Controlled Remote Preparation of a Two-Qubit State via an Asymmetric Quantum Channel

I present a new scheme for probabilistic remote preparation of a general two-qubit state from a sender to either of two receivers. The quantum channel is composed of a partial entangled tripartite Greenberger-Horne-Zeilinger (GHZ) state and a W-type state. I try to realize the remote two-qubit preparation by using the usual projective measurement and the method of positive operator-valued measure, respectively. The corresponding success probabilities of the scheme with different methods as well as the total classical communicationcost required in this scheme are also calculated.     

The Quantum Harmonic Oscillator in the ESR Model

The ESR model proposes a new theoretical perspective which incorporates the mathematical formalism of standard (Hilbert space) quantum mechanics (QM) in a noncontextual framework, reinterpreting quantum probabilities as conditional on detection instead of absolute. We have provided in some previous papers mathematical representations of the physical entities introduced by the ESR model, namely observables, properties, pure states, proper and improper mixtures, together with rules for calculating conditional and overall probabilities, and for describing transformations of states induced by measurements. We study in this paper the relevant physical case of the quantum harmonic oscillator in our mathematical formalism. We reinterpret the standard quantum rules for probabilities, provide new expressions for absolute probabilities, and show how the standard state transformations must be modified according to the ESR model.     

Position operators as integrals with respect to Euclidean systems of covariance

Position operators (p.o.) for relativistic elementary quantum systems are constructed as operator-valued integrals with respect to Euclidean systems of covariance (ESC), i.e., positive operator-valued (POV) measures being covariant under the Euclidean group, and are expressed in terms of the generators of the Poincaré transformations. These p.o. are partly well-known in the literature where they are found by other methods.     

On the relation between quantum mechanical probabilities and event frequencies

The probability `measure' for measurements at two consecutive moments of time is non-additive. These probabilities, on the other hand, may be determined by the limit of relative frequency of measured events, which are by nature additive. We demonstrate that there are only two ways to resolve this problem. The first solution places emphasis on the precise use of the concept of conditional probability for successive measurements. The physically correct conditional probabilities define additive probabilities for two-time measurements. These probabilities depend explicitly on the resolution of the physical device and do not, therefore, correspond to a function of the associated projection operators. It follows that quantum theory distinguishes between physical events and propositions about events, the latter are not represented by projection operators and that the outcomes of two-time experiments cannot be described by quantum logic. The alternative explanation is rather radical: it is conceivable that the relative frequencies for two-time measurements do not converge, unless a particular consistency condition is satisfied. If this is true, a strong revision of the quantum mechanical formalism may prove necessary. We stress that it is possible to perform experiments that will distinguish the two alternatives.     

Controlled Remote Preparation of a Two-Qubit State via Positive Operator-Valued Measure and Two Three-Qubit Entanglements

Utilizing a proper positive operator-valued measure (POVM) instead of usual projective measurement, we present a new scheme for remote preparation of a general two-qubit state in either distant ministrant’s place. In the scheme, we exploit a GHZ-type state and a W-type state as the shared quantum channel, and the remote two-qubit preparation can be realized in a probabilistic manner via two ministrants’ collaboration. Furthermore, we also explore the scheme’s applications to some special ensembles of states in detail. The success probabilities in different cases are calculated, respectively, which show that the remote preparation can be achieved with higher probability provided that the prepared state belongs to the special ensembles.     

Condition for any realistic theory of quantum systems

In quantum physics, the density operator completely describes the state. Instead, in classical physics the mean value of physical quantities is evaluated by means of a probability distribution. We study the possibility to describe pure quantum states and events with classical probability distributions and conditional probabilities and prove that the distributions have to be nonlinear functions of the density operator. Some examples are considered. Finally, we deal with the exponential complexity problem of quantum physics and introduce the concept of classical dimension for a quantum system.     

Generalized Householder Transformations

The Householder transformation, allowing a rewrite of probabilities into expectations of dichotomic observables, is generalized in terms of its spectral decomposition. The dichotomy is modulated by allowing more than one negative eigenvalue or by abandoning binaries altogether, yielding generalized operator-valued arguments for contextuality. We also discuss a form of contextuality by the variation of the functional relations of the operators, in particular by additivity.     

Conditional expectations,conditional distributions,anda posteriori ensembles in generalized probability theory

A general probabilistic framework containing the essential mathematical structure of any statistical physical theory is reviewed and enlarged to enable the generalization of some concepts of classical probability theory. In particular, generalized conditional probabilities of effects and conditional distributions of observables are introduced and their interpretation is discussed in terms of successive measurements. The existence of generalized conditional distributions is proved, and the relation to M. Ozawa'sa posteriori states is investigated. Examples concerning classical as well as quantum probability are discussed.     

Q operators for the simple quantum relativistic Toda Chain

We investigate the simple quantum relativistic Toda chain. The ultralocal simple Weyl algebra pair is associated with each site of the chain. Weyl’s q is considered to be inside a unit circle. Both independent Baxter operators Q are constructed explicitly as series in local Weyl generators. The operator-valued Wronskian of Q-s is also calculated.     

Cramér-Rao inequalities for operator-valued measures in quantum mechanics

The theory of estimation of parameters of quantum-mechanical density operators is expressed in terms of the measurement of operator-valued measures. Lower bounds on mean-square errors of parameter estimates are set by two quantum-mechanical forms of the Cramér-Rao inequality of classical statistics, derived here in terms of such measures. The results are exemplified by the simultaneous estimation of the real and imaginary parts of the complex amplitude of a coherent oscillation in the presence of thermal noise.This research was supported by grant NSF GK-33811 from the National Science Foundation.     

Microlocal Analysis and¶Interacting Quantum Field Theories:¶Renormalization on Physical Backgrounds

We present a perturbative construction of interacting quantum field theories on smooth globally hyperbolic (curved) space-times. We develop a purely local version of the Stückelberg–Bogoliubov–Epstein–Glaser method of renormalization by using techniques from microlocal analysis. Relying on recent results of Radzikowski, K?hler and the authors about a formulation of a local spectrum condition in terms of wave front sets of correlation functions of quantum fields on curved space-times, we construct time-ordered operator-valued products of Wick polynomials of free fields. They serve as building blocks for a local (perturbative) definition of interacting fields. Renormalization in this framework amounts to extensions of expectation values of time-ordered products to all points of space-time. The extensions are classified according to a microlocal generalization of Steinmann scaling degree corresponding to the degree of divergence in other renormalization schemes. As a result, we prove that the usual perturbative classification of interacting quantum field theories holds also on curved space-times. Finite renormalizations are deferred to a subsequent paper. As byproducts, we describe a perturbative construction of local algebras of observables, present a new definition of Wick polynomials as operator-valued distributions on a natural domain, and we find a general method for the extension of distributions which were defined on the complement of some surface. Received: 31 March 1999 / Accepted: 10 June 1999     

A Generalized Quantum Theory

In quantum mechanics, the selfadjoint Hilbert space operators play a triple role as observables, generators of the dynamical groups and statistical operators defining the mixed states. One might expect that this is typical of Hilbert space quantum mechanics, but it is not. The same triple role occurs for the elements of a certain ordered Banach space in a much more general theory based upon quantum logics and a conditional probability calculus (which is a quantum logical model of the Lüders-von Neumann measurement process). It is shown how positive groups, automorphism groups, Lie algebras and statistical operators emerge from one major postulate—the non-existence of third-order interference [third-order interference and its impossibility in quantum mechanics were discovered by Sorkin (Mod Phys Lett A 9:3119–3127, 1994)]. This again underlines the power of the combination of the conditional probability calculus with the postulate that there is no third-order interference. In two earlier papers, its impact on contextuality and nonlocality had already been revealed.     

A discussion on the origin of quantum probabilities

We study the origin of quantum probabilities as arising from non-Boolean propositional-operational structures. We apply the method developed by Cox to non distributive lattices and develop an alternative formulation of non-Kolmogorovian probability measures for quantum mechanics. By generalizing the method presented in previous works, we outline a general framework for the deduction of probabilities in general propositional structures represented by lattices (including the non-distributive case).     

Incompatible stochastic processes and complex probabilities

The definition of conditional probabilities is based upon the existence of a joint probability. However, a reconstruction of the joint probability from given conditional probabilities imposes certain constraints upon the latter, so that if several conditional probabilities are chosen arbitrarily, the corresponding joint probability may not exist. Such an incompleteness in conditional probabilities can be eliminated by introducing complex probabilities. The physical meaning of the new mathematical formalism, as well as its relation to quantum probabilities, is discussed.     

Different Types of Conditional Expectation and the Lüders—von Neumann Quantum Measurement

In operator algebra theory, a conditional expectation is usually assumed to be a projection map onto a sub-algebra. In the paper, a further type of conditional expectation and an extension of the Lüders—von Neumann measurement to observables with continuous spectra are considered; both are defined for a single operator and become a projection map only if they exist for all operators. Criteria for the existence of the different types of conditional expectation and of the extension of the Lüders—von Neumann measurement are presented, and the question whether they coincide is studied. All this is done in the general framework of Jordan operator algebras. The examples considered include the type I and type II operator algebras, the standard Hilbert space model of quantum mechanics, and a no-go result concerning the conditional expectation of observables that satisfy the canonical commutator relation.     

Noether Charges for self‐interacting quantum field theories in curved spacetimes with a Killing‐vector

We consider a self‐interacting, perturbative Klein‐Gordon quantum field in a curved spacetime admitting a Killing vector field. We show that the action of this spacetime symmetry on interacting field operators can be implemented by a Noether charge which arises, in a certain sense, as a surface integral over the time‐component of some interacting Noether current‐density associated with the Killing field. The proof of this involves the demonstration of a corresponding set of Ward identities. Our work is based on the perturbative construction by Brunetti and Fredenhagen (Commun. Math. Phys. 208 (2000) 623—661) of self‐interacting quantum field theories in general globally hyperbolic spacetimes.     

Multibit gates for quantum computing

We present a general technique to implement products of many qubit operators communicating via a joint harmonic oscillator degree of freedom in a quantum computer. By conditional displacements and rotations we can implement Hamiltonians which are trigonometric functions of qubit operators. With such operators we can effectively implement higher order gates such as Toffoli gates and C(n)-NOT gates, and we show that the entire Grover search algorithm can be implemented in a direct way.     

Continual measurements in the spacetime formation of nonrelativistic quantum mechanics

We study the nature and effects of some continual measurements in nonrelativistic quantum mechanics, a concept introduced by R. P. Feynman in his path integral formulation of quantum mechanics. We prove the existence in various senses of mathematically rigorous objects formally equivalent to the path integrals associated with such measurements, by means of both a limiting definition and more general techniques; and we derive some quantitative results concerning the effects on the wave-function and the numerical value of certain associated probabilities.