Conditional expectations associated with quantum states

An extension of the conditional expectations (those under a given subalgebra of events and not the simple ones under a single event) from the classical to the quantum case is presented. In the classical case, the conditional expectations always exist; in the quantum case, however, they exist only if a certain weak compatibility criterion is satisfied. This compatibility criterion was introduced among others in a recent paper by the author. Then, state-independent conditional expectations and quantum Markov processes are studied. A classical Markov process is a probability measure, together with a system of random variables, satisfying the Markov property and can equivalently be described by a system of Markovian kernels (often forming a semigroup). This equivalence is partly extended to quantum probabilities. It is shown that a dynamical (semi)group can be derived from a given system of quantum observables satisfying the Markov property, and the group generators are studied. The results are presented in the framework of Jordan operator algebras, and a very general type of observables (including the usual real-valued observables or self-adjoint operators) is considered.Comment: 10 pages, the original publication is available at http://www.aip.or

A hierarchy of compatibility and comeasurability levels in quantum logics with unique conditional probabilities

In the quantum mechanical Hilbert space formalism, the probabilistic interpretation is a later ad-hoc add-on, more or less enforced by the experimental evidence, but not motivated by the mathematical model itself. A model involving a clear probabilistic interpretation from the very beginning is provided by the quantum logics with unique conditional probabilities. It includes the projection lattices in von Neumann algebras and here probability conditionalization becomes identical with the state transition of the Lueders - von Neumann measurement process. This motivates the definition of a hierarchy of five compatibility and comeasurability levels in the abstract setting of the quantum logics with unique conditional probabilities. Their meanings are: the absence of quantum interference or influence, the existence of a joint distribution, simultaneous measurability, and the independence of the final state after two successive measurements from the sequential order of these two measurements. A further level means that two elements of the quantum logic (events) belong to the same Boolean subalgebra. In the general case, the five compatibility and comeasurability levels appear to differ, but they all coincide in the common Hilbert space formalism of quantum mechanics, in von Neumann algebras, and in some other cases.Comment: 12 page

Simplified methods of assessing the impact of grid frequency dynamics upon generating plants

The frequency of the national electricity grid is affected by fluctuations in supply and demand, and so continually "judders" in an essentially unpredictable fashion around 50 Hz. At present such perturbations do not seemingly affect Nuclear Electric as most of their plant is run at more or less constant load, but they would like to be able to offer the national grid a mode of operation in which they "followed" the grid frequency: i.e., as the frequency rose above or fell below 50 Hz, the plant's output would be adjusted so as to tend to restore the frequency to 50 Hz. The aim is to maintain grid frequency within 0.2 Hz of its notional value. Such a mode of operation, however, would cause a certain amount of damage to plant components owing to the consequent continual changes in temperature and pressure within them. Nuclear Electric currently have complex computational models of how plants will behave under these conditions, which allows them to compute plant data (e.g., reactor temperatures) from given grid frequency data. One approach to damage assessment would require several years'-worth of real grid data to be fed into this model and the corresponding damage computed (via "cycle distributions" created by their damage experts). The results of this analysis would demonstrate one of three possibilities: the damage may be acceptable under all reasonable operating conditions; or it may be acceptable except in the case of an exceptional abrupt change in grid frequency (caused by power transmission line failure, or another power station suddenly going off-line, for instance), in which case some kind of backup supply (e.g., gas boilers) would be required; or it may simply be unacceptable. However, their current model runs in approximately real time, making it inappropriate for such a large amount of data: our problem was to suggest alternative approaches. Specifically, we were asked the following questions: - Can component damage be reliably estimated directly from cycle distributions of grid frequency? i.e., are there maps from frequency cycle distributions to plant parameter cycle distributions? - Can a simple model of plant dynamics be used to assess the potential for such maps? - What methods can be used to select representative samples of grid frequency behaviour? - What weightings should be applied to the selections? - Is it possible to construct a "cycle transform" (Fourier transform) which will capture the essential features of grid frequency and which can then be inverted to generate simulated frequency transients? We did not consider this last question, other than to say "probably not". We were supplied with data of the actual grid frequency measurements for the evening of 29/7/95, and the corresponding plant responses (obtained using Nuclear Electric's current computational model). A simplified nonlinear mathematical model of the plant was also provided. Two main approaches were considered: statistical prediction and analytical modelling via a reduction of the simplified plant model

Ensemble averaged entanglement of two-particle states in Fock space

Recent results, extending the Schmidt decomposition theorem to wavefunctions of identical particles, are reviewed. They are used to give a definition of reduced density operators in the case of two identical particles. Next, a method is discussed to calculate time averaged entanglement. It is applied to a pair of identical electrons in an otherwise empty band of the Hubbard model, and to a pair of bosons in the the Bose-Hubbard model with infinite range hopping. The effect of degeneracy of the spectrum of the Hamiltonian on the average entanglement is emphasised.Comment: 19 pages Latex, changed title, references added in the conclusion

A Representation of Quantum Measurement in Nonassociative Algebras

Starting from an abstract setting for the Lueders - von Neumann quantum measurement process and its interpretation as a probability conditionalization rule in a non-Boolean event structure, the author derived a certain generalization of operator algebras in a preceding paper. This is an order-unit space with some specific properties. It becomes a Jordan operator algebra under a certain set of additional conditions, but does not own a multiplication operation in the most general case. A major objective of the present paper is the search for such examples of the structure mentioned above that do not stem from Jordan operator algebras; first natural candidates are matrix algebras over the octonions and other nonassociative rings. Therefore, the case when a nonassociative commutative multiplication exists is studied without assuming that it satisfies the Jordan condition. The characteristics of the resulting algebra are analyzed. This includes the uniqueness of the spectral resolution as well as a criterion for its existence, subalgebras that are Jordan algebras, associative subalgebras, and more different levels of compatibility than occurring in standard quantum mechanics. However, the paper cannot provide the desired example, but contribute to the search by the identification of some typical differences between the potential examples and the Jordan operator algebras and by negative results concerning some first natural candidates. The possibility that no such example exists cannot be ruled out. However, this would result in an unexpected new characterization of Jordan operator algebras, which would have a significant impact on quantum axiomatics since some customary axioms (e.g., powerassociativity or the sum postulate for observables) might turn out to be redundant then.Comment: 14 pages, the original publication is available at http://www.springerlink.co

Different Types of Conditional Expectation and the Lueders - 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 Lueders - 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 Lueders - 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.Comment: 10 pages, the original publication is available at http://www.springerlink.co

Boundedness of completely additive measures with application to 2-local triple derivations

We prove a Jordan version of Dorofeev's boundedness theorem for completely additive measues and use it to show that every (not necessarily linear nor continuous) 2-local triple derivation on a continuous JBW*-triple is a triple derivation.Comment: 30 page

Three-slit experiments and quantum nonlocality

An interesting link between two very different physical aspects of quantum mechanics is revealed; these are the absence of third-order interference and Tsirelson's bound for the nonlocal correlations. Considering multiple-slit experiments - not only the traditional configuration with two slits, but also configurations with three and more slits - Sorkin detected that third-order (and higher-order) interference is not possible in quantum mechanics. The EPR experiments show that quantum mechanics involves nonlocal correlations which are demonstrated in a violation of the Bell or CHSH inequality, but are still limited by a bound discovered by Tsirelson. It now turns out that Tsirelson's bound holds in a broad class of probabilistic theories provided that they rule out third-order interference. A major characteristic of this class is the existence of a reasonable calculus of conditional probability or, phrased more physically, of a reasonable model for the quantum measurement process.Comment: 9 pages, no figur

On non-completely positive quantum dynamical maps on spin chains

The new arguments based on Majorana fermions indicating that non-completely positive maps can describe open quantum evolution are presented.Comment: published; small change

Non-Boolean probabilities and quantum measurement

A non-Boolean extension of the classical probability model is proposed. The non-Boolean probabilities reproduce typical quantum phenomena. The proposed model is more general and more abstract, but easier to interpret, than the quantum mechanical Hilbert space formalism and exhibits a particular phenomenon (state-independent conditional probabilities) which may provide new opportunities for an understanding of the quantum measurement process. Examples of the proposed model are provided, using Jordan operator algebras.Comment: 12 pages, the original publication is available at http://www.iop.or