POVMs: a small but important step beyond standard quantum mechanics

It is the purpose of the present contribution to demonstrate that the generalization of the concept of a quantum mechanical observable from the Hermitian operator of standard quantum mechanics to a positive operator-valued measure is not a peripheral issue, allegedly to be understood in terms of a trivial nonideality of practical measurement procedures, but that this generalization touches the very core of quantum mechanics, viz. complementarity and violation of the Bell inequalities.Comment: Contribution to Proceedings of the Workshop `Beyond the quantum', Leiden, May/June 200

The Haroche-Ramsey experiment as a generalized measurement

A number of atomic beam experiments, related to the Ramsey experiment and a recent experiment by Brune et al., are studied with respect to the question of complementarity. Three different procedures for obtaining information on the state of the incoming atom are compared. Positive operator-valued measures are explicitly calculated. It is demonstrated that, in principle, it is possible to choose the experimental arrangement so as to admit an interpretation as a joint non-ideal measurement yielding interference and ``which-way'' information. Comparison of the different measurements gives insight into the question of which information is provided by a (generalized) quantum mechanical measurement. For this purpose the subspaces of Hilbert-Schmidt space, spanned by the operators of the POVM, are determined for different measurement arrangements and different values of the parameters.Comment: REVTeX, 22 pages, 5 figure

Quantum state tomography using a single apparatus

The density matrix of a two-level system (spin, atom) is usually determined by measuring the three non-commuting components of the Pauli vector. This density matrix can also be obtained via the measurement data of two commuting variables, using a single apparatus. This is done by coupling the two-level system to a mode of radiation field, where the atom-field interaction is described with the Jaynes--Cummings model. The mode starts its evolution from a known coherent state. The unknown initial state of the atom is found by measuring two commuting observables: the population difference of the atom and the photon number of the field. We discuss the advantages of this setup and its possible applications.Comment: 7 pages, 8 figure, Phys. Rev.

Channel kets, entangled states, and the location of quantum information

The well-known duality relating entangled states and noisy quantum channels is expressed in terms of a channel ket, a pure state on a suitable tripartite system, which functions as a pre-probability allowing the calculation of statistical correlations between, for example, the entrance and exit of a channel, once a framework has been chosen so as to allow a consistent set of probabilities. In each framework the standard notions of ordinary (classical) information theory apply, and it makes sense to ask whether information of a particular sort about one system is or is not present in another system. Quantum effects arise when a single pre-probability is used to compute statistical correlations in different incompatible frameworks, and various constraints on the presence and absence of different kinds of information are expressed in a set of all-or-nothing theorems which generalize or give a precise meaning to the concept of ``no-cloning.'' These theorems are used to discuss: the location of information in quantum channels modeled using a mixed-state environment; the $CQ$ (classical-quantum) channels introduced by Holevo; and the location of information in the physical carriers of a quantum code. It is proposed that both channel and entanglement problems be classified in terms of pure states (functioning as pre-probabilities) on systems of $p\geq 2$ parts, with mixed bipartite entanglement and simple noisy channels belonging to the category $p=3$, a five-qubit code to the category $p=6$, etc.; then by the dimensions of the Hilbert spaces of the component parts, along with other criteria yet to be determined.Comment: Latex 32 pages, 4 figures in text using PSTricks. Version 3: Minor typographical errors correcte

Informationally complete joint measurements on finite quantum systems

We show that there are informationally complete joint measurements of two conjugated observables on a finite quantum system, meaning that they enable to identify all quantum states from their measurement outcome statistics. We further demonstrate that it is possible to implement a joint observable as a sequential measurement. If we require minimal noise in the joint measurement, then the joint observable is unique. If the dimension d is odd, then this observable is informationally complete. But if d is even, then the joint observable is not informationally complete and one has to allow more noise in order to obtain informational completeness

Brownian Entanglement

We show that for two classical brownian particles there exists an analog of continuous-variable quantum entanglement: The common probability distribution of the two coordinates and the corresponding coarse-grained velocities cannot be prepared via mixing of any factorized distributions referring to the two particles in separate. This is possible for particles which interacted in the past, but do not interact in the present. Three factors are crucial for the effect: 1) separation of time-scales of coordinate and momentum which motivates the definition of coarse-grained velocities; 2) the resulting uncertainty relations between the coordinate of the brownian particle and the change of its coarse-grained velocity; 3) the fact that the coarse-grained velocity, though pertaining to a single brownian particle, is defined on a common context of two particles. The brownian entanglement is a consequence of a coarse-grained description and disappears for a finer resolution of the brownian motion. We discuss possibilities of its experimental realizations in examples of macroscopic brownian motion.Comment: 18 pages, no figure

Linear representations of probabilistic transformations induced by context transitions

By using straightforward frequency arguments we classify transformations of probabilities which can be generated by transition from one preparation procedure (context) to another. There are three classes of transformations corresponding to statistical deviations of different magnitudes: (a) trigonometric; (b) hyperbolic; (c) hyper-trigonometric. It is shown that not only quantum preparation procedures can have trigonometric probabilistic behaviour. We propose generalizations of ${\bf C}$-linear space probabilistic calculus to describe non quantum (trigonometric and hyperbolic) probabilistic transformations. We also analyse superposition principle in this framework.Comment: Added a physical discussion and new reference

On the Consequences of Retaining the General Validity of Locality in Physical Theory

The empirical validity of the locality (LOC) principle of relativity is used to argue in favour of a local hidden variable theory (HVT) for individual quantum processes. It is shown that such a HVT may reproduce the statistical predictions of quantum mechanics (QM), provided the reproducibility of initial hidden variable states is limited. This means that in a HVT limits should be set to the validity of the notion of counterfactual definiteness (CFD). This is supported by the empirical evidence that past, present, and future are basically distinct. Our argumentation is contrasted with a recent one by Stapp resulting in the opposite conclusion, i.e. nonlocality or the existence of faster-than-light influences. We argue that Stapp's argumentation still depends in an implicit, but crucial, way on both the notions of hidden variables and of CFD. In addition, some implications of our results for the debate between Bohr and Einstein, Podolsky and Rosen are discussed.Comment: revtex, 11 page