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 $s<~{(1+p}^{n\ensuremath{-}1}{)}^{\ensuremath{-}1}.$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 $s<~{(1+p}^{n\ensuremath{-}1}{)}^{\ensuremath{-}1}.$

arXiv:quant-ph/0001014Physical Review AAug 18, 200054 citations

Physically realizable transformations of density matrices

A. Kitaev, Alexander Khanievich Shen', M. Vyalyi

Graduate studies in mathematicsMay 31, 20020 citations

A new representation for the density matrix

G. Ramachandran, M. V. N. Murthy

Nuclear Physics AMar 1, 198016 citations

Structure of nonlinear gauge transformations

Marek Czachor

Nonlinear Doebner-Goldin [Phys. Rev. A 54, 3764 (1996)] gauge transformations (NGT's) defined in terms of a wave function $\ensuremath{\psi}(x)$ do not form a group. To get a group property one has to consider transformations that act differently on different branches of the complex argument function and the knowledge of the value of $\ensuremath{\psi}(x)$ is not sufficient for a well-defined NGT. NGT's that are well defined in terms of $\ensuremath{\psi}(x)$ form a semigroup parametrized by a real number \ensuremath{\gamma} and a nonzero \ensuremath{\lambda} that is either an integer or $\ensuremath{-}1<~\ensuremath{\lambda}<~1.$ An extension of NGT's to projectors and general density matrices leads to NGT's with complex \ensuremath{\gamma}. Both the linearity of evolution and Hermiticity of density matrices are gauge-dependent properties.Nonlinear Doebner-Goldin [Phys. Rev. A 54, 3764 (1996)] gauge transformations (NGT's) defined in terms of a wave function $\ensuremath{\psi}(x)$ do not form a group. To get a group property one has to consider transformations that act differently on different branches of the complex argument function and the knowledge of the value of $\ensuremath{\psi}(x)$ is not sufficient for a well-defined NGT. NGT's that are well defined in terms of $\ensuremath{\psi}(x)$ form a semigroup parametrized by a real number \ensuremath{\gamma} and a nonzero \ensuremath{\lambda} that is either an integer or $\ensuremath{-}1<~\ensuremath{\lambda}<~1.$ An extension of NGT's to projectors and general density matrices leads to NGT's with complex \ensuremath{\gamma}. Both the linearity of evolution and Hermiticity of density matrices are gauge-dependent properties.

arXiv:quant-ph/9711053Physical Review AApr 1, 19983 citations

Separability and entanglement of two qubits density matrices using Lorentz transformations

Y. Ben-Aryeh, A. Mann

Explicit separability of general two qubits density matrices is related to Lorentz transformations. We use the 4-dimensional form [Formula: see text] of the Hilbert–Schmidt (HS) decomposition of the density matrix. For the generic case, in which Lorentz transformations diagonalize [Formula: see text] into [Formula: see text], we give relations between the [Formula: see text] and the [Formula: see text]. In particular, we consider two cases: (a) Two qubits density matrices with one pair of linear terms in the HS decomposition. (b) Two qubits density matrices with two or three symmetric pairs of linear terms. Some of the theoretical results are demonstrated by numerical calculations. The four non-generic cases (which may be reduced to case (a) are analyzed and the non-generic property is related explicitly to Lorentz velocity [Formula: see text] which is not reachable physically.Explicit separability of general two qubits density matrices is related to Lorentz transformations. We use the 4-dimensional form [Formula: see text] of the Hilbert–Schmidt (HS) decomposition of the density matrix. For the generic case, in which Lorentz transformations diagonalize [Formula: see text] into [Formula: see text], we give relations between the [Formula: see text] and the [Formula: see text]. In particular, we consider two cases: (a) Two qubits density matrices with one pair of linear terms in the HS decomposition. (b) Two qubits density matrices with two or three symmetric pairs of linear terms. Some of the theoretical results are demonstrated by numerical calculations. The four non-generic cases (which may be reduced to case (a) are analyzed and the non-generic property is related explicitly to Lorentz velocity [Formula: see text] which is not reachable physically.

arXiv:1707.03696International Journal of Quantum InformationSep 1, 20170 citations

A new representation for the density matrix

G. Ramachandran, M. V. N. Murthy

Nuclear Physics AJul 1, 197921 citations

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 $s<~{(1+p}^{n\ensuremath{-}1}{)}^{\ensuremath{-}1}.$