10,780 research outputs found
Quantum Set Theory Extending the Standard Probabilistic Interpretation of Quantum Theory (Extended Abstract)
The notion of equality between two observables will play many important roles
in foundations of quantum theory. However, the standard probabilistic
interpretation based on the conventional Born formula does not give the
probability of equality relation for a pair of arbitrary observables, since the
Born formula gives the probability distribution only for a commuting family of
observables. In this paper, quantum set theory developed by Takeuti and the
present author is used to systematically extend the probabilistic
interpretation of quantum theory to define the probability of equality relation
for a pair of arbitrary observables. Applications of this new interpretation to
measurement theory are discussed briefly.Comment: In Proceedings QPL 2014, arXiv:1412.810
Quantum Set Theory Extending the Standard Probabilistic Interpretation of Quantum Theory
The notion of equality between two observables will play many important roles in foundations of quantum theory. However, the standard probabilistic interpretation based on the conventional Born formula does not give the probability of equality between two arbitrary observables, since the Born formula gives the probability distribution only for a commuting family of observables. In this paper, quantum set theory developed by Takeuti and the present author is used to systematically
extend the standard probabilistic interpretation of quantum theory to define the probability of equality between two arbitrary observables in an arbitrary state. We apply this new interpretation to quantum measurement theory, and establish a logical basis for the difference between simultaneous measurability and simultaneous determinateness
A derivation of quantum theory from physical requirements
Quantum theory is usually formulated in terms of abstract mathematical
postulates, involving Hilbert spaces, state vectors, and unitary operators. In
this work, we show that the full formalism of quantum theory can instead be
derived from five simple physical requirements, based on elementary assumptions
about preparation, transformations and measurements. This is more similar to
the usual formulation of special relativity, where two simple physical
requirements -- the principles of relativity and light speed invariance -- are
used to derive the mathematical structure of Minkowski space-time. Our
derivation provides insights into the physical origin of the structure of
quantum state spaces (including a group-theoretic explanation of the Bloch ball
and its three-dimensionality), and it suggests several natural possibilities to
construct consistent modifications of quantum theory.Comment: 16 pages, 2 figures. V3: added alternative formulation of Requirement
5, extended abstract, some minor modification
Quartic quantum theory: an extension of the standard quantum mechanics
We propose an extended quantum theory, in which the number K of parameters
necessary to characterize a quantum state behaves as fourth power of the number
N of distinguishable states. As the simplex of classical N-point probability
distributions can be embedded inside a higher dimensional convex body of mixed
quantum states, one can further increase the dimensionality constructing the
set of extended quantum states. The embedding proposed corresponds to an
assumption that the physical system described in N dimensional Hilbert space is
coupled with an auxiliary subsystem of the same dimensionality. The extended
theory works for simple quantum systems and is shown to be a non-trivial
generalisation of the standard quantum theory for which K=N^2. Imposing certain
restrictions on initial conditions and dynamics allowed in the quartic theory
one obtains quadratic theory as a special case. By imposing even stronger
constraints one arrives at the classical theory, for which K=N.Comment: 30 pages in latex with 6 figures included; ver.2: several
improvements, new references adde
Quantum Information Theory of Entanglement and Measurement
We present a quantum information theory that allows for a consistent
description of entanglement. It parallels classical (Shannon) information
theory but is based entirely on density matrices (rather than probability
distributions) for the description of quantum ensembles. We find that quantum
conditional entropies can be negative for entangled systems, which leads to a
violation of well-known bounds in Shannon information theory. Such a unified
information-theoretic description of classical correlation and quantum
entanglement clarifies the link between them: the latter can be viewed as
``super-correlation'' which can induce classical correlation when considering a
tripartite or larger system. Furthermore, negative entropy and the associated
clarification of entanglement paves the way to a natural information-theoretic
description of the measurement process. This model, while unitary and causal,
implies the well-known probabilistic results of conventional quantum mechanics.
It also results in a simple interpretation of the Kholevo theorem limiting the
accessible information in a quantum measurement.Comment: 26 pages with 6 figures. Expanded version of PhysComp'96 contributio
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