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{{Short description|Noncommutative probability framework for quantum mechanics and measurement}}
{{Short description|Noncommutative probability framework for quantum mechanics and measurement}}
{{Quantum book backlink|Foundations}}
{{Quantum book backlink|Foundations}}
 
{{Quantum article nav|previous=Physics:Quantum superposition|previous label=Superposition|next=Physics:Quantum Mathematical Foundations of Quantum Theory|next label=Mathematical Foundations of Quantum Theory}}
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'''Quantum probability''' was developed in the 1980s as a noncommutative analog of the Kolmogorovian theory of stochastic processes.<ref name=Accardi82>{{cite journal
'''probability''' 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 filtering and feedback control theory as applications of quantum stochastic calculus. Orthodox quantum mechanics has two seemingly contradictory mathematical descriptions: 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.
|author1=L. Accardi |author2=A. Frigerio |author3=J.T. Lewis  |last-author-amp=yes | title = Quantum stochastic processes
| journal = Publ. Res. Inst. Math. Sci.
| volume = 18
| year = 1982
| pages = 97–133
| issue = 1
| doi = 10.2977/prims/1195184017
| url = https://art.torvergata.it/bitstream/2108/83328/1/Ac84_Quantum%20Stochastic%20Processes.pdf
}}</ref><ref name=Hudson-Parthasarathy84>{{cite journal
| author = R.L. Hudson, K.R. Parthasarathy
| title = Quantum Ito's formula and stochastic evolutions
| journal = Comm. Math. Phys.
| volume = 93
| year = 1984
| pages = 301–323
| issue = 3
| bibcode = 1984CMaPh..93..301H
| last2 = Parthasarathy
| doi = 10.1007/BF01258530
}}</ref><ref name=Parthasarathy92>{{cite book
| author = K.R. Parthasarathy
| title = An introduction to quantum stochastic calculus
| series = Monographs in Mathematics
| volume = 85
| publisher = Birkhäuser Verlag
| location = Basel
| year = 1992
}}</ref><ref name=Voiculescu92>{{cite book
|author1=D. Voiculescu |author2=K. Dykema |author3=A. Nica | title = Free random variables. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups
| series = CRM Monograph Series
| volume = 1
| publisher = American Mathematical Society
| location = Providence, RI
| year = 1992
}}</ref><ref name=Meyer93>{{cite book
| author = P.-A. Meyer
| title = Quantum probability for probabilists
| series = Lecture Notes in Mathematics
| volume = 1538
| year = 1993
}}</ref> One of its aims is to clarify the mathematical foundations of [[Physics:Quantum mechanics|quantum theory]] and its statistical interpretation.<ref name=Neumann29>{{cite journal
| author = John von Neumann
| title = Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren
| journal = Mathematische Annalen
| volume = 102
| pages = 49–131
| year = 1929
| doi=10.1007/BF01782338
}}</ref><ref name=Neumann32>{{cite book
| author = John von Neumann
| title = Mathematische Grundlagen der Quantenmechanik
| series = Die Grundlehren der Mathematischen Wissenschaften, Band 38
| location = Berlin
| publisher = Springer
| year = 1932
}}</ref>
 
A significant recent application to [[Physics:Physics|physics]] is the dynamical solution of the quantum measurement problem,<ref name=Belavkin95>{{cite journal
| author = V. P. Belavkin
| title = A Dynamical Theory of Quantum Measurement and Spontaneous Localization
| journal = Russian Journal of Mathematical Physics
| volume = 3
| year = 1995
| pages = 3–24
| arxiv =  math-ph/0512069 |bibcode = 2005math.ph..12069B
| issue = 1 }}</ref><ref name=Belavkin2000>{{cite journal
| author = V. P. Belavkin
| title = Dynamical Solution to the Quantum Measurement Problem, Causality, and Paradoxes of the Quantum Century
| journal = Open Systems and Information Dynamics
| volume = 7
| pages = 101–129
| year = 2000
| doi = 10.1023/A:1009663822827
| arxiv = quant-ph/0512187
| issue = 2}}</ref> by giving constructive models of quantum observation processes which resolve many famous paradoxes of [[Physics:Quantum mechanics|quantum mechanics]].
 
Some recent advances are based on [[Belavkin equation|quantum filtering]]<ref name=Belavkin99>{{cite journal
| author = V. P. Belavkin
| title = Measurement, filtering and control in quantum open dynamical systems
| journal = Reports on Mathematical Physics
| volume = 43
| pages = A405–A425
| year = 1999
| doi = 10.1016/S0034-4877(00)86386-7
| arxiv = quant-ph/0208108|bibcode = 1999RpMP...43A.405B
| issue = 3 | citeseerx = 10.1.1.252.701
}}</ref> and feedback control theory as applications of [[Quantum stochastic calculus|quantum stochastic calculus]].
</div>
</div>


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Orthodox [[Physics:Quantum mechanics|quantum mechanics]] has two seemingly contradictory mathematical descriptions:
Orthodox [[Physics:Quantum mechanics|quantum mechanics]] has two seemingly contradictory mathematical descriptions:


# deterministic [[Unitary operator|unitary]] [[Time evolution|time evolution]] (governed by the [[Physics:Schrödinger equation|Schrödinger equation]]) and
# deterministic unitary time evolution (governed by the Schrödinger equation) and
# [[Stochastic|stochastic]] (random) wavefunction collapse.
# 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., [[Physics:Schrödinger's cat|Schrödinger's cat]], an isolated atom) do paradoxes seem to occur.
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 [[Belavkin equation|quantum filtering]] theory (see Bouten et al.<ref>{{Cite journal|last=Bouten|first=Luc|last2=Van Handel|first2=Ramon|last3=James|first3=Matthew R.|date=2007|title=An Introduction to Quantum Filtering|journal=SIAM Journal on Control and Optimization|language=en-US|volume=46|issue=6|pages=2199–2241|doi=10.1137/060651239|issn=0363-0129|arxiv=math/0601741}}</ref><ref name=Bouten2009>{{cite journal
Orthodox quantum mechanics can be reformulated in a quantum-probabilistic framework, where quantum filtering theory (see Bouten et al.<ref>{{Cite journal|last=Bouten|first=Luc|last2=Van Handel|first2=Ramon|last3=James|first3=Matthew R.|date=2007|title=An Introduction to Quantum Filtering|journal=SIAM Journal on Control and Optimization|language=en-US|volume=46|issue=6|pages=2199–2241|doi=10.1137/060651239|issn=0363-0129|arxiv=math/0601741}}</ref><ref name=Bouten2009>{{cite journal
  |author1=Luc Bouten |author2=Ramon van Handel |author3=Matthew R. James | title = A discrete invitation to quantum filtering and feedback control
  |author1=Luc Bouten |author2=Ramon van Handel |author3=Matthew R. James | title = A discrete invitation to quantum filtering and feedback control
  | journal = SIAM Review
  | journal = SIAM Review
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  | arxiv = math/0606118
  | arxiv = math/0606118
|bibcode = 2009SIAMR..51..239B
|bibcode = 2009SIAMR..51..239B
  | issue = 2 }}</ref> for introduction or [[Biography:Viacheslav Belavkin|Belavkin]], 1970s<ref name="Belavkin72">{{cite journal|author=V. P. Belavkin|year=1972–1974|title=Optimal linear randomized filtration of quantum boson signals|url=|journal=Problems of Control and Information Theory|volume=3|issue=1|pages=47–62|via=}}</ref><ref name=Belavkin75>{{cite journal
  | issue = 2 }}</ref> for introduction or Belavkin, 1970s<ref name="Belavkin72">{{cite journal|author=V. P. Belavkin|year=1972–1974|title=Optimal linear randomized filtration of quantum boson signals|url=|journal=Problems of Control and Information Theory|volume=3|issue=1|pages=47–62|via=}}</ref><ref name=Belavkin75>{{cite journal
  | author = V. P. Belavkin
  | author = V. P. Belavkin
  | title = Optimal multiple quantum statistical hypothesis testing
  | title = Optimal multiple quantum statistical hypothesis testing
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=== Motivation ===
=== Motivation ===
In classical [[Probability theory|probability theory]], information is summarized by the [[Sigma-algebra|sigma-algebra]] ''F'' of events in a classical [[Probability space|probability space]] (Ω, ''F'','''P'''). For example, ''F'' could be the σ-algebra σ(''X'') generated by a [[Random variable|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|*-algebra]]. A (unital) *- algebra is a complex vector space ''A'' of operators on a Hilbert space ''H'' that
In classical probability theory, 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


* contains the identity ''I'' and
* contains the identity ''I'' and
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:'''Definition : Quantum probability space.'''
:'''Definition : Quantum probability space.'''


A quantum probability space is a pair (''A'', '''P'''), where ''A'' is a [[*-algebra|*-algebra]] and '''P''' is a state.
A quantum probability space is a pair (''A'', '''P'''), where ''A'' is a *-algebra and '''P''' is a state.


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 almost everywhere bounded complex-valued measurable functions{{citation needed|date=March 2018}}.
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 almost everywhere bounded complex-valued measurable functions{{citation needed|date=March 2018}}.
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== External links ==
== External links ==
* [https://sites.google.com/site/associationqp/home Association for Quantum Probability and Infinite Dimensional Analysis (AQPIDA)]
* [https://sites.google.com/site/associationqp/home Association for Quantum Probability and Infinite Dimensional Analysis (AQPIDA)]
{{Quantum mechanics topics|state=expanded}}
[[Category:Quantum mechanics]]
[[Category:Quantum mechanics]]
[[Category:Exotic probabilities]]
[[Category:Exotic probabilities]]

Latest revision as of 23:35, 23 May 2026

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Next : Mathematical Foundations of Quantum Theory →

probability 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 filtering and feedback control theory as applications of quantum stochastic calculus. Orthodox quantum mechanics has two seemingly contradictory mathematical descriptions: 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.

Quantum probability describes probabilistic structure in quantum mechanics using noncommutative mathematical models.

Orthodox quantum mechanics

Orthodox quantum mechanics 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 theory (see Bouten et al.[1][2] for introduction or Belavkin, 1970s[3][4][5]) 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 theory, 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

  • 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 quantum probability space is a pair (A, P), where A is a *-algebra and P is a state.

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 almost everywhere bounded complex-valued measurable functions[citation needed].

The idempotents p ∈ A are the events in A, and P(p) gives the probability of the event p.

See also

Table of contents (217 articles)

Index

Full contents

References

  1. ↑ Bouten, Luc; Van Handel, Ramon; James, Matthew R. (2007). "An Introduction to Quantum Filtering" (in en-US). SIAM Journal on Control and Optimization 46 (6): 2199–2241. doi:10.1137/060651239. ISSN 0363-0129. 
  2. ↑ Luc Bouten; Ramon van Handel; Matthew R. James (2009). "A discrete invitation to quantum filtering and feedback control". SIAM Review 51 (2): 239–316. doi:10.1137/060671504. Bibcode: 2009SIAMR..51..239B. 
  3. ↑ V. P. Belavkin (1972–1974). "Optimal linear randomized filtration of quantum boson signals". Problems of Control and Information Theory 3 (1): 47–62. 
  4. ↑ V. P. Belavkin (1975). "Optimal multiple quantum statistical hypothesis testing". Stochastics 1 (1–4): 315–345. doi:10.1080/17442507508833114. 
  5. ↑ V. P. Belavkin (1978). "Optimal quantum filtration of Makovian signals [In Russian]". Problems of Control and Information Theory 7 (5): 345–360. 


Author: Harold Foppele


Source attribution: Physics:Quantum probability

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