QUANTUM MEASUREMENT AND FOUNDATIONS
Quantum Measurement Theory (QMT)
goes back to the work of von Neumann in the early 30's
on mathematical foundations of quantum theory
where he introduced quantum observables as self-adjoint operators
in Hilbert space and his projection postulate to describe the
reduced quantum states for discrete observables
(see for details my recent review paper [1]).
Much later it has been understood that this definition
is not sufficient for the description of quantum observations
such as measurements of quantum time [2],
and that not all self-adjoint operators
should be considered as observables.
The latter follows from the dynamical models [3, 4]
for Hamiltonian interpretation of the projection postulate,
quantum jumps and spontaneous localizations
as interaction of the measured system and measurement apparatus.
This development enabled the derivation [5] of the notorious
quantum collapse from the Stochastic Schroedinger Equation.
The modern QMT gives the mathematical interpretation of
quantum theory which is free of paradoxes and inordinately useful
for distinguishing statistical features from mechanical features.
For this reason, it offers clarity of thought sorely missing
in quantum probability, information and physics.
Quantum Operational Foundations (QOF)
play the fundamental role in mathematical foundations
of modern physics. After realizing the restrictiveness of the von Neumann's
QMT there have been several developments in operational analysis
and axiomatics of quantum measurement starting from Ludwig's and Davies-Lewis
operational approaches in quantum physics and quantum probability.
The dilations theorems for completely positive operations establish
the equivalence of the minimal axiomatic operational QMT
with the constructive stochastic dynamical QMT [5]
Which adds to modern quantum theory
a new quantum causality non-demolition principle [6].
Among the successes of the dynamical QMT based on this principle
are the constructions of the quantum stochastic models explaining
the quantum phase jumps [7] under continuous observation,
and the Zeno paradox for continuous trajectories of quantum particles [8].
It gives the firm basis for the operational theory of
Continuous Quantum Measurements and their stochastic realizations [9],
and has taken off worldwide particularly in the last
ten years, partly as a result of the discovery
in Nottingham of proper calculus for quantum noise
and quantum stochastic nondemolition processes of observation [10].
My 10 Relevant Publications:
- V. P. Belavkin:
Dynamical Solution to the Quantum Measurement Problem, Causality, and Paradoxes of the Quantum Century.
Open Systems and Information Dynamics 7 101-129 (2000).
quant-ph/0512187,
PDF.
- V. P. Belavkin & M. G. Perkins:
The Nondemolition Measurement of Quantum Time.
International Journal of Theoretical Physics 37 (1) 219-226 (1998).
quant-ph/0512205,
PDF.
- V. P. Belavkin & R. L. Stratonovich:
Dynamical Interpretation of the Quantum Measurement Projection Postulate.
Intern. Journal of Theoretical Physics 35 (11) 2215-2228 (1996).
quant-ph/0512196,
PDF.
- V. P. Belavkin and O. Melsheimer:
A Stochastic Hamiltonian Approach for Quantum Jumps, Spontaneous Localizations, and Continuous Trajectories.
Quantum Semiclass. Opt. 8 167-187 (1996).
quant-ph/0512192,
PDF.
- V. P. Belavkin:
A Dynamical Theory of Quantum Measurement and Spontaneous Localization.
Russian J of Math Phys 3 (1) 3-24 (1995).
math-ph/0512069,
PDF.
- V. P. Belavkin:
Nondemolition Principle of Quantum Measurement Theory.
Foundations of Physics 24 (5) 685-713 (1994).
quant-ph/0512188,
PDF.
- V. P. Belavkin and Ch. Bendjaballah:
Continuous Measurements of Quantum Phase.
Quantum Optics 6 169-186 (1994).
PDF.
- V. P. Belavkin and P. Staszewski:
Nondemolition Observation of a Free Quantum Particle.
Phys Rev A
45 (3) 1347-1357 (1992).
quant-ph/0512138,
PDF.
- A. Barchielli and V. P. Belavkin:
Measurements Continuous in Time and Posteriori States in Quantum Mechanics.
J Phys A Math Gen 24(12) 1495-1514 (1991).
quant-ph/0512189,
PDF.
- V. P. Belavkin:
A Continuous Counting Observation and Posterior Quantum Dynamics.
J Phys A Math Gen 22(3) L 1109-L 1114 (1989).