Note on Separability of the Werner states in arbitrary dimensions

Pittenger, A O, Rubin, M H

Published Feb 1, 2000• 35 citations

Abstract

Recently, a great deal of progress has been made in proving sufficient conditions for Werner density matrices on the direct product space of n d-level systems (qudits). In this brief note we complete the process by proving the sufficient condition for general n and d.

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Note on Separability of the Werner states in arbitrary dimensions

Pittenger, A O, Rubin, M H

Published Feb 1, 2000• 35 citations

OpenAlex

Abstract

Related papers

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A. O. Pittenger, Morton H. Rubin

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Using the finite Fourier transform, we introduce a generalization of Pauli-spin matrices for d-dimensional spaces, and the resulting set of unitary matrices

S(d)

is a basis for $d\ifmmode\times\else\texttimes\fi{}d$ matrices. If ${N=d}_{1}\ifmmode\times\else\texttimes\fi{}{d}_{2}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}\ifmmode\times\else\texttimes\fi{}{d}_{b}$ and ${H}^{[N]}=\ensuremath{\bigotimes}{H}^{[{d}_{k}]},$ we give a sufficient condition for separability of a density matrix $\ensuremath{\rho}$ relative to the

{H}^{[{d}_{k}]}

in terms of the

{L}_{1}

norm of the spin coefficients of $\ensuremath{\rho}.$ Since the spin representation depends on the form of the tensor product, the theory applies to both full and partial separability on a given space

{H}^{[N]}.

It follows from this result that for a prescribed form of separability, there is always a neighborhood of the normalized identity in which every density matrix is separable. We also show that for every prime p and

n>1,

the generalized Werner density matrix

{W}^{[{p}^{n}]}(s)

is fully separable if and only if $s<~{(1+p}^{n\ensuremath{-}1}{)}^{\ensuremath{-}1}.$

We propose a natural generalization of bipartite Werner and isotropic states to multipartite systems consisting of an arbitrary even number of

d

-dimensional subsystems (qudits). These generalized states are invariant under the action of local unitary operations. We study basic properties of multipartite invariant states and present necessary and sufficient separability criteria.