Quantum probability
Encyclopedia
Quantum probability was developed in the 1980s as a noncommutative analog of the Kolmogorovian theory of stochastic processes. One of its aims is to clarify the mathematical foundations of quantum theory
and its statistical interpretation.
A significant recent application to physics
is the dynamical solution of the quantum measurement problem, by giving constructive models of quantum observation processes which resolve many famous paradoxes of quantum mechanics
.
Some recent advances are based on quantum stochastic filtering
and feedback control theory as applications of quantum stochastic calculus
.
has two seemingly contradictory mathematical descriptions:
1. deterministic unitary
time evolution
(governed by the Schrödinger equation
) and
2. stochastic
(random) wavefunction collapse
.
Most physicists are not concerned with this apparent problem. Physical intuition usually provides the answer, and only in unphysical systems (e.g., Schrödinger's cat
, an isolated atom) do paradoxes seem to occur.
Orthodox quantum mechanics can be reformulated in a quantum-probabilistic framework, where quantum filtering (2005 (or Belavkin
, 1970s)) gives the natural description of the measurement process. This new framework encapsulates the standard postulates of quantum mechanics, and thus all of the science involved in the orthodox postulates.
, information is summarized by the sigma-algebra
F of events in a classical probability space
(Ω ,F,P). For example, F could be the σ-algebra σ(X) generated by a random variable
X, which contains all the information on the values taken by X. We wish to describe quantum information in similar algebraic terms, in such a way as to capture the non-commutative features and the information made available in an experiment. The appropriate algebraic structure for observables, or more generally operators, is a *-algebra. A (unital) *- algebra is a complex vector space A of operators on a Hilbert space H that
A state P on A is a linear functional P : A → C (where C is the field of complex numbers) such that 0 ≤ P(a* a) for all a ∈ A (positivity) and P(I) = 1 (normalization). A projection is an element p ∈ A such that p2 = p = p*.
A pair (A , P), where A is a *-algebra and P is a state, is called a quantum probability space.
This definition is a generalization of the definition of a probability space in Kolmogorovian probability theory, in the sense that every (classical) probability space gives rise to a quantum probability space if A is chosen as the *-algebra of bounded complex-valued measurable functions on it.
The projections p ∈ A are the events in A , and P(p) gives the probability of the event p.
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
and its statistical interpretation.
A significant recent application to physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
is the dynamical solution of the quantum measurement problem, by giving constructive models of quantum observation processes which resolve many famous paradoxes of quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
.
Some recent advances are based on quantum stochastic filtering
Filtering problem (stochastic processes)
In the theory of stochastic processes, the filtering problem is a mathematical model for a number of filtering problems in signal processing and the like. The general idea is to form some kind of "best estimate" for the true value of some system, given only some observations of that system...
and feedback control theory as applications of quantum stochastic calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
.
Orthodox quantum mechanics
Orthodox quantum mechanicsQuantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
has two seemingly contradictory mathematical descriptions:
1. deterministic unitary
Unitary operator
In functional analysis, a branch of mathematics, a unitary operator is a bounded linear operator U : H → H on a Hilbert space H satisfyingU^*U=UU^*=I...
time evolution
Time evolution
Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state . In this formulation, time is not required to be a continuous parameter, but may be discrete or even finite. In classical physics, time evolution of a collection of rigid bodies...
(governed by the Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
) and
2. stochastic
Stochastic
Stochastic refers to systems whose behaviour is intrinsically non-deterministic. A stochastic process is one whose behavior is non-deterministic, in that a system's subsequent state is determined both by the process's predictable actions and by a random element. However, according to M. Kac and E...
(random) wavefunction collapse
Wavefunction collapse
In quantum mechanics, wave function collapse is the phenomenon in which a wave function—initially in a superposition of several different possible eigenstates—appears to reduce to a single one of those states after interaction with an observer...
.
Most physicists are not concerned with this apparent problem. Physical intuition usually provides the answer, and only in unphysical systems (e.g., Schrödinger's cat
Schrödinger's cat
Schrödinger's cat is a thought experiment, usually described as a paradox, devised by Austrian physicist Erwin Schrödinger in 1935. It illustrates what he saw as the problem of the Copenhagen interpretation of quantum mechanics applied to everyday objects. The scenario presents a cat that might be...
, an isolated atom) do paradoxes seem to occur.
Orthodox quantum mechanics can be reformulated in a quantum-probabilistic framework, where quantum filtering (2005 (or Belavkin
Viacheslav Belavkin
Viacheslav Pavlovich Belavkin is a professor in applied mathematics at the University of Nottingham. He was born in Lwów, and graduated from Moscow State University in 1970. In 1996, he and Ruslan L. Stratonovich were awarded the Main State Prize of the Russian Federation for outstanding...
, 1970s)) gives the natural description of the measurement process. This new framework encapsulates the standard postulates of quantum mechanics, and thus all of the science involved in the orthodox postulates.
Motivation
In classical probability theoryProbability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, information is summarized by the sigma-algebra
Sigma-algebra
In mathematics, a σ-algebra is a technical concept for a collection of sets satisfying certain properties. The main use of σ-algebras is in the definition of measures; specifically, the collection of sets over which a measure is defined is a σ-algebra...
F of events in a classical probability space
Probability space
In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind...
(Ω ,F,P). For example, F could be the σ-algebra σ(X) generated by a random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
X, which contains all the information on the values taken by X. We wish to describe quantum information in similar algebraic terms, in such a way as to capture the non-commutative features and the information made available in an experiment. The appropriate algebraic structure for observables, or more generally operators, is a *-algebra. A (unital) *- algebra is a complex vector space A of operators on a Hilbert space H that
- contains the identity I and
- is closed under composition (a multiplication) and adjoint (an involution *): a ∈ A implies a* ∈ A.
A state P on A is a linear functional P : A → C (where C is the field of complex numbers) such that 0 ≤ P(a* a) for all a ∈ A (positivity) and P(I) = 1 (normalization). A projection is an element p ∈ A such that p2 = p = p*.
Mathematical definition
The basic definition in quantum probability is that of a quantum probability space, sometimes also referred to as an algebraic or noncommutative probability space.- Definition : Quantum probability space.
A pair (A , P), where A is a *-algebra and P is a state, is called a quantum probability space.
This definition is a generalization of the definition of a probability space in Kolmogorovian probability theory, in the sense that every (classical) probability space gives rise to a quantum probability space if A is chosen as the *-algebra of bounded complex-valued measurable functions on it.
The projections p ∈ A are the events in A , and P(p) gives the probability of the event p.