${\ensuremath{\tau}}_{0},$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}}_{0}.$ If representations of ${\ensuremath{\tau}}_{0}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}}_{0}$ as convex combinations of separable projections are known, then the entanglement of ${\ensuremath{\rho}}_{0}$ can be detected by local measurements. G\"uhne et al. [Phys. Rev. A 66, 062305 (2002)] obtain the minimum number of measurement settings required for a class of two-qubit states. We use our geometric approach to generalize their result to the corresponding two-qudit case when d is prime and obtain the minimum number of measurement settings. In those particular bipartite cases, ${\ensuremath{\tau}}_{0}={\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}}_{0}.$ We illustrate our general approach with a two-parameter family of three-qubit bound entangled states for which ${\ensuremath{\tau}}_{0}\ensuremath{\ne}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}}_{0}$ and we show that our approach works for n qubits. We elaborated earlier [A. O. Pittenger, Linear Algebr. App. 359, 235 (2003)] on the role of a ``far face'' of the separable states relative to a bound entangled state ${\ensuremath{\rho}}_{0}$ constructed from an orthogonal unextendible product base. In this paper the geometric approach leads to an entanglement witness expressible in terms of a constant times I and a separable density ${\ensuremath{\mu}}_{0}$ on the far face from ${\ensuremath{\rho}}_{0}.$ Up to a normalization this coincides with the witness obtained by G\"uhne et al. for the particular example analyzed there.
Randomized measurement protocols give experimental access to low-degree polynomial invariants of a quantum state rather than to the state itself, which raises the question of what the best entanglement criterion is that can be built from finitely many such invariants. In this work we answer this question through convex duality. For each degree we introduce the cone of local unitary trace inequalities that are valid for separable bipartite states in all local dimensions, together with its one-matrix counterparts for Hermitian and positive semidefinite matrices. Each cone is the polar dual of a moment cone; we also determine the minimal combinatorial parametrization of the separable cone. The framework organizes the known moment criteria: the PPT criterion using the first three moments of the partial transposition is the image under an embedding of a Hankel determinant from the positive semidefinite cone to the bipartite cone, which we show to be an extremal ray, and we determine the first nontrivial separable cone completely, finding exactly four extremal rays. The framework allows us to analyze the realignment criterion and its centered (or enhanced) variant. We prove that the even singular value moments of the (centered) realignment matrix are trace invariants, associated with explicit fixed-point-free involutions. Extracting the optimal trace-norm bound from mmm such moments thus becomes a dimension-free truncated moment problem, which we solve in closed form for m=2m = 2m=2 and relax to a semidefinite program of size O(m)O(m)O(m) for arbitrary mmm. The dual solutions are polynomial minorants of x\sqrt{x}x​ on [0,1][0,1][0,1], and two explicit families, binomial and L1L^1L1-weighted, yield hierarchies of separability inequalities. Finally, we compare the criteria against one another and on PPT entangled states.
We present an analysis of the properties and characteristics of weakly optimal entanglement witnesses, that is witnesses whose expectation value vanishes on at least one product vector. Any weakly optimal entanglement witness can be written as the form of Wwopt=σ−cσmaxIW^{wopt}=\sigma-c_{\sigma}^{max} IWwopt=σ−cσmax​I, where cσmaxc_{\sigma}^{max}cσmax​ is a non-negative number and III is the identity matrix. We show the relation between the weakly optimal witness WwoptW^{wopt}Wwopt and the eigenvalues of the separable states σ\sigmaσ. Further we give an application of weakly optimal witnesses for constructing entanglement witnesses in a larger Hilbert space by extending the result of [P. Badzi\c{a}g {\it et al}, Phys. Rev. A {\bf 88}, 010301(R) (2013)], and we examine their geometric properties.
From the physical point of view entanglement witnesses (EWs) define a universal tool for analysis and classification of quantum entangled states. From the mathematical perspective bipartite EWs provide highly non-trivial generalization of positive operators and establish elegant correspondence with the theory of positive maps in matrix algebras. We concentrate on theoretical analysis of various important notions like decomposability, atomicity, optimality, extremality and exposedness. Several methods of construction are provided as well. Our discussion is illustrated by many examples enabling the reader to see the intricate structure of these objects. It is shown that the theory of EWs finds elegant geometric formulation in terms of convex cones and related geometric structures.