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arXiv:quant-ph/9912116 · Physical Review A

Complete separability and Fourier representations of n-qubit states

A. O. Pittenger, Morton H. Rubin

Published Sep 1, 2000• 17 citations
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Abstract

Necessary conditions for separability are most easily expressed in the computational basis, while sufficient conditions are most conveniently expressed in the spin basis. We use the Hadamard matrix to define the relationship between these two bases and to emphasize its interpretation as a Fourier transform. We then prove a general sufficient condition for complete separability in terms of the spin coefficients and give necessary and sufficient conditions for the complete separability of a class of generalized Werner densities. As a further application of the theory, we give necessary and sufficient conditions for full separability for a particular set of n-qubit states whose densities all satisfy the Peres condition.

DOI
10.1103/physreva.62.042306

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Necessary conditions for separability are most easily expressed in the computational basis, while sufficient conditions are most conveniently expressed in the spin basis. We use the Hadamard matrix to define the relationship between these two bases and to emphasize its interpretation as a Fourier transform. We then prove a general sufficient condition for complete separability in terms of the spin coefficients and give necessary and sufficient conditions for the complete separability of a class of generalized Werner densities. As a further application of the theory, we give necessary and sufficient conditions for full separability for a particular set of n-qubit states whose densities all satisfy the Peres condition.
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)S(d)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[dk]{H}^{[{d}_{k}]}H[dk​] in terms of the L1{L}_{1}L1​ 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].{H}^{[N]}.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,n>1,n>1, the generalized Werner density matrix W[pn](s){W}^{[{p}^{n}]}(s)W[pn](s) is fully separable if and only if $s<~{(1+p}^{n\ensuremath{-}1}{)}^{\ensuremath{-}1}.$
Explicit separable density matrices, for mixed two qubits states, are derived by the use of Hilbert Schmidt decompositions and Peres Horodecki criterion. A strongly separable two qubits mixed state is defined by multiplications of two density matrices given with pure states while weakly separable two qubits state is defined by multiplications of two density matrices which includes non-pure states. We find the sufficient and necessary condition for separability of two-qubits density matrices and show that under this condition the two-qubit density matrices are strongly separable.
We study the separability of bipartite quantum systems in arbitrary dimensions using the Bloch representation of their density matrix. This approach enables us to find an alternative characterization of the separability problem, from which we derive a necessary condition and sufficient conditions for separability. For a certain class of states the necessary condition and a sufficient condition turn out to be equivalent, therefore yielding a necessary and sufficient condition. The proofs of the sufficient conditions are constructive, thus providing decompositions in pure product states for the states that satisfy them. We provide examples that show the ability of these conditions to detect entanglement. In particular, the necessary condition is proved to be strong enough to detect bound entangled states.
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