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arXiv:0802.0291 · Reports on Mathematical Physics

Integral representations of separable states

Bronisław Jakubczyk, Gabriel Pietrzkowski

Published Feb 1, 2009• 2 citations
arXivHTMLPDF
DOI
10.1016/s0034-4877(09)90008-8

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SOV approach for integrable quantum models associated with general representations on spin-1/2 chains of the 8-vertex reflection algebra

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The analysis of the transfer matrices associated to the most general representations of the 8-vertex reflection algebra on spin-1/2 chains is here implemented by introducing a quantum separation of variables (SOV) method which generalizes to these integrable quantum models the method first introduced by Sklyanin. More in detail, for the representations reproducing in their homogeneous limits the open XYZ spin-1/2 quantum chains with the most general integrable boundary conditions, we explicitly construct representations of the 8-vertex reflection algebras for which the transfer matrix spectral problem is separated. Then, in these SOV representations we get the complete characterization of the transfer matrix spectrum (eigenvalues and eigenstates) and its non-degeneracy. Moreover, we present the first fundamental step toward the characterization of the dynamics of these models by deriving determinant formulae for the matrix elements of the identity on separated states, which apply in particular to transfer matrix eigenstates. The comparison of our analysis for the 8-vertex reflection algebra with that of [1, 2] for the 6-vertex one leads to the interesting remark that a profound similarity in both the characterization of the spectral problems and of the scalar products exists for these two different realizations of the reflection algebra once they are described by SOV method. As it will be shown in a future publication, this remarkable similarity will be at the basis of the simultaneous determination of form factors of local operators of integrable quantum models associated to general reflection algebra representations of both 8-vertex and 6-vertex type.

arXiv:1307.5531•Journal of Physics A Mathematical and Theoretical•Feb 28, 2014•46 citations
In this paper we give a general integral representation for separable states in the tensor product of infinite dimensional Hilbert spaces and provide the first example of separable states that are not countably decomposable. We also prove the structure theorem for the quantum communication channels that are entanglement-breaking, generalizing the finite-dimensional result of M. Horodecki, Ruskai and Shor. In the finite dimensional case such channels can be characterized as having the Kraus representation with operators of rank 1. The above example implies existence of infinite-dimensional entanglement-breaking channels having no such representation.
A density operator (state) on a tensor product H ⊗ K of Hilbert spaces is separable if it is in the convex closure of the subset of all tensor product states. Non-separable states are called entangled. These concepts are of great importance in quantum information theory, but they have been studied in depth only in the finite-dimensional context [1]. In this note we give a general integral representation for separable states and provide the first example of separable states that are not countably decomposable. We also prove a structure theorem for quantum communication channels that are entanglement-breaking, generalizing the finite-dimensional result of [2]. In the finite-dimensional case such channels can be characterized as having a Stinespring–Kraus representation (3) with operators Vj of rank 1. The above example implies the existence of infinite-dimensional entanglement-breaking channels having no such representation. In what follows, H, K ,... are separable Hilbert spaces, T(H) is the Banach space of trace-class operators and S(H) is the convex subset of all density operators on H. For brevity we shall also call them states, having in mind that a density operator ρ uniquely determines a normal state on the algebra B(H) of all bounded operators on H. Equipped with the trace-norm topology, S(H) is a complete separable metric space. If π is a Borel probability measure on S(H), then the Bochner integral
The analysis of the transfer matrices associated to the most general representations of the 8-vertex reflection algebra on spin-1/2 chains is here implemented by introducing a quantum separation of variables (SOV) method which generalizes to these integrable quantum models the method first introduced by Sklyanin. More in detail, for the representations reproducing in their homogeneous limits the open XYZ spin-1/2 quantum chains with the most general integrable boundary conditions, we explicitly construct representations of the 8-vertex reflection algebras for which the transfer matrix spectral problem is separated. Then, in these SOV representations we get the complete characterization of the transfer matrix spectrum (eigenvalues and eigenstates) and its non-degeneracy. Moreover, we present the first fundamental step toward the characterization of the dynamics of these models by deriving determinant formulae for the matrix elements of the identity on separated states, which apply in particular to transfer matrix eigenstates. The comparison of our analysis for the 8-vertex reflection algebra with that of [1, 2] for the 6-vertex one leads to the interesting remark that a profound similarity in both the characterization of the spectral problems and of the scalar products exists for these two different realizations of the reflection algebra once they are described by SOV method. As it will be shown in a future publication, this remarkable similarity will be at the basis of the simultaneous determination of form factors of local operators of integrable quantum models associated to general reflection algebra representations of both 8-vertex and 6-vertex type.