We construct representation of the Separated Variables (SoV) for the quantum SL(2,R) Heisenberg closed spin chain and obtain the integral representation for the eigenfunctions of the model. We calculate explicitly the Sklyanin measure defining the scalar product in the SoV representation and demonstrate that the language of Feynman diagrams is extremely useful in establishing various properties of the model. The kernel of the unitary transformation to the SoV representation is described by the same "pyramid diagram" as appeared before in the SoV representation for the SL(2,C) spin magnet. We argue that this kernel is given by the product of the Baxter Q-operators projected onto a special reference state.We construct representation of the Separated Variables (SoV) for the quantum SL(2,R) Heisenberg closed spin chain and obtain the integral representation for the eigenfunctions of the model. We calculate explicitly the Sklyanin measure defining the scalar product in the SoV representation and demonstrate that the language of Feynman diagrams is extremely useful in establishing various properties of the model. The kernel of the unitary transformation to the SoV representation is described by the same "pyramid diagram" as appeared before in the SoV representation for the SL(2,C) spin magnet. We argue that this kernel is given by the product of the Baxter Q-operators projected onto a special reference state.

Jan 1, 200221 citations

The Weyl Operator and its Generalization

Leon W. Cohen

Springer eBooksDec 11, 201277 citations

Nongeneric positive partial transpose states of rank five in $3\times 3$ dimensionsNongeneric positive partial transpose states of rank five in $3\times 3$ dimensions

Leif Ove Hansen, Jan Myrheim

In $3\times 3$ dimensions, entangled mixed states that are positive under partial transposition (PPT states) must have rank at least four. They are well understood. We say that they have rank $(4,4)$ since a state $ρ$ and its partial transpose $ρ^P$ both have rank four. The next problem is to understand the extremal PPT states of rank $(5,5)$ . We call two states $\textrm{SL}\otimes\textrm{SL}$ -equivalent if they are related by a product transformation. A generic rank $(5,5)$ PPT state $ρ$ is extremal, and $ρ$ and $ρ^P$ both have six product vectors in their ranges, and no product vectors in their kernels. The three numbers $\{6,6;0\}$ are $\textrm{SL}\otimes\textrm{SL}$ -invariants that help us classify the state. We have studied numerically a few types of nongeneric rank five PPT states, in particular states with one or more product vectors in their kernels. We find an interesting new analytical construction of all rank four extremal PPT states, up to $\textrm{SL}\otimes\textrm{SL}$ -equivalence, where they appear as boundary states on one single five dimensional face on the set of normalized PPT states. We say that a state $ρ$ is $\textrm{SL}\otimes\textrm{SL}$ -symmetric if $ρ$ and $ρ^P$ are $\textrm{SL}\otimes\textrm{SL}$ -equivalent, and is genuinely $\textrm{SL}\otimes\textrm{SL}$ -symmetric if it is $\textrm{SL}\otimes\textrm{SL}$ -equivalent to a state $τ$ with $τ=τ^P$ . Genuine $\textrm{SL}\otimes\textrm{SL}$ -symmetry implies a special form of $\textrm{SL}\otimes\textrm{SL}$ -symmetry. We have produced numerically a random sample of rank $(5,5)$ $\textrm{SL}\otimes\textrm{SL}$ -symmetric states. About fifty of these are of type $\{6,6;0\}$ , among those all are extremal and about half are genuinely $\textrm{SL}\otimes\textrm{SL}$ -symmetric.In $3\times 3$ dimensions, entangled mixed states that are positive under partial transposition (PPT states) must have rank at least four. They are well understood. We say that they have rank $(4,4)$ since a state $ρ$ and its partial transpose $ρ^P$ both have rank four. The next problem is to understand the extremal PPT states of rank $(5,5)$. We call two states $\textrm{SL}\otimes\textrm{SL}$-equivalent if they are related by a product transformation. A generic rank $(5,5)$ PPT state $ρ$ is extremal, and $ρ$ and $ρ^P$ both have six product vectors in their ranges, and no product vectors in their kernels. The three numbers $\{6,6;0\}$ are $\textrm{SL}\otimes\textrm{SL}$-invariants that help us classify the state. We have studied numerically a few types of nongeneric rank five PPT states, in particular states with one or more product vectors in their kernels. We find an interesting new analytical construction of all rank four extremal PPT states, up to $\textrm{SL}\otimes\textrm{SL}$-equivalence, where they appear as boundary states on one single five dimensional face on the set of normalized PPT states. We say that a state $ρ$ is $\textrm{SL}\otimes\textrm{SL}$-symmetric if $ρ$ and $ρ^P$ are $\textrm{SL}\otimes\textrm{SL}$-equivalent, and is genuinely $\textrm{SL}\otimes\textrm{SL}$-symmetric if it is $\textrm{SL}\otimes\textrm{SL}$-equivalent to a state $τ$ with $τ=τ^P$. Genuine $\textrm{SL}\otimes\textrm{SL}$-symmetry implies a special form of $\textrm{SL}\otimes\textrm{SL}$-symmetry. We have produced numerically a random sample of rank $(5,5)$ $\textrm{SL}\otimes\textrm{SL}$-symmetric states. About fifty of these are of type $\{6,6;0\}$, among those all are extremal and about half are genuinely $\textrm{SL}\otimes\textrm{SL}$-symmetric.

arXiv:1706.07671Jun 23, 20170 citations

A New Generalized Schur-Weyl Duality

Qiang Lei, Haijiang Yu, Junde Wu

International Journal of Theoretical PhysicsJan 12, 20150 citations

$(q;l,λ)-$ deformed Heisenberg algebra: representations, special functions and quantization$(q;l,λ)-$deformed Heisenberg algebra: representations, special functions and quantization

Mahouton Norbert Hounkonnou, Sama Arjika, Ezinvi Baloïtcha

This paper addresses a new characterization of Sudarshan's diagonal representation of the density matrix elements $ρ(z',z)$ , derivedfrom $(q;l,λ)-$ deformed boson coherent states.The induced $ρ(z',z)$ self-reproducing property with the associated self-reproducing kernel $K(z',z)$ is computed and analyzed. An explicit construction of novel classes of generalized continuous $(q;l,λ)-$ Hermite polynomials is provided with the corresponding recursion relations and exact resolution of the moment problems giving their orthogonality weight functions. Besides, the Berezin-Klauder-Toeplitz quantization of classical phase space observables and relevant normal and anti-normal forms are investigated and discussed.This paper addresses a new characterization of Sudarshan's diagonal representation of the density matrix elements $ρ(z',z)$, derivedfrom $(q;l,λ)-$deformed boson coherent states.The induced $ρ(z',z)$ self-reproducing property with the associated self-reproducing kernel $K(z',z)$ is computed and analyzed. An explicit construction of novel classes of generalized continuous $(q;l,λ)-$Hermite polynomials is provided with the corresponding recursion relations and exact resolution of the moment problems giving their orthogonality weight functions. Besides, the Berezin-Klauder-Toeplitz quantization of classical phase space observables and relevant normal and anti-normal forms are investigated and discussed.

arXiv:1301.7028Jan 29, 20130 citations

Nongeneric positive partial transpose states of rank five in

3\times 3

dimensions

3\times 3

dimensions, entangled mixed states that are positive under partial transposition (PPT states) must have rank at least four. They are well understood. We say that they have rank

(4,4)

since a state

ρ

and its partial transpose

ρ^P

both have rank four. The next problem is to understand the extremal PPT states of rank

(5,5)

. We call two states

\textrm{SL}\otimes\textrm{SL}

-equivalent if they are related by a product transformation. A generic rank

(5,5)

PPT state

ρ

is extremal, and

ρ

and

ρ^P

both have six product vectors in their ranges, and no product vectors in their kernels. The three numbers

\{6,6;0\}

are

\textrm{SL}\otimes\textrm{SL}

-invariants that help us classify the state. We have studied numerically a few types of nongeneric rank five PPT states, in particular states with one or more product vectors in their kernels. We find an interesting new analytical construction of all rank four extremal PPT states, up to

\textrm{SL}\otimes\textrm{SL}

-equivalence, where they appear as boundary states on one single five dimensional face on the set of normalized PPT states. We say that a state

ρ

\textrm{SL}\otimes\textrm{SL}

-symmetric if

ρ

and

ρ^P

are

\textrm{SL}\otimes\textrm{SL}

-equivalent, and is genuinely

\textrm{SL}\otimes\textrm{SL}

-symmetric if it is

\textrm{SL}\otimes\textrm{SL}

-equivalent to a state

τ

with

τ=τ^P

. Genuine

\textrm{SL}\otimes\textrm{SL}

-symmetry implies a special form of

\textrm{SL}\otimes\textrm{SL}

-symmetry. We have produced numerically a random sample of rank

(5,5)

\textrm{SL}\otimes\textrm{SL}

-symmetric states. About fifty of these are of type

\{6,6;0\}

, among those all are extremal and about half are genuinely

\textrm{SL}\otimes\textrm{SL}

-symmetric.

(q;l,λ)-

deformed Heisenberg algebra: representations, special functions and quantization

This paper addresses a new characterization of Sudarshan's diagonal representation of the density matrix elements

ρ(z',z)

, derivedfrom

(q;l,λ)-

deformed boson coherent states.The induced

ρ(z',z)

self-reproducing property with the associated self-reproducing kernel

K(z',z)

is computed and analyzed. An explicit construction of novel classes of generalized continuous

(q;l,λ)-

Hermite polynomials is provided with the corresponding recursion relations and exact resolution of the moment problems giving their orthogonality weight functions. Besides, the Berezin-Klauder-Toeplitz quantization of classical phase space observables and relevant normal and anti-normal forms are investigated and discussed.