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In this paper, we consider all possible variants of Choi matrices of linear maps, and show that they are determined by non-degenerate bilinear forms on the domain space. We will do this in the setting of finite dimensional vector spaces. In case of matrix algebras, we characterize all variants of Choi matrices which retain the usual correspondences between k-superpositivity and Schmidt number ≤ k as well as k-positivity and k-block-positivity. We also compare de Pillis' definition [Pac. J. Math. 23, 129–137 (1967)] and Choi's definition [Linear Algebra Appl. 10, 285–290 (1975)], which arise from different bilinear forms. [ABSTRACT FROM AUTHOR]

Since K(h, )-1 (X) <= (X ) for m <= X <= M and h = M/m by Ref. 1, Lemma 4.3, it follows thatTr[ ( )]=Tr[ ( )1/2 (( -1/2 -1/2) ) ( )1/2]>=K(h2, )-1Tr[ ( )1/2 ( -1/2 -1/2) ( )1/2]=K(h2, )-1Tr[ ( ) ( )],where (X) = ( )-1/2 ( 1/2X 1/2) ( )-1/2, and hence, we have the corrected Theorem 7.4.Furthermore, since Ref. 3, Lemma 8.1 is derived from Ref. 3, Lemma 5.2, it is not valid in general. By the corrected Lemma 5.1, since I t i SP I i sp is operator convex for I i [-1, 0) [1, 2], the condition for inequality (5.2) in Ref. [3], Lemma 5.2 to hold is modified to I i [-1, 0] [1, 2]. (5.3) Therefore, it follows from the corrected Lemma 5.2 that Ref. 3, Theorem 5.3 (data processing inequality for -Rényi divergence) holds for the following restricted conditions. [Extracted from the article]

A linear map between matrix algebras corresponds to the Choi matrix in the tensor product of two matrix algebras whose definition depends on matrix units. Paulsen and Shultz [J. Math. Phys. 54, 072201 (2013)] considered the question if one can replace matrix units by another basis of matrix algebras in the definition of the Choi matrix to retain the correspondence between the complete positivity of maps and the positivity of Choi matrices and gave a sufficient condition on basis under which this is true. In this note, we provide necessary and sufficient conditions to see that the Paulsen–Shultz condition is also necessary. [ABSTRACT FROM AUTHOR]

We study the probability that all the eigenvalues of n × n Hermitian matrices, from the Laguerre unitary ensemble with the weight x γ e − 4 n x , x ∈ 0 , ∞ , γ > − 1 , lie in the interval [0, α]. By using previous results for finite n obtained by the ladder operator approach of orthogonal polynomials, we derive the large n asymptotics of the largest eigenvalue distribution function with α ranging from 0 to the soft edge. In addition, at the soft edge, we compute the constant conjectured by Tracy and Widom [Commun. Math. Phys. 159, 151–174 (1994)] and later proved by Deift, Its, and Krasovsky [Commun. Math. Phys. 278, 643–678 (2008)]. Our conclusions are reduced to those of Deift et al. when γ = 0. It should be pointed out that our derivation is straightforward but not rigorous, and hence, the above results are stated as conjectures. [ABSTRACT FROM AUTHOR]

Starting from a four-by-four matrix spectral problem, an integrable coupling family of the Toda lattice systems is derived. A pair of discrete Hamiltonian operators is presented, and a sequence of the corresponding Hamiltonian functionals is constructed by using the discrete variational identity. Then, the bi-Hamiltonian structure of the obtained integrable coupling family is established. Finally, a nonisospectral integrable lattice family associated with the obtained integrable coupling family is given through nonisospectral discrete zero curvature representation. [ABSTRACT FROM AUTHOR]

We give a short, operator-theoretic proof of the asymptotic independence (including a first correction term) of the minimal and maximal eigenvalue of the n×n Gaussian unitary ensemble in the large matrix limit n→∞. This is done by representing the joint probability distribution of the extreme eigenvalues as the Fredholm determinant of an operator matrix that asymptotically becomes diagonal. As a corollary, we get that the correlation of the extreme eigenvalues asymptotically behaves like n-2/3/4σ2, where σ2 denotes the variance of the Tracy–Widom distribution. While we conjecture that the extreme eigenvalues are asymptotically independent for Wigner random Hermitian matrix ensembles, in general, the actual constant in the asymptotic behavior of the correlation turns out to be specific and can thus be used to distinguish the Gaussian unitary ensemble statistically from certain other Wigner ensembles. [ABSTRACT FROM AUTHOR]

We investigate the quantum integrability of the Landau–Lifshitz (LL) model and solve the long-standing problem of finding the local quantum Hamiltonian for the arbitrary n-particle sector. The particular difficulty of the LL model quantization, which arises due to the ill-defined operator product, is dealt with by simultaneously regularizing the operator product and constructing the self-adjoint extensions of a very particular structure. The diagonalizibility difficulties of the Hamiltonian of the LL model, due to the highly singular nature of the quantum-mechanical Hamiltonian, are also resolved in our method for the arbitrary n-particle sector. We explicitly demonstrate the consistency of our construction with the quantum inverse scattering method due to Sklyanin [Lett. Math. Phys. 15, 357 (1988)] and give a prescription to systematically construct the general solution, which explains and generalizes the puzzling results of Sklyanin for the particular two-particle sector case. Moreover, we demonstrate the S-matrix factorization and show that it is a consequence of the discontinuity conditions on the functions involved in the construction of the self-adjoint extensions. [ABSTRACT FROM AUTHOR]

We construct C-algebras for a class of surfaces that are inverse images of certain polynomials of arbitrary degree. By using the directed graph associated with a matrix, the representation theory can be understood in terms of “loop” and “string” representations, which are closely related to the dynamics of an iterated map in the plane. As a particular class of algebras, we introduce the “Hénon algebras,” for which the dynamical map is a generalized Hénon map, and give an example where irreducible representations of all dimensions exist. [ABSTRACT FROM AUTHOR]

A new class of dynamical systems are presented, together with their solutions. Some of these models are isochronous, namely, their generic solutions are all completely periodic with the same period; others are characterized by friction, all solutions vanishing in the remote future; and still others are “asymptotically isochronous,” approaching an isochronous behavior in the remote future. [ABSTRACT FROM AUTHOR]

Our starting point is a class of braid matrices, presented in a previous paper, constructed on a basis of a nested sequence of projectors. Statistical models associated to such N2×N2 matrices for odd N are studied here. Presence of 1/2(N+3)(N-1) free parameters is the crucial feature of our models, setting them apart from other well-known ones. There are N possible states at each site. The trace of the transfer matrix is shown to depend on 1/2(N-1) parameters. For order r, N eigenvalues constitute the trace and the remaining (Nr-N) eigenvalues involving the full range of parameters come in zero-sum multiplets formed by the rth roots of unity, or lower dimensional multiplets corresponding to factors of the order r when r is not a prime number. The modulus of any eigenvalue is of the form eμθ, where μ is a linear combination of the free parameters, θ being the spectral parameter. For r a prime number an amusing relation of the number of multiplets with a theorem of Fermat is pointed out. Chain Hamiltonians and potentials corresponding to factorizable S-matrices are constructed starting from our braid matrices. Perspectives are discussed. [ABSTRACT FROM AUTHOR]

A basis of N2 projectors, each an N2×N2 matrix with constant elements, is implemented to construct a class of braid matrices R⁁(θ), θ being the spectral parameter. Only odd values of N are considered here. Our ansatz for the projectors Pα appearing in the spectral decomposition of R⁁(θ) leads to exponentials exp(mαθ) as the coefficient of Pα. The sums and differences of such exponentials on the diagonal and the antidiagonal, respectively, provide the (2N2-1) nonzero elements of R⁁(θ). One element at the center is normalized to unity. A class of supplementary constraints imposed by the braid equation leaves 1/2(N+3)(N-1) free parameters mα. The diagonalizer of R⁁(θ) is presented for all N. Transfer matrices t(θ) and L(θ) operators corresponding to our R⁁(θ) are studied. Our diagonalizer signals specific combinations of the components of the operators that lead to a quadratic algebra of N2 constant N×N matrices. The θ dependence factors out for such combinations. R⁁(θ) is developed in a power series in θ. The basic difference arising for even dimensions is made explicit. Some special features of our R⁁(θ) are discussed in a concluding section. [ABSTRACT FROM AUTHOR]

We introduce a condition for memoryless quantum channels which, when satisfied guarantees the multiplicativity of the maximal ℓp-norm with p a fixed integer. By applying the condition to qubit channels, it can be shown that it is not a necessary condition, although some known results for qubits can be recovered. When applied to the Werner-Holevo channel, which is known to violate multiplicativity when p is large relative to the dimension d, the condition suggests that multiplicativity holds when d>=2p-1. This conjecture is proved explicitly for p=2,3,4. Finally, a new class of channels is considered which generalizes the depolarizing channel to maps which are combinations of the identity channel and a noisy one whose image is an arbitrary density matrix. It is shown that these channels are multiplicative for p=2. [ABSTRACT FROM AUTHOR]

We present new classes of exact solutions with noncommutative symmetries constructed in vacuum Einstein gravity (in general, with nonzero cosmological constant), five-dimensional (5D) gravity and (anti) de Sitter gauge gravity. Such solutions are generated by anholonomic frame transforms and parametrized by generic off-diagonal metrics. For certain particular cases, the new classes of metrics have explicit limits with Killing symmetries but, in general, they may be characterized by certain anholonomic noncommutative matrix geometries. We argue that different classes of noncommutative symmetries can be induced by exact solutions of the field equations in commutative gravity modeled by a corresponding moving real and complex frame geometry. We analyze two classes of black ellipsoid solutions (in the vacuum case and with cosmological constant) in four-dimensional gravity and construct the analytic extensions of metrics for certain classes of associated frames with complex valued coefficients. The third class of solutions describes 5D wormholes which can be extended to complex metrics in complex gravity models defined by noncommutative geometric structures. The anholonomic noncommutative symmetries of such objects are analyzed. We also present a descriptive account how the Einstein gravity can be related to gauge models of gravity and their noncommutative extensions and discuss such constructions in relation to the Seiberg–Witten map for the gauge gravity. Finally, we consider a formalism of vielbeins deformations subjected to noncommutative symmetries in order to generate solutions for noncommutative gravity models with Moyal (star) product. [ABSTRACT FROM AUTHOR]

Motivated by existence problems for dissipative systems arising naturally in lattice models from quantum statistical mechanics, we consider the following C*-algebraic setting: A given Hermitian dissipative mapping δ is densely defined in a unital C*-algebra A. The identity element in A is also in the domain of δ. Completely dissipative maps δ are defined by the requirement that the induced maps, (aij)→(δ(aij)), are dissipative on the n-by-n complex matrices over A for all n. We establish the existence of different types of maximal extensions of completely dissipative maps. If the enveloping von Neumann algebra of A is injective, we show the existence of an extension of δ which is the infinitesimal generator of a quantum dynamical semigroup of completely positive maps in the von Neumann algebra. If δ is a given well-behaved *-derivation, then we show that each of the maps ±δ is completely dissipative. [ABSTRACT FROM AUTHOR]

The previous supersymmetric generalization of the unitary Harish-Chandra integral prompted the conjecture that the Harish-Chandra formula should have an extension to superspaces. We prove this conjecture for the unitary orthosymplectic supermanifold UOSp(k1/2k2). To this end, we construct and solve an eigenvalue equation. [ABSTRACT FROM AUTHOR]

As an extension of the theory of Dyson’s Brownian motion models for the standard Gaussian random-matrix ensembles, we report a systematic study of Hermitian matrix-valued processes and their eigenvalue processes associated with the chiral and nonstandard random-matrix ensembles. In addition to the noncolliding Brownian motions, we introduce a one-parameter family of temporally homogeneous noncolliding systems of the Bessel processes and a two-parameter family of temporally inhomogeneous noncolliding systems of Yor’s generalized meanders and show that all of the ten classes of eigenvalue statistics in the Altland–Zirnbauer classification are realized as particle distributions in the special cases of these diffusion particle systems. As a corollary of each equivalence in distribution of a temporally inhomogeneous eigenvalue process and a noncolliding diffusion process, a stochastic-calculus proof of a version of the Harish–Chandra (Itzykson–Zuber) formula of integral over unitary group is established. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

By means of complex representation and companion vector, this paper studies the problems of eigenvalues and eigenvectors of quaternion matrices, and gives a technique of computing the eigenvalues and eigenvectors of the quaternion matrices in quaternionic quantum mechanics. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

By applying higher powers of Bogoliubov’s transformation operator b†=ν*a+μ*a† to the even and odd coherent states we construct mathematically two new types of quantum states. Various special functions and mathematical formulations are used to obtain the matrix elements for the states in the coherent state representation. The nonclassical nature of these states is found to be related to their quantum coherence in phase space. The interference effects can also be regarded as a consequence of the oscillatory behavior of the special functions in the structure of the states. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Given a family of monodromy matrices T0,T1,...,TK-1 corresponding to integrable anisotropic vertex models of Anμ-1 type, μ=0,1,...,K-1, we built up a related mixed vertex model by means of gluing the lattices on which they are defined, in such a way that integrability property is preserved. The gluing process is implemented through one-dimensional representations of rectangular quantum matrix algebras A(Rnμ-1:Rnμ), namely, the gluing matrices ζμ. Algebraic Bethe ansatz is applied on a pseudovacuum space with a selected basis and, for each element of this basis, it yields a set of nested Bethe ansatz equations matching up to the ones corresponding to an Am-1 quasiperiodic model, with m equal to minμ∈ZK{rank ζμ}. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Fisher–Hartwig asymptotics refers to the large n form of a class of Toeplitz determinants with singular generating functions. This class of Toeplitz determinants occurs in the study of the spin–spin correlations for the two-dimensional Ising model, and the ground state density matrix of the impenetrable Bose gas, amongst other problems in mathematical physics. We give a new application of the original Fisher–Hartwig formula to the asymptotic decay of the Ising correlations above Tc, while the study of the Bose gas density matrix leads us to generalize the Fisher–Hartwig formula to the asymptotic form of random matrix averages over the classical groups and the Gaussian and Laguerre unitary matrix ensembles. Another viewpoint of our generalizations is that they extend to Hankel determinants the Fisher–Hartwig asymptotic form known for Toeplitz determinants.© 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The homogeneous Bloch equation reduces to the Riccati equation. By linearizing the Riccati equation, a set of three solutions of the homogeneous Bloch equation is found. The fundamental matrix becomes singular. We clarify the utility and limitation of our approach to solve the homogeneous Bloch equation. © 2003 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Discrete transformation for three-wave problems is constructed in explicit form. Generalization of this system on the matrix case in three dimensional space together with corresponding discrete transformation is presented also. © 2003 American Institute of Physics. [ABSTRACT FROM AUTHOR]

We study the mathematical structure of covariant phase observables. Such observables can alternatively be expressed as phase matrices, as sequences of unit vectors, as sequences of phase states, or as equivalence classes of covariant trace-preserving operations. Covariant generalized operator measures are defined by structure matrices which form a W[sup *]-algebra with phase matrices as its subset. The properties of the Radon–Nikody´m derivatives of phase probability measures are studied. © 2002 American Institute of Physics. [ABSTRACT FROM AUTHOR]

An additive braided coproduct (also be called braided coaddition) is introduced on the quantized braided matrices (QBMs). It gives QBMs another braided-Hopf algebra structure. The coaddition is also shown to be compatible with the existing coproduct such that they together form a so-called quantized-braided ring. And some quantized braided differential operator bialgebras (Hopf algebras) relating to this braided coaddition are constructed. These give a unification and generalization of the known results about braided and quantum matrices. Moreover, the coaddition construction and the related differential calculi on the QBMs are further extended to a kind of more general quantum matrix algebraic systems. Some examples are also given. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]