Measures on Hilbert space and quantum states

A representation of quantum mechanics of one particle by means of integration in Hilbert space H is embedded into a multi-particle theory on boson Fock space F₊. We prove that the grand canonical statistical operator on F₊ can be represented by means of a gaussian measure on the one-particle Hilbert space H.     

On the quantum mechanical superposition of macroscopically distinguishable states

We consider the superposition of macroscopically distinguishable states for a measuring process whose time evolution is described by the Schrödinger equation. We ask whether it is possible to observe interference effects due to the above mentioned superposition and how to observe them, taking into consideration an experiment performed by other authors. We find a necessary condition in order to be able to observe these effects. We also point out some very serious difficulties in observing them and analyse the connection between some of these difficulties and the Wigner-Araki-Yanase theory. We try to explain why the whole problem seems to us to be far from having a solution and we suggest some paths we think worthy of further investigation.     

Measures on the Hilbert space of a quantum system

The paper is the first in a series of papers on the use of measures and generalized measures in quantum theory. In particular, a survey of the proofs of equivalence of various definitions of the density operator is presented. The exposition is of algebraic nature, and analytic assumptions are usually omitted.     

Embedding quantum logics in Hilbert space

We show that an orthomodular lattice is embeddable in a Hilbert space if and only if states of a certain kind exist. A physical motivation for the existence of such states is given and a connection is provided between the quantum logic, algebraic, and operational approaches to quantum mechanics.     

Teleporting a quantum state from a subset of the whole Hilbert space

We investigate the lower bound of the amount of entanglement for faithfully teleporting a quantum state belonging to a subset of the whole Hilbert space. Moreover, when the quantum state belongs to a set composed of two states, a probabilistic teleportation scheme is presented using a non-maximally entangled state as the quantum channel.     

Comments on “On the quantum mechanical superposition of macroscopically distinguishable states”

The substance of the authors' disagreement with the views of D. Gutkowski and M. V. Valdes Franco is presented.     

Relativistic resonances as non-orthogonal states in Hilbert space

We analyze the energy-momentum properties of relativistic short-lived particles with the result that they are characterized by two 4-vectors: in addition to the familiar energy-momentum vector (timelike) there is an energy-momentum `spread vector' (spacelike). The wave functions in space and time for unstable particles are constructed. For the relativistic properties of unstable states we refer to Wigner's method of Poincaré group representations that are induced by representations of the space-time translation and rotation groups. If stable particles, unstable particles and resonances are treated as elementary objects that are not fundamentally different one has to take into account that they will not generally be orthogonal to each other in their state space. The scalar product between a stable and an unstable state with otherwise identical properties is calculated in a particular Lorentz frame. The spin of an unstable particle is not infinitely sharp but has a `spin spread' giving rise to `spin neighbors'. This opens the possibility of a non-zero scalar product between states with unequal spin. - A first practical application of non-orthogonal states is seen in diffraction dissociation reactions whose large cross-sections are attributed to interference of states that are `partially identical'. Received: 28 June 2002 / Revised version: 13 November 2002 / Published online: 14 March 2003 RID="a" ID="a" e-mail: walter.blum@cern.ch RID="b" ID="b" e-mail: hns@mppmu.mpg.de     

Minimal supports in quantum logics and Hilbert space

It is shown that if a fully atomic, complete orthomodular lattice satisfies a minimal support condition (m.s.c.), then it satisfies Piron's axioms, and is thereby shown to be the projection lattice of a generalized Hilbert space. It is shown, conversely, that m.s.c. holds in Hilbert space subspace lattices. The physical justification for m.s.c. is provided in the context of a property lattice (A, ) for a realistic entity (A, ) in the sense of Foulis-Piron-Randall. This context provides a clear focus on key issues in the debate over the appropriateness of requiring quantum logics to be represented over Hilbert spaces.     

A note on the Hilbert space formalism of quantum mechanics

In [5] M.J. Maczyński showed that the Hilbert space formalism of quantum mechanics (see [3]) can be derived from a set of seven axioms involving only the probability function p(A, α, E) and the complex field postulate. The purpose of this note is to give the next axiom that is equivalent to the analogous real field postulate.     

Hilbert space representation of time evolution of pure states

Under a reasonable assumption it is shown that the set Tot_A of pure states of a physical system provided with the Scott information topology is homeomorphic to a subset of the Cantor discontinuum. Moreover, any smooth nonlinear time evolution with the set Tot_A as the phase space has a linear Hilber space representation.     

Admissible states in quantum phase space

We address the question of which phase space functionals might represent a quantum state. We derive necessary and sufficient conditions for both pure and mixed phase space quantum states. From the pure state quantum condition we obtain a formula for the momentum correlations of arbitrary order and derive explicit expressions for the wave functions in terms of time-dependent and independent Wigner functions. We show that the pure state quantum condition is preserved by the Moyal (but not by the classical Liouville) time evolution and is consistent with a generic stargenvalue equation. As a by-product Baker's converse construction is generalized both to an arbitrary stargenvalue equation, associated to a generic phase space symbol, as well as to the time-dependent case. These results are properly extended to the mixed state quantum condition, which is proved to imply the Heisenberg uncertainty relations. Globally, this formalism yields the complete characterization of the kinematical structure of Wigner quantum mechanics. The previous results are then succinctly generalized for various quasi-distributions. Finally, the formalism is illustrated through the simple examples of the harmonic oscillator and the free Gaussian wave packet. As a by-product, we obtain in the former example an integral representation of the Hermite polynomials.     

Discrete model of adsorption with a finite number of states

A discrete model of adsorption with allowance for recharge of the state and lateral interaction between molecules is constructed in the framework of the theory of probabilistic cellular automata. It is found that this model admits of the regular (ordered) behavior of the system accompanied by the global synchronization of the system’s parameters. The turbulent (chaotic)-ordered transition takes place through the occurrence of local ordered areas due to the appearance of local leading centers (pacemakers) and helical waves. The ordered behavior originates from intrinsic instability in the system. The ordering is related to the collective behavior of the subsystems constituting the entire system. The model can be extended to the case of chemical reactions between an adsorbate and the surface.     

Plane frame functions and pure states in Hilbert space

An elementary proof of Gleason's theorem for pure states in a Hilbert space is given. This theorem follows from a general result concerning plane frame functions in real inner product spaces.     

States in the Hilbert space formulation and in the phase space formulation of quantum mechanics

We consider the problem of testing whether a given matrix in the Hilbert space formulation of quantum mechanics or a function considered in the phase space formulation of quantum theory represents a quantum state. We propose several practical criteria for recognising states in these two versions of quantum physics. After minor modifications, they can be applied to check positivity of any operators acting in a Hilbert space or positivity of any functions from an algebra with a ∗$\ast$-product of Weyl type.     

Generation of a superposition of odd photon number states for quantum information networks

We report on the experimental observation of quantum-network-compatible light described by a nonpositive Wigner function. The state is generated by photon subtraction from a squeezed vacuum state produced by a continuous wave optical parametric amplifier. Ideally, the state is a coherent superposition of odd photon number states, closely resembling a superposition of weak coherent states |alpha > - |-alpha >. In the limit of low squeezing the state is basically a single photon state. Light is generated with about 10,000 and more events per second in a nearly perfect spatial mode with a Fourier-limited frequency bandwidth which matches well atomic quantum memory requirements. The generated state of light is an excellent input state for testing quantum memories, quantum repeaters, and linear optics quantum computers.     

Naturalness of the space of states in quantum mechanics

We show how certain constructions of quantum mechanics, like monopoles, instantons, and the Schrödinger-von Neumann equation, are related to geometric functors which are representable. We study the differential geometry of the projective bundle associated with an infinite-dimensional separable Hilbert space, and we construct a universal connection which, is described as a subspace of skwe-Hermitian operators. This connection is responsible for the Berry phase.     

希尔伯特空间有限区域成像积分算子的谱

本文讨论了希尔伯特空间中有限区域成像积分算子的本征值和本征函数,研究了非对称核的情形,并用微扰和有限秩方法计算了衍射受限和离焦情况下的本征值和本征函数。