The physical interpretation of partial traces: Two nonstandard views

Mixed states are introduced in physics to express our ignorance about the actual state of a physical system and are represented in standard quantum mechanics by density operators. Such operators also appear if we consider a (pure) entangled state of a compound system Ω and take partial traces on the projection operator representing it. But because the coefficients in the convex sums expressing them never bear the ignorance interpretation in this case, they represent not mixed states (proper mixtures) but improper mixtures of the subsystems. Hence, states cannot be attributed to the subsystems of a compound physical system in an entangled state (the subentity problem). We discuss two alternative proposals that can be developed in the Brussels and the Lecce approaches. We firstly summarize the general framework provided by the Brussels approach, which suggests that improper mixtures can be regarded as new pure states. We then show that improper mixtures can also be regarded as true (but nonpure) states according to the Lecce approach. Despite their different terminologies, the two proposals seem compatible. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 2, pp. 248–264, August, 2007.     

Representation and interpretation of quantum mixtures in the ESR model

The interpretation of mixtures is problematic in quantum mechanics (QM) because physical properties are nonobjective in this theory. An extended semantic realism model was recently developed, restoring objectivity by reinterpreting quantum probabilities as conditional on detection and embodying the QM mathematical formalism in a broader noncontextual (hence local) framework. In this model, each generalized observable is represented by a family of positive operator-valued measures parameterized by the pure states of the considered physical system Ω. We here propose a new proof that each proper mixture is represented by a family of density operators parameterized by the macroscopic properties characterizing Ω. We then show that this representation implies some predictions differing from the QM predictions and avoids the problems following from the standard QM representation of proper mixtures. We also recall that the state transformations induced by idealized nondestructive measurements can be obtained using a nontrivial generalization of the Lüders postulate.     

Statistical Algebraic Approach to Quantum Mechanics

A scheme for constructing quantum mechanics not based on the Hilbert space and linear operators as primary elements of the theory is proposed. A particular variant of the algebraic approach is discussed. The elements of a noncommutative algebra (i.e., the observables) and the nonlinear functionals on this algebra (i.e., the physical states) serve as the primary components of the theory. The functionals are associated with the results of a single measurement. The ensembles of physical states are suggested for the role of quantum states in the standard quantum mechanics. It is shown that the mathematical formalism of the standard quantum mechanics can be fully recovered within this scheme.     

Schrödinger Uncertainty Relations and Physical Features of Correlated-Coherent States

We show that the difference between the Schrödinger uncertainty relations (UR) and the Heisenberg UR is fundamental. We propose a modified version of stochastic mechanics that allows clearly demonstrating that the contributions from the anticommutator and the commutator to the Schrödinger UR are equally important. A classification of quantum states minimizing the Schrödinger UR at an arbitrary instant is proposed. We show that the correlation of the coordinate and momentum fluctuations in such correlated-coherent states (CCS) is largely determined by the contributions from not only the commutator but also the anticommutator of the corresponding operators. We demonstrate that the character of this correlation changes qualitatively in time from the antiphase correlation typical for the Heisenberg UR to the inphase correlation for which the contribution from the anticommutator is decisive. We comparatively analyze properties of a free microparticle and a quantum oscillator in CCS and show that the CCS correspond to traveling-standing de Broglie waves in both models.     

A state-space approach to quantum permutations

In this exposition of quantum permutation groups, an alternative to the ‘Gelfand picture’ of compact quantum groups is proposed. This point of view is inspired by algebraic quantum mechanics and interprets the states of an algebra of continuous functions on a quantum permutation group as quantum permutations. This interpretation allows talk of an element of a quantum permutation group, and allows a clear understanding of the difference between deterministic, random, and quantum permutations. The interpretation is illustrated by various quantum permutation group phenomena.     

Fuzzy expectation and energy states in fuzzy media

This paper is concerned with energy levels and the density of states in fuzzy crystals. The determination of eigenvalue spectrum, or the density of states, for a particle in a fuzzy crystal, is obtained by using the concepts developed in fuzzy statistics. Various analytical results are found, using the fuzzy expectation; these results can be applied to a variety of fields such as decision making under uncertainty, pattern analysis, and quantum mechanics.     

A tensor product matrix approximation problem in quantum physics

We consider a matrix approximation problem arising in the study of entanglement in quantum physics. This notion represents a certain type of correlations between subsystems in a composite quantum system. The states of a system are described by a density matrix, which is a positive semidefinite matrix with trace one. The goal is to approximate such a given density matrix by a so-called separable density matrix, and the distance between these matrices gives information about the degree of entanglement in the system. Separability here is expressed in terms of tensor products. We discuss this approximation problem for a composite system with two subsystems and show that it can be written as a convex optimization problem with special structure. We investigate related convex sets, and suggest an algorithm for this approximation problem which exploits the tensor product structure in certain subproblems. Finally some computational results and experiences are presented.     

Finiteness of the discrete spectrum of the three-particle schrödinger equation on a lattice

Consideration is given to the Hamiltonian of a system of three identical quantum particles on a lattice that interact via pairwise contact attractive potentials. Finiteness of the three-particle bound states is proved for the three-dimensional discrete Schrödinger operator on the condition that the operators describing the two-particle subsystems have no virtual levels. For high dimensions (v ≥ 5), the finiteness of three-particle bound states is also proved in the presence of virtual levels.     

Two-particle wave function as an integral operator and the random field approach to quantum correlations

We propose a new interpretation of the wave function Ψ (x, y) of a two-particle quantum system, interpreting it not as an element of the functional space L ₂ of square-integrable functions, i.e., as a vector, but as the kernel of an integral (Hilbert-Schmidt) operator. The first part of the paper is devoted to expressing quantum averages including the correlations in two-particle systems using the wave-function operator. This is a new mathematical representation in the framework of conventional quantum mechanics. But the new interpretation of the wave function not only generates a new mathematical formalism for quantum mechanics but also allows going beyond quantum mechanics, i.e., representing quantum correlations (including those in entangled systems) as correlations of (Gaussian) random fields.     

Classical dynamics,arrow of time,and genesis of the Heisenberg commutation relations

Based on the assumption that time evolves only in one direction and mechanical systems can be described by Lagrangeans, a dynamical C*-algebra is presented for non-relativistic particles at atomic scales. Without presupposing any quantization scheme, this algebra is inherently non-commutative and comprises a large set of dynamics. In contrast to other approaches, the generating elements of the algebra are not interpreted as observables, but as operations on the underlying system; they describe the impact of temporary perturbations caused by the surroundings. In accordance with the doctrine of Niels Bohr, the operations carry individual names of classical significance. Without stipulating from the outset their “quantization”, their concrete implementation in the quantum world emerges from the inherent structure of the algebra. In particular, the Heisenberg commutation relations for position and velocity measurements are derived from it. Interacting systems can be described within the algebraic setting by a rigorous version of the interaction picture. It is shown that Hilbert space representations of the algebra lead to the conventional formalism of quantum mechanics, where operations on states are described by time-ordered exponentials of interaction potentials. It is also discussed how the familiar statistical interpretation of quantum mechanics can be recovered from operations.     

Generating M-Indeterminate Probability Densities by Way of Quantum Mechanics

Probability densities that are not uniquely determined by their moments are said to be “moment-indeterminate,” or “M-indeterminate.” Determining whether or not a density is M-indeterminate, or how to generate an M-indeterminate density, is a challenging problem with a long history. Quantum mechanics is inherently probabilistic, yet the way in which probability densities are obtained is dramatically different in comparison with standard probability theory, involving complex wave functions and operators, among other aspects. Nevertheless, the end results are standard probabilistic quantities, such as expectation values, moments and probability density functions. We show that the quantum mechanics procedure to obtain densities leads to a simple method to generate an infinite number of M-indeterminate densities. Different self-adjoint operators can lead to new classes of M-indeterminate densities. Depending on the operator, the method can produce densities that are of the Stieltjes class or new formulations that are not of the Stieltjes class. As such, the method complements and extends existing approaches and opens up new avenues for further development. The method applies to continuous and discrete probability densities. A number of examples are given.

     

Center-of-mass tomography and probability representation of quantum states for tunneling

We develop a representation of quantum states in which the states are described by fair probability distribution functions instead of wave functions and density operators. We present a one-random-variable tomography map of density operators onto the probability distributions, the random variable being analogous to the center-of-mass coordinate considered in reference frames rotated and scaled in the phase space. We derive the evolution equation for the quantum state probability distribution and analyze the properties of the map. To illustrate the advantages of the new tomography representations, we describe a new method for simulating nonstationary quantum processes based on the tomography representation. The problem of the nonstationary tunneling of a wave packet of a composite particle, an exciton, is considered in detail.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 2, pp. 371–387, February, 2005.     

Dynamic effect algebras

We introduce the so-called tense operators in lattice effect algebras. Tense operators express the quantifiers “it is always going to be the case that” and “it has always been the case that” and hence enable us to express the dimension of time in the logic of quantum mechanics. We present an axiomatization of these tense operators and prove that every lattice effect algebra whose underlying lattice is complete can be equipped with tense operators. Such an effect algebra is called dynamic since it reflects changes of quantum events from past to future.     

The possibility of reconciling quantum mechanics with classical probability theory

We describe a scheme for constructing quantum mechanics in which a quantum system is considered as a collection of open classical subsystems. This allows using the formal classical logic and classical probability theory in quantum mechanics. Our approach nevertheless allows completely reproducing the standard mathematical formalism of quantum mechanics and identifying its applicability limits. We especially attend to the quantum state reduction problem. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 3, pp. 457–472, December, 2006.     

Clocks and fisher information

In a broad sense, any parametric family of quantum states can be viewed as a quantum clock. The time, which is the parameter, is encoded in the corresponding quantum states. The quality of such a clock depends on how precisely we can distinguish the states or, equivalently, estimate the parameter. In view of the quantum Cramér—Rao inequalities, the quality of quantum clocks can be characterized by the quantum Fisher information. We address the issue of quantum clock synchronization in terms of quantum Fisher information and demonstrate its fundamental difference from the classical paradigm. The key point is the superadditivity of Fisher information, which always holds in the classical case but can be violated in quantum mechanics. The violation can occur for both pure and mixed states. Nevertheless, we establish the superadditivity of quantum Fisher information for any classical-quantum state. We also demonstrate an alternative form of superadditivity and propose a weak form of superadditivity. The violation of superadditivity can be exploited to enhance quantum clock synchronization.     

Tomography of Spin States, the Entanglement Criterion, and Bell's Inequalities

We review the method of spin tomography of quantum states in which we use the standard probability distribution functions to describe spin projections on selected directions, which provides the same information about states as is obtained by the density matrix method. In this approach, we show that satisfaction or violation of Bell's inequalities can be understood as properties of tomographic functions for joint probability distributions for two spins. We compare results obtained using the methods of classical probability theory with those obtained in the framework of traditional quantum mechanics. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 172–185, January, 2006.     

A quantum analogue of Hunt's representation theorem for the generator of convolution semigroups on Lie groups

Summary In quantum mechanics certain operator-valued measures are introduced, called instruments, which are an analogue of the probability measures of classical probability theory. As in the classical case, it is interesting to study convolution semigroups of instruments on groups and the associated semigroups of probability operators. In this paper the case is considered of a finite-dimensional Hilbert space (n-level quantum system) and of instruments defined on a finite-dimensional Lie group. Then, the generator of a continuous semigroup of (quantum) probability operators is characterized. In this way a quantum analogue of Hunt's representation theorem for the generator of convolution semigroups on Lie groups is obtained.     

Separability for Positive Operators on Tensor Product of Hilbert Spaces

The separability and the entanglement(that is, inseparability) of the composite quantum states play important roles in quantum information theory. Mathematically, a quantum state is a trace-class positive operator with trace one acting on a complex separable Hilbert space. In this paper,in more general frame, the notion of separability for quantum states is generalized to bounded positive operators acting on tensor product of Hilbert spaces. However, not like the quantum state case, there are different kinds of separability for positive operators with different operator topologies. Four types of such separability are discussed; several criteria such as the finite rank entanglement witness criterion,the positive elementary operator criterion and PPT criterion to detect the separability of the positive operators are established; some methods to construct separable positive operators by operator matrices are provided. These may also make us to understand the separability and entanglement of quantum states better, and may be applied to find new separable quantum states.     

Using the wigner function for quantum transport in device simulation

The Wigner function was introduced as a generalization of the concept of distribution function for quantum statistics. The aim of this work is pushing further the formal analogy between quantum and classical approaches. The Wigner function is defined as an ensemble average, i.e., in terms of a mixture of pure states. From the point of view of basic physics, it would be very appealing to be able to define a Wigner function also for pure states and the associated expectation values for quantum observables, in strict analogy with the definition of mean value of a physical quantity in classical mechanics; then correct results for any quantum system should be recovered as appropriate superpositions of such “pure-state” quantities. We will show that this is actually possible, at the cost of dealing with generalized functions in place of proper functions.