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Repair Quantum Collection B backlink template

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{{Quantum book backlink|Foundations}}

'''Quantum probability''' was developed in the 1980s as a noncommutative analog of the Kolmogorovian theory of stochastic processes.<ref name=Accardi82>{{cite journal
|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]].

== Orthodox quantum mechanics ==
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
# [[Stochastic|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.

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
|author1=Luc Bouten |author2=Ramon van Handel |author3=Matthew R. James | title = A discrete invitation to quantum filtering and feedback control
| journal = SIAM Review
| volume = 51
| pages = 239–316
| year = 2009
| doi = 10.1137/060671504
| arxiv = math/0606118
|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
| author = V. P. Belavkin
| title = Optimal multiple quantum statistical hypothesis testing
| journal = Stochastics
| volume = 1
| issue = 1–4
| pages = 315–345
| year = 1975
| doi = 10.1080/17442507508833114
}}</ref><ref name=Belavkin78>{{cite journal
| author = V. P. Belavkin
| title = Optimal quantum filtration of Makovian signals [In Russian]
| journal = Problems of Control and Information Theory
| volume = 7
| pages = 345–360
| year = 1978
| issue = 5
}}</ref>) 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|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

* 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 ''p''2 = ''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|*-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}}.

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

== References ==
<references />

== External links ==
* [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:Exotic probabilities]]

{{Sourceattribution|Quantum probability|1}}

Physics:Quantum probability - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template