104 research outputs found
A quantum logical and geometrical approach to the study of improper mixtures
We study improper mixtures from a quantum logical and geometrical point of
view. Taking into account the fact that improper mixtures do not admit an
ignorance interpretation and must be considered as states in their own right,
we do not follow the standard approach which considers improper mixtures as
measures over the algebra of projections. Instead of it, we use the convex set
of states in order to construct a new lattice whose atoms are all physical
states: pure states and improper mixtures. This is done in order to overcome
one of the problems which appear in the standard quantum logical formalism,
namely, that for a subsystem of a larger system in an entangled state, the
conjunction of all actual properties of the subsystem does not yield its actual
state. In fact, its state is an improper mixture and cannot be represented in
the von Neumann lattice as a minimal property which determines all other
properties as is the case for pure states or classical systems. The new lattice
also contains all propositions of the von Neumann lattice. We argue that this
extension expresses in an algebraic form the fact that -alike the classical
case- quantum interactions produce non trivial correlations between the
systems. Finally, we study the maps which can be defined between the extended
lattice of a compound system and the lattices of its subsystems.Comment: submitted to the Journal of Mathematical Physic
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 -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
Energy-Sensitive and "Classical-like" Distances Between Quantum States
We introduce the concept of the ``polarized'' distance, which distinguishes
the orthogonal states with different energies. We also give new inequalities
for the known Hilbert-Schmidt distance between neighbouring states and express
this distance in terms of the quasiprobability distributions and the normally
ordered moments. Besides, we discuss the distance problem in the framework of
the recently proposed ``classical-like'' formulation of quantum mechanics,
based on the symplectic tomography scheme. The examples of the Fock, coherent,
``Schroedinger cats,'' squeezed, phase, and thermal states are considered.Comment: 23 pages, LaTex, 2 eps figures, to appear in Physica Script
Relational Quantum Mechanics
I suggest that the common unease with taking quantum mechanics as a
fundamental description of nature (the "measurement problem") could derive from
the use of an incorrect notion, as the unease with the Lorentz transformations
before Einstein derived from the notion of observer-independent time. I suggest
that this incorrect notion is the notion of observer-independent state of a
system (or observer-independent values of physical quantities). I reformulate
the problem of the "interpretation of quantum mechanics" as the problem of
deriving the formalism from a few simple physical postulates. I consider a
reformulation of quantum mechanics in terms of information theory. All systems
are assumed to be equivalent, there is no observer-observed distinction, and
the theory describes only the information that systems have about each other;
nevertheless, the theory is complete.Comment: Substantially revised version. LaTeX fil
Classical interventions in quantum systems. I. The measuring process
The measuring process is an external intervention in the dynamics of a
quantum system. It involves a unitary interaction of that system with a
measuring apparatus, a further interaction of both with an unknown environment
causing decoherence, and then the deletion of a subsystem. This description of
the measuring process is a substantial generalization of current models in
quantum measurement theory. In particular, no ancilla is needed. The final
result is represented by a completely positive map of the quantum state
(possibly with a change of the dimensions of ). A continuous limit of the
above process leads to Lindblad's equation for the quantum dynamical semigroup.Comment: Final version, 14 pages LaTe
Contextual-value approach to the generalized measurement of observables
We present a detailed motivation for and definition of the contextual values
of an observable, which were introduced by Dressel et al. [Phys. Rev. Lett. 104
240401 (2010)]. The theory of contextual values extends the well-established
theory of generalized state measurements by bridging the gap between partial
state collapse and the observables that represent physically relevant
information about the system. To emphasize the general utility of the concept,
we first construct the full theory of contextual values within an operational
formulation of classical probability theory, paying special attention to
observable construction, detector coupling, generalized measurement, and
measurement disturbance. We then extend the results to quantum probability
theory built as a superstructure on the classical theory, pointing out both the
classical correspondences to and the full quantum generalizations of both
L\"uder's rule and the Aharonov-Bergmann-Lebowitz rule in the process. We find
in both cases that the contextual values of a system observable form a
generalized spectrum that is associated with the independent outcomes of a
partially correlated and generally ambiguous detector; the eigenvalues are a
special case when the detector is perfectly correlated and unambiguous. To
illustrate the approach, we apply the technique to both a classical example of
marble color detection and a quantum example of polarization detection. For the
quantum example we detail two devices: Fresnel reflection from a glass
coverslip, and continuous beam displacement from a calcite crystal. We also
analyze the three-box paradox to demonstrate that no negative probabilities are
necessary in its analysis. Finally, we provide a derivation of the quantum weak
value as a limit point of a pre- and postselected conditioned average and
provide sufficient conditions for the derivation to hold.Comment: 36 pages, 5 figures, published versio
Generally covariant dynamical reduction models and the Hadamard condition
We recall and review earlier work on dynamical reduction models, both non-relativistic and relativistic, and discuss how they may relate to suggestions which have been made (including the matter-gravity entanglement hypothesis of one of us) for how quantum gravity could be connected to the resolution of the quantum-mechanical measurement problem. We then provide general guidelines for generalizing dynamical reduction models to curved spacetimes and propose a class of generally covariant relativistic versions of the GRW model. We anticipate that the collapse operators of our class of models may play a r\^ole in a yet-to-be-formulated theory of semiclassical gravity with collapses. We show explicitly that the collapse operators map a dense domain of states that are initially Hadamard to final Hadamard states -- a property that we expect will be needed for the construction of such a semiclassical theory. Finally, we provide a simple example in which we explicitly compute the violations in energy-momentum due to the state reduction process and conclude that this violation is of the order of a parameter of the model -- supposed to be small. We briefly discuss how this work may, upon further development of a suitable semiclassical gravity theory with collapses, enable further progress to be made on earlier work one of us and collaborators on the explanation of structure-formation in a homogeneous and isotropic quantum universe and on a possible resolution of the black hole information loss puzzle
Is symmetry identity?
Wigner found unreasonable the "effectiveness of mathematics in the natural
sciences". But if the mathematics we use to describe nature is simply a coded
expression of our experience then its effectiveness is quite reasonable. Its
effectiveness is built into its design. We consider group theory, the logic of
symmetry. We examine the premise that symmetry is identity; that group theory
encodes our experience of identification. To decide whether group theory
describes the world in such an elemental way we catalogue the detailed
correspondence between elements of the physical world and elements of the
formalism. Providing an unequivocal match between concept and mathematical
statement completes the case. It makes effectiveness appear reasonable. The
case that symmetry is identity is a strong one but it is not complete. The
further validation required suggests that unexpected entities might be
describable by the irreducible representations of group theory
Ensembles and experiments in classical and quantum physics
A philosophically consistent axiomatic approach to classical and quantum
mechanics is given. The approach realizes a strong formal implementation of
Bohr's correspondence principle. In all instances, classical and quantum
concepts are fully parallel: the same general theory has a classical
realization and a quantum realization.
Extending the `probability via expectation' approach of Whittle to
noncommuting quantities, this paper defines quantities, ensembles, and
experiments as mathematical concepts and shows how to model complementarity,
uncertainty, probability, nonlocality and dynamics in these terms. The approach
carries no connotation of unlimited repeatability; hence it can be applied to
unique systems such as the universe.
Consistent experiments provide an elegant solution to the reality problem,
confirming the insistence of the orthodox Copenhagen interpretation on that
there is nothing but ensembles, while avoiding its elusive reality picture. The
weak law of large numbers explains the emergence of classical properties for
macroscopic systems.Comment: 56 page
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