Trace Class Operators Research Articles

Alpha-Beta Log-Determinant Divergences Between Positive Definite Trace Class Operators

This work presents a parametrized family of divergences, namely Alpha-Beta Log-Determinant (Log-Det) divergences, between positive definite unitized trace class operators on a Hilbert space. This is a generalization of the Alpha-Beta Log-Determinant divergences between symmetric, positive definite matrices to the infinite-dimensional setting. The family of Alpha-Beta Log-Det divergences is highly general and contains many divergences as special cases, including the recently formulated infinite-dimensional affine-invariant Riemannian distance and the infinite-dimensional Alpha Log-Det divergences between positive definite unitized trace class operators. In particular, it includes a parametrized family of metrics between positive definite trace class operators, with the affine-invariant Riemannian distance and the square root of the symmetric Stein divergence being special cases. For the Alpha-Beta Log-Det divergences between covariance operators on a Reproducing Kernel Hilbert Space (RKHS), we obtain closed form formulas via the corresponding Gram matrices.

Infinite-dimensional Log-Determinant divergences between positive definite Hilbert–Schmidt operators

The current work generalizes the author’s previous work on the infinite-dimensional Alpha Log-Determinant (Log-Det) divergences and Alpha-Beta Log-Det divergences, defined on the set of positive definite unitized trace class operators on a Hilbert space, to the entire Hilbert manifold of positive definite unitized Hilbert–Schmidt operators. This generalization is carried out via the introduction of the extended Hilbert–Carleman determinant for unitized Hilbert–Schmidt operators, in addition to the previously introduced extended Fredholm determinant for unitized trace class operators. The resulting parametrized family of Alpha-Beta Log-Det divergences is general and contains many divergences between positive definite unitized Hilbert–Schmidt operators as special cases, including the infinite-dimensional affine-invariant Riemannian distance and the infinite-dimensional generalization of the symmetric Stein divergence.

A class of stochastic products on the convex set of quantum states

We introduce the notion of stochastic product as a binary operation on the convex set of quantum states (the density operators) that preserves the convex structure, and we investigate its main consequences. We consider, in particular, stochastic products that are covariant wrt a symmetry action of a locally compact group. We then construct an interesting class of group-covariant, associative stochastic products, the so-called twirled stochastic products. Every binary operation in this class is generated by a triple formed by a square integrable projective representation of a locally compact group, by a probability measure on that group and by a fiducial density operator acting in the carrier Hilbert space of the representation. The salient properties of such a product are studied. It is argued, in particular, that, extending this binary operation from the density operators to the whole Banach space of trace class operators, this space becomes a Banach algebra, a so-called twirled stochastic algebra. This algebra is shown to be commutative in the case where the relevant group is abelian. In particular, the commutative stochastic products generated by the Weyl system are treated in detail. Finally, the physical interpretation of twirled stochastic products and various interesting connections with the literature are discussed.

On the analyticity of the fredholm determinant

We prove that the Fredholm determinant is a real analytic function on the space of trace class operators defined on a separable Hilbert space H. This allows us to measure the size of sets of trace class and Schatten p-class operators which are not invertible and give an explicit description of their Aronszajn decomposition.

On Accumulated Cohen’s Class Distributions and Mixed-State Localization Operators

Recently we introduced mixed-state localization operators associated with a density operator and a (compact) domain in phase space. We continue the investigations of their eigenvalues and eigenvectors. Our main focus is the definition of a time-frequency distribution that is based on the Cohen class distribution associated with the density operator and the eigenvectors of the mixed-state localization operator. This time-frequency distribution is called the accumulated Cohen class distribution. If the trace class operator is a rank-one operator, then the mixed-state localization operators and the accumulated Cohen class distribution reduce to Daubechies’ localization operators and the accumulated spectrogram. We extend all the results about the accumulated spectrogram to the accumulated Cohen class distribution. The techniques used in the case of spectrograms cannot be adapted to other distributions in Cohen’s class since they rely on the reproducing kernel property of the short-time Fourier transform. Our approach is based on quantum harmonic analysis on phase space, which also provides the tools and notions to introduce the analogues of the accumulated spectrogram for mixed-state localization operators, the accumulated Cohen’s class distributions.

Mixed-State Localization Operators: Cohen’s Class and Trace Class Operators

We study mixed-state localization operators from the perspective of Werner’s operator convolutions which allows us to extend known results from the rank-one case to trace class operators. The idea of localizing a signal to a domain in phase space is approached from various directions such as bounds on the spreading function, probability densities associated to mixed-state localization operators, positive operator valued measures, positive correspondence rules and variants of Tauberian theorems for operator translates. Our results include a rigorous treatment of multiwindow-STFT filters and a characterization of mixed-state localization operators as positive correspondence rules. Furthermore we provide a description of the Cohen class in terms of Werner’s convolution of operators and deduce consequences on positive Cohen class distributions, an uncertainty principle, uniqueness and phase retrieval for general elements of Cohen’s class.

Decompositions of completely bounded maps into completely positive maps involving trace class operators

Decompositions of completely bounded maps into completely positive maps involving trace class operators

On the positivity of trace class operators

The characterization of positivity properties of Weyl operators is a notoriously difficult problem, and not much progress has been made since the pioneering work of Kastler, Loupias, and Miracle-Sole (KLM). In this paper we begin by reviewing and giving simpler proofs of some known results for trace-class Weyl operators; the latter play an essential role in quantum mechanics. We then apply time-frequency analysis techniques to prove a phase space version of the KLM condition; the main tools are Gabor frames and the Wigner formalism. Finally, discrete approximations of the KLM condition, which are tractable numerically, are provided.

Infinite dimensional generalizations of Choi’s Theorem

Abstract In this paper we give a simple sequence of necessary and sufficient finite dimensional conditions for a positive map between certain subspaces of bounded linear operators on separable Hilbert spaces to be completely positive. These criterions are natural generalization of Choi’s characterization for completely positive maps between pairs of linear operators on finite dimensional Hilbert spaces. We apply our conditions to a completely positive map between two trace class operators on separable Hilbert spaces. A completely positive map μ is called a quantum channel, if it is trace preserving, and μ is called a quantum subchannel if it decreases the trace of a positive operator.We give simple neccesary and sufficient condtions for μ to be a quantum subchannel.We show that μ is a quantum subchannel if and only if it hasHellwig-Kraus representation. The last result extends the classical results of Kraus and the recent result of Holevo for characterization of a quantum channel.

Radial Schur multipliers on some generalisations of trees

We give a characterisation of radial Schur multipliers on finite products of trees. The equivalent condition is that a certain generalised Hankel matrix involving the discrete derivatives of the radial function is a trace class operator. This extends Haagerup, Steenstrup and Szwarc's result for trees. The same condition can be expressed in terms of Besov spaces on the torus. We also prove a similar result for products of hyperbolic graphs and provide a sufficient condition for a function to define a radial Schur multiplier on a finite dimensional CAT(0) cube complex.

Estimating the spectral gap of a trace-class Markov operator

The utility of a Markov chain Monte Carlo algorithm is, in large part, determined by the size of the spectral gap of the corresponding Markov operator. However, calculating (and even approximating) the spectral gaps of practical Monte Carlo Markov chains in statistics has proven to be an extremely difficult and often insurmountable task, especially when these chains move on continuous state spaces. In this paper, a method for accurate estimation of the spectral gap is developed for general state space Markov chains whose operators are non-negative and trace-class. The method is based on the fact that the second largest eigenvalue (and hence the spectral gap) of such operators can be bounded above and below by simple functions of the power sums of the eigenvalues. These power sums often have nice integral representations. A classical Monte Carlo method is proposed to estimate these integrals, and a simple sufficient condition for finite variance is provided. This leads to asymptotically valid confidence intervals for the second largest eigenvalue (and the spectral gap) of the Markov operator. In contrast with previously existing techniques, our method is not based on a near-stationary version of the Markov chain, which, paradoxically, cannot be obtained in a principled manner without bounds on the spectral gap. On the other hand, it can be quite expensive from a computational standpoint. The efficiency of the method is studied both theoretically and empirically.

Generators of Quantum One-Dimensional Diffusions

\bad Quantum dynamical semigroups represent a noncommutative analogue of (sub)Markov semigroups in classical probability: while the latter are semigroups of maps in functional spaces, the former are semigroups of maps in operator algebras having certain properties of positivity and normalization. In this paper we describe quantum dynamical semigroups, which are the noncommutative analogues of classical diffusions on $R$ and $R_{+}$, and demonstrate various properties of the semigroup and its generator depending on the boundary condition. We also give a proof of a result describing the domain of the generator of “noncommutative diffusion on $R_{+}$ with extinction at 0” and give an explicit example of the trace-class operator in this domain, which does not belong to the domain of closure of the initial operator.

The $$\lambda $$ λ -Function in the Space of Trace Class Operators

Let $$C_1(H)$$ denote the space of all trace class operators on an arbitrary complex Hilbert space H. We prove that $$C_1(H)$$ satisfies the $$\lambda $$ -property, and we determine the form of the $$\lambda $$ -function of Aron and Lohman on the closed unit ball of $$C_1(H)$$ by showing that $$\begin{aligned} \lambda (a) = \frac{1 - \Vert a\Vert _1 + 2 \Vert a\Vert _{\infty }}{2}, \end{aligned}$$ for every a in $${C_1(H)}$$ with $$\Vert a\Vert _1 \le 1$$ . This is a non-commutative extension of the formula established by Aron and Lohman for $$\ell _1$$ .

The Correct Formulation of Gleason’s Theorem in Quaternionic Hilbert Spaces

From the viewpoint of the theory of orthomodular lattices of elementary propositions, Quantum Theories can be formulated in real, complex or quaternionic Hilbert spaces as established in Sol\'er's theorem. The said lattice eventually coincides with the lattice of all orthogonal projectors on a separable Hilbert space over R, C, or over the algebra of quaternions H. Quantum states are $\sigma$-additive probability measures on that non-Boolean lattice. Gleason's theorem proves that, if the Hilbert space is separable with dimension >2 and the Hilbert space is either real or complex, then states are one-to-one with standard density matrices (self-adjoint, positive, unit-trace, trace-class operators). The extension of this result to quaternionic Hilbert spaces was obtained by Varadarajan in 1968. Unfortunately, even if the hard part of the proof is correct, the formulation of this extension is mathematically incorrect. This is due to some peculiarities of the notion of trace in quaternionic Hilbert spaces, e.g., basis dependence, making the theory of trace-class operators in quaternionic Hilbert spaces different from the standard theory in real and complex Hilbert spaces. A minor issue also affects Varadarajan's statement for real Hilbert space formulation. This paper is mainly devoted to present Gleason-Varadarajan's theorem into a technically correct form valid for the three types of Hilbert spaces. After having develped part of the general mathematical technology of trace-class operators in (generally non-separable) quaternionic Hilbert spaces, we prove that only the {\em real part} of the trace enters the formalism of quantum theories (also dealing with unbounded observables and symmetries) and it can be safely used to formulate and prove a common statement of Gleason's theorem.

Iterated stochastic integrals in infinite dimensions: approximation and error estimates

Higher order numerical schemes for stochastic partial differential equations that do not possess commutative noise require the simulation of iterated stochastic integrals. In this work, we extend the algorithms derived by Kloeden et al. (Stoch Anal Appl 10(4):431–441, 1992. https://doi.org/10.1080/07362999208809281) and by Wiktorsson (Ann Appl Probab 11(2):470–487, 2001. https://doi.org/10.1214/aoap/1015345301) for the approximation of two-times iterated stochastic integrals involved in numerical schemes for finite dimensional stochastic ordinary differential equations to an infinite dimensional setting. These methods clear the way for new types of approximation schemes for SPDEs without commutative noise. Precisely, we analyze two algorithms to approximate two-times iterated integrals with respect to an infinite dimensional Q-Wiener process in case of a trace class operator Q given the increments of the Q-Wiener process. Error estimates in the mean-square sense are derived and discussed for both methods. In contrast to the finite dimensional setting, which is contained as a special case, the optimal approximation algorithm cannot be uniquely determined but is dependent on the covariance operator Q. This difference arises as the stochastic process is of infinite dimension.

The C-Numerical Range in Infinite Dimensions

ABSTRACTIn infinite dimensions and on the level of trace-class operators C rather than matrices, we show that the closure of the C-numerical range is always star-shaped with respect to the set , where denotes the essential numerical range of the bounded operator T. Moreover, the closure of is convex if either C is normal with collinear eigenvalues or if T is essentially self-adjoint. In the case of compact normal operators, the C-spectrum of T is a subset of the C-numerical range, which itself is a subset of the convex hull of the closure of the C-spectrum. This convex hull coincides with the closure of the C-numerical range if, in addition, the eigenvalues of C or T are collinear.

Gram matrix associated to controlled frames

Controlled frames have been recently introduced in Hilbert spaces to improve the numerical efficiency of interactive algorithms for inverting the frame operator. In this paper, unlike the cross-Gram matrix of two different sequences which is not always a diagnostic tool, we define the controlled-Gram matrix of a sequence as a practical implement to diagnose that a given sequence is a controlled Bessel, frame or Riesz basis. Also, we discuss the cases that the operator associated to controlled Gram matrix will be bounded, invertible, Hilbert–Schmidt or a trace-class operator. Similar to standard frames, we present an explicit structure for controlled Riesz bases and show that every [Formula: see text]-controlled Riesz basis [Formula: see text] is in the form [Formula: see text], where [Formula: see text] is a bijective operator on [Formula: see text]. Furthermore, we propose an equivalent accessible condition to the sequence [Formula: see text] being a [Formula: see text]-controlled Riesz basis.

On Spectral Asymptotics for a Family of Finite Dimensional Perturbations of Operators of Trace Class

On Spectral Asymptotics for a Family of Finite Dimensional Perturbations of Operators of Trace Class

On Spectral Asymptotics for a Family of Finite-Dimensional Perturbations of Operators of Trace Class

Spectral asymptotics for a family of finite-dimensional perturbations of operators of trace class are found. The results are used to find exact asymptotics of small ball probabilities in the Hilbert norm for finitedimensional perturbations of Gaussian functions. As an example, Durbin processes appearing in the study of empirical processes with estimated parameters are considered.

A survey on Tingley’s problem for operator algebras

We survey the most recent results on extension of isometries between special subsets of the unit spheres of C$^*$-algebras, von Neumann algebras, trace class operators, preduals of von Neumann algebras, and $p$-Schatten-von Neumann spaces, with special interest on Tingley's problem.