QUANTUM CHAOS, ITO ALGEBRAS AND NOISE CALCULUS
Quantum Noise Analysis (QNA)
is the infinite dimensional operator analysis in Fock space giving a rigorous mathematical framework
for Quantum White Noise (QWN) calculus. QWN, which first appeared in quantum optics,
was introduced into quantum probability and statistics in [1] as a quantum stochastic model
for the description of input-output processes in a Langevin system of open quantum oscillators
modeling a quantum communication system. Hudson and Parthasarathy included into the canonical
system of QWN the quantum exchange process creating a possibility to unify the Gaussian QWN with the
quantum Poisson process which is important in quantum optics and radiophysics.
In [2] I developed the quantum stochastic calculus of nondemolition output processes
for such unified input QWN, and solved the general problem of quantum nonlinear filtering
of the unified QWN. This solution, applied in [3] to the Gaussian quantum states
on an algebra of infinite-dimensional CCR, reproduced the quantum Kalman linear filter obtained in [1].
Quantum Chaotic States (QCS)
are the Quantum White Noise states (QWN) characterized by infinitely divisible generating functionals.
The canonical GNS-type decomposition of the logarithms of such generating functionals
and the corresponding Fock space representation of the QWN was constructed in [4].
It generalizes the Araki-Woods, Parthasarathy-Smidt and Schurmann decompositions and
explains the appearance of the canonical triangular-matrix representation
in the the star-algebraic approach to QSC.
The non-stationary version of this dilation theorem extended also to quantum white fields,
and the corresponding kernel stochastic differential calculus reduced to the "heaven" finite-difference
calculus in Pseudo-Fock space is published in [5].
The QNA related to the QCS of the particular pseudo-Poisson type such as in [6],
defined in [5] by the linear logarithmic generating functions, is yet to be developed.
The dilation theorem for quantum stochastic logarithmic derivatives of positive-definite exponents,
extending these canonical representations has been published recently in [7].
Quantum Ito Algebras (QIA)
are the associative involutive complex algebras without identity but with
a fixed annihilating pseudo-Hermitian element having all products zero,
and with normalized separating state.
The simplest example is the Newtonian one-dimensional nilpotent algebra
of classical differential calculus. The simplest non Newtonian diffusive calculus
is described by two-dimensional second-order nilpotent commutative algebra,
and the direct sum of Newtonian and unital one-dimensional c-algebra is
another non Newtonian Ito algebra of Poisson stochastic calculus.
Each such algebra has no faithful Hilbert space representation,
but it has the canonical triangular-matrix representation
in a complex (in general infinitely-dimensional) Minkowski space.
The classification of the general (noncommutative) QIA,
and the representation theorem is published in [8,9].
The quantum Levy-Chinchin type decomposition theorem is proved in [8]
for finite-dimensional QIA, and in [10] for the infinite-dimensional case.
My 10 Relevant Publications:
- V. P. Belavkin:
Optimal Filtering of Markov Signals with Quantum White Noise.
Radio Eng Electron Physics 25 1445--1453 (1980).
quant-ph/0512091,
PDF.
- V. P. Belavkin:
Stochastic Calculus of Input-Output Quantum Processes and Non-Demolition Filtering.
In Reviews on Newest Achievements in Science and Technology
36 29--67, Ed: A. S. Holevo, VINITI Moscow, 1989.
- V. P. Belavkin:
Quantum Continual Measurements and a Posteriori Collapse on CCR.
Communications Mathematical Physics 146 611--635 (1992).
math-ph/0512070,
PDF.
- V. P. Belavkin:
Kernel Representations of *-Semigroups Associated with Infinitely Divisible States.
In Quantum Probability and Related Topics 7
31--50. World Scientific, Singapore 1992.
- V. P. Belavkin:
Chaotic States and Stochastic Integrations in Quantum Systems.
Usp. Math Nauk (Russian Math Surveys) 47 47--106 (1992).
math.PR/0512265,
PDF.
- V. P. Belavkin O. Hirota and R. L. Hudson:
The World of Quantum Noise and the Fundamental Output Process},
In Quantum Communications and Measurement 3--19,
Plenum Press, New-York and London 1995.
quant-ph/0510030,
PDF.
- V. P. Belavkin:
On Stochastic Generators of Positive-Definite Exponents.
In Transactions of a Japanese-German Symposium
Infinite Dimensional Harmonic Analysis 84--92, Eds: H. Heyer et al,
Kyoto, 2000.
math.PR/0512290,
PDF.
- V. P. Belavkin:
Quantum Ito B*-Algebras, their Classification and Decomposition.
In Quantum Probability
43 63--70 Banach Center Publications, Warszawa 1998.
math.OA/0512508,
PDF.
- V. P. Belavkin:
On Quantum Ito Algebras and Their Decompositions.
Letters in Mathematical Physics 45 131-145 (1998).
math-ph/0512071,
PDF.
- V. P. Belavkin:
Infinite Dimensional Ito Algebras of Quantum White Noise.
In Trends in Contemporary Infinite Dimensional Analysis and
Quantum Probability 57--80, Instituto Italiano di Cultura, Kyoto, 2000.
math.PR/0512288,
PDF.