Space Of Trace Class Operators Research Articles

Trace class operators and states in p-adic quantum mechanics

Within the framework of quantum mechanics over a quadratic extension of the non-Archimedean field of p-adic numbers, we provide a definition of a quantum state relying on a general algebraic approach and on a p-adic model of probability theory. As in the standard complex case, a distinguished set of physical states are related to a notion of trace for a certain class of bounded operators, and in fact, we show that one can define a suitable space of trace class operators in the non-Archimedean setting, as well. The analogies—but also the several (highly non-trivial) differences—with respect to the case of standard quantum mechanics in a complex Hilbert space are analyzed.

Dual Convolution for the Affine Group of the Real Line

The Fourier algebra of the affine group of the real line has a natural identification, as a Banach space, with the space of trace-class operators on L^2({{mathbb {R}}}^times , dt/ |t|). In this paper we study the “dual convolution product” of trace-class operators that corresponds to pointwise product in the Fourier algebra. Answering a question raised in work of Eymard and Terp, we provide an intrinsic description of this operation which does not rely on the identification with the Fourier algebra, and obtain a similar result for the connected component of this affine group. In both cases we construct explicit derivations on the corresponding Banach algebras, verifying the derivation identity directly without requiring the inverse Fourier transform. We also initiate the study of the analogous Banach algebra structure for trace-class operators on L^p({{mathbb {R}}}^times , dt/ |t|) for pin (1,2)cup (2,infty ).

Quantum Stochastic Products and the Quantum Convolution

A quantum stochastic product is a binary operation on the space of quantum states preserving the convex structure. We describe a class of associative stochastic products, the twirled products, that have interesting connections with quantum measurement theory. Constructing such a product involves a square integrable group representation, a probability measure and a fiducial state. By extending a twirled product to the full space of trace class operators, one obtains a Banach algebra. This algebra is commutative if the underlying group is abelian. In the case of the group of translations on phase space, one gets a quantum convolution algebra, a quantum counterpart of the classical phase-space convolution algebra. The peculiar role of the fiducial state characterizing each quantum convolution product is highlighted.

Covariant stochastic products of quantum states

A notion of stochastic product of quantum states — a binary operation on the set of density operators preserving the convex structure — is discussed. We describe, in particular, a class of group-covariant, associative stochastic products: the twirled products. Each binary operation in this class can be constructed by means of a square integrable projective representation of a locally compact group, a probability measure on this group and a fiducial density operator in the Hilbert space of the representation. By suitably extending this operation from the convex set of density operators to the full Banach space of trace class operators, one obtains a Banach algebra, which is commutative in the case where the relevant group is abelian.

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.

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 $$\ell _1$$ .

Preduals for spaces of operators involving Hilbert spaces and trace-class operators

Continuing the study of preduals of spaces L(H,Y) of bounded, linear maps, we consider the situation that H is a Hilbert space. We establish a natural correspondence between isometric preduals of L(H,Y) and isometric preduals of Y. The main ingredient is a Tomiyama-type result which shows that every contractive projection that complements L(H,Y) in its bidual is automatically a right L(H)-module map. As an application, we show that isometric preduals of L(S1), the algebra of operators on the space of trace-class operators, correspond to isometric preduals of S1 itself (and there is an abundance of them). On the other hand, the compact operators are the unique predual of S1 making its multiplication separately weak⁎ continuous.

Scattering Theory for Lindblad Master Equations

We study scattering theory for a quantum-mechanical system consisting of a particle scattered off a dynamical target that occupies a compact region in position space. After taking a trace over the degrees of freedom of the target, the dynamics of the particle is generated by a Lindbladian acting on the space of trace-class operators. We study scattering theory for a general class of Lindbladians with bounded interaction terms. First, we consider models where a particle approaching the target is always re-emitted by the target. Then we study models where the particle may be captured by the target. An important ingredient of our analysis is a scattering theory for dissipative operators on Hilbert space.

Peller's problem concerning Koplienko–Neidhardt trace formulae: the unitary case

We prove the existence of a C2-function f:T→C defined on the unit circle, a unitary operator U and a self-adjoint operator Z in the Hilbert–Schmidt class S2, such thatf(eiZU)−f(U)−ddt(f(eitZU))|t=0∉S1, the space of trace class operators. This resolves a problem of Peller concerning the validity of the Koplienko–Neidhardt trace formula for unitaries.

Integrability of multivariate subordinated Lévy processes in Hilbert space

We investigate multivariate subordination of Lévy processes which was first introduced by Barndorff-Nielsen et al. [O.E. Barndorff-Nielsen, F.E. Benth, and A. Veraart, Modelling electricity forward markets by ambit fields, J. Adv. Appl. Probab. (2010)], in a Hilbert space valued setting which has been introduced in Pérez-Abreu and Rocha-Arteaga [V. Pérez-Abreu and A. Rocha-Arteaga, Covariance-parameter Lévy processes in the space of trace-class operators, Infin. Dimens. Anal. Quantum Probab. Related Top. 8(1) (2005), pp. 33–54]. The processes are explicitly characterized and conditions for integrability and martingale properties are derived under various assumptions of the Lévy process and subordinator. As an application of our theory we construct explicitly some Hilbert space valued versions of Lévy processes which are popular in the univariate and multivariate case. In particular, we define a normal inverse Gaussian Lévy process in Hilbert space. The resulting process has the property that at each time all its finite dimensional projections are multivariate normal inverse Gaussian distributed as introduced in Rydberg [T. Rydberg, The normal inverse Gaussian Lévy process: Simulation and approximation, Commun. Stat. Stochastic Models 13 (1997), pp. 887–910].

Duality, cohomology, and geometry of locally compact quantum groups

In this paper, we study various convolution-type algebras associated with a locally compact quantum group from cohomological and geometrical points of view. The quantum group duality endows the space of trace class operators over a locally compact quantum group with two products which are operator versions of convolution and pointwise multiplication, respectively; we investigate the relation between these two products, and derive a formula linking them. Furthermore, we define some canonical module structures on these convolution algebras, and prove that certain topological properties of a quantum group, can be completely characterized in terms of cohomological properties of these modules. We also prove a quantum group version of a theorem of Hulanicki characterizing group amenability. Finally, we study the Radon–Nikodym property of the L1-algebra of locally compact quantum groups. In particular, we obtain a criterion that distinguishes discreteness from the Radon–Nikodym property in this setting.

A description of the evolution of quantum states by means of the kinetic equation

We develop a rigorous formalism for describing the evolution of states of quantum many-particle systems in terms of a one-particle density operator. For initial states which are specified in terms of a one-particle density operator, the equivalence of the description of the evolution of quantum many-particle states by the Cauchy problem of the quantum Bogolyubov–Born–Green–Kirkwood–Yvon hierarchy, and by the Cauchy problem of the generalized quantum kinetic equation, together with a sequence of explicitly defined functionals of a solution of a stated kinetic equation, is established in the space of trace-class operators. The links between the specific quantum kinetic equations with the generalized quantum kinetic equation are discussed.

Induced and coinduced Banach Lie–Poisson spaces and integrability

The Poisson induction and coinduction procedures are used to construct Banach Lie–Poisson spaces as well as related systems of integrals in involution. This general method applied to the Banach Lie–Poisson space of trace class operators leads to infinite Hamiltonian systems of k-diagonal trace class operators which have infinitely many integrals. The bidiagonal case is investigated in detail.

On the structure of tensor products ofℓp-spaces

A Banach space X is prime if every infinite-dimensional complemented subspace contains a further subspace which is isomorphic to X. A Banach space X is said to be primary if whenever X — Y © Z, X is isomorphic to either Y or Z. The classical examples of prime spaces are the spaces ipj 1 < p < oc. Many spaces derived from the £p-spaces in various ways are primary (see for example [AEO] and [CL]). The primarity of B(H) was shown by Blower [B] in 1990, and Arias [A] has recently developed further techniques which are used to prove the primarity of Cι, the space of trace class operators (this was first shown by Arazy [Arl, Ar2]). It has become clear that these techniques are not naturally confined to a Hubert space context; in the present paper we wish to extend the results to a variety of tensor products and operator spaces of ^-spaces (and in some cases £p-spaces). We also include some related results. Some of the intermediate propositions (on factoring operators through the identity) may actually be true for a wider class of Banach spaces (those with unconditional bases which have nontrivial lower and upper estimates). In fact, the combinatorial aspects of the factorization can be applied quite generally, and may have other applications. The proofs of primarity, however, rely on Pelczyήski's decomposition method which is not so readily extended. We have thus kept mainly to the case of injective and projective tensor products of tv spaces throughout. The results we obtain apply to the growing study of polynomials on Banach spaces since polynomials may be considered as symmetric multilinear operators with an equivalent norm (see [FJ], [M],

Constructing quantum measurement processes via classical stochastic calculus

A class of linear stochastic differential equations in Hilbert spaces is studied, which allows to construct probability densities and to generate changes in the probability measure one started with. Related linear equations for trace-class operators are discussed. Moreover, some analogue of filtering theory gives rise to related non-linear stochastic differential equations in Hilbert spaces and in the space of trace-class operators. Finally, it is shown how all these equations represent a new formulation and a generalization of the theory of measurements continuous in time in quantum mechanics.

𝒞₁ is uniformly Kadec-Klee

A dual Banach space X X is Kadec-Klee in the weak * topology if weak * and norm convergence of sequences coincide in the unit sphere of X X . We shall consider a stronger, uniform version of this property. A dual Banach space X X is uniformly Kadec-Klee in the weak * topology (UKK*) if for each ε &gt; 0 \varepsilon &gt; 0 we can find a δ \delta in ( 0 , 1 ) (0,1) such that every weak ∗ * -compact, convex subset C C of the unit ball of X X whose measure of norm compactness exceeds ε \varepsilon must meet the ( 1 − δ ) (1 - \delta ) -ball of X X . We show in this paper that C 1 ( H ) {C_1}(\mathcal {H}) , the space of trace class operators on an arbitrary infinite-dimensional Hilbert space H \mathcal {H} is UKK*. Consequently C 1 ( H ) {C_1}(\mathcal {H}) has weak ∗ * -normal structure. This answers affirmatively a question of A. T. Lau and P. F. Mah. From this it follows that C 1 ( H ) {C_1}(\mathcal {H}) has the weak ∗ * -fixed point property.

The simplex structure of the classical states of the quantum harmonic oscillator

It is shown that the convex set of classical states of the quantum harmonic oscillator is a simplex generated as the closed convex hull of the coherent states in the weak topology of the Banach space of trace class operators.