3,466 research outputs found
Multivariate Krawtchouk polynomials and composition birth and death processes
This paper defines the multivariate Krawtchouk polynomials, orthogonal on the
multinomial distribution, and summarizes their properties as a review. The
multivariate Krawtchouk polynomials are symmetric functions of orthogonal sets
of functions defined on each of N multinomial trials. The dual multivariate
Krawtchouk polynomials, which also have a polynomial structure, are seen to
occur naturally as spectral orthogonal polynomials in a Karlin and McGregor
spectral representation of transition functions in a composition birth and
death process. In this Markov composition process in continuous time there are
N independent and identically distributed birth and death processes each with
support 0,1, .... The state space in the composition process is the number of
processes in the different states 0,1,... Dealing with the spectral
representation requires new extensions of the multivariate Krawtchouk
polynomials to orthogonal polynomials on a multinomial distribution with a
countable infinity of states
Quantum Measurements Are Noncontextual
Quantum measurements are noncontextual, with outcomes independent of which
other commuting observables are measured at the same time, when consistently
analyzed using principles of Hilbert space quantum mechanics rather than
classical hidden variables.Comment: Minor update of previous version, with comments on the BKS theorem
added towards the en
Quantum Information: What Is It All About?
This paper answers Bell's question: What does quantum information refer to?
It is about quantum properties represented by subspaces of the quantum Hilbert
space, or their projectors, to which standard (Kolmogorov) probabilities can be
assigned by using a projective decomposition of the identity (PDI or framework)
as a quantum sample space. The single framework rule of consistent histories
prevents paradoxes or contradictions. When only one framework is employed,
classical (Shannon) information theory can be imported unchanged into the
quantum domain. A particular case is the macroscopic world of classical physics
whose quantum description needs only a single quasiclassical framework.
Nontrivial issues unique to quantum information, those with no classical
analog, arise when aspects of two or more incompatible frameworks are compared.Comment: 14 pages. v2:Minor changes in title, abstract, Sec. 7. References
added and correcte
The New Quantum Logic
It is shown how all the major conceptual difficulties of standard (textbook)
quantum mechanics, including the two measurement problems and the (supposed)
nonlocality that conflicts with special relativity, are resolved in the
consistent or decoherent histories interpretation of quantum mechanics by using
a modified form of quantum logic to discuss quantum properties (subspaces of
the quantum Hilbert space), and treating quantum time development as a
stochastic process. The histories approach in turn gives rise to some
conceptual difficulties, in particular the correct choice of a framework
(probabilistic sample space) or family of histories, and these are discussed.
The central issue is that the principle of unicity, the idea that there is a
unique single true description of the world, is incompatible with our current
understanding of quantum mechanics.Comment: Minor changes and corrections to bring into conformity with published
versio
Consistent Histories and Quantum Reasoning
A system of quantum reasoning for a closed system is developed by treating
non-relativistic quantum mechanics as a stochastic theory. The sample space
corresponds to a decomposition, as a sum of orthogonal projectors, of the
identity operator on a Hilbert space of histories. Provided a consistency
condition is satisfied, the corresponding Boolean algebra of histories, called
a {\it framework}, can be assigned probabilities in the usual way, and within a
single framework quantum reasoning is identical to ordinary probabilistic
reasoning. A refinement rule, which allows a probability distribution to be
extended from one framework to a larger (refined) framework, incorporates the
dynamical laws of quantum theory. Two or more frameworks which are incompatible
because they possess no common refinement cannot be simultaneously employed to
describe a single physical system.Comment: Latex, 31 page
- …