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arXiv:1511.02110 · Physical Review A

Visualizing extremal positive maps in unital and trace-preserving form

Leif Ove Hansen, Jan Myrheim

Published Oct 5, 2015• 3 citations
arXivHTMLPDF

Abstract

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 $\mathcal{D}$, the set of density matrices, fills out $\mathcal{D}$ 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.

DOI
10.1103/physreva.92.042306

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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×\else\texttimes\fi33\ifmmode\times\else\texttimes\fi{}33\ifmmode×\else\texttimes\fi3 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.
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 demonstrate a general procedure to construct entanglement witnesses for any entangled state. This procedure is based on the trace inequality and a general form of entanglement witnesses, which is in the form $W=\ensuremath{\rho}\ensuremath{-}{c}_{\ensuremath{\rho}}I$, where $\ensuremath{\rho}$ is a density matrix, ${c}_{\ensuremath{\rho}}$ is a non-negative number related to $\ensuremath{\rho}$, and III is the identity matrix. The general form of entanglement witnesses is deduced from Choi-Jamio\l{}kowski isomorphism, that can be reinterpreted as that all quantum states can be obtained by a maximally quantum entangled state pass through certain completely positive maps. Furthermore, we provide the necessary and sufficient condition of the entanglement witness $W=\ensuremath{\rho}\ensuremath{-}{c}_{\ensuremath{\rho}}I$ in operation, as well as in theory.
We consider entanglement witnesses arising from positive linear maps which generate exposed extremal rays. We show that every entangled state can be detected by one of these witnesses, and this witness detects a unique set of entangled states among those. Therefore, they provide a minimal set of witnesses to detect any kind of entanglement in a sense. Furthermore, if those maps are indecomposable then they detect large classes of entangled states with positive partial transposes which have nonempty relative interiors in the cone generated by all PPT states. We also provide a one-parameter family of indecomposable positive linear maps which generate exposed extremal rays. This gives the first examples of such maps in three-dimensional matrix algebra.
An entanglement witness (EW) is an operator that allows the detection of entangled states. We give necessary and sufficient conditions for such operators to be optimal, i.e., to detect entangled states in an optimal way. We show how to optimize general EW, and then we particularize our results to the nondecomposable ones; the latter are those that can detect positive partial transpose entangled states (PPTES's). We also present a method to systematically construct and optimize this last class of operators based on the existence of ``edge'' PPTES's, i.e., states that violate the range separability criterion [Phys. Lett. A 232, 333 (1997)] in an extreme manner. This method also permits a systematic construction of nondecomposable positive maps (PM's). Our results lead to a sufficient condition for entanglement in terms of nondecomposable EW's and PM's. Finally, we illustrate our results by constructing optimal EW acting on $H={C}^{2}\ensuremath{\bigotimes}{C}^{4}.$ The corresponding PM's constitute examples of PM's with minimal ``qubit'' domains, or---equivalently---minimal Hermitian conjugate codomains.