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Quantum probability theory is a generalization of [[Probability theory|probability theory]] in which random variables are not assumed to commute. An alternative name is non-commutative probability theory. It developed in the 1970's from an urge to apply probabilistic concepts, such as "independence" , "noise" and "process" , to quantum mechanics. The mathematical ingredients of quantum probability theory derive from the theory of operator algebras, as founded by J. von Neumann and developed by M.A. Naimark, J. Dixmier, R.V. Kadison [[#References|[a1]]], M. Tomita, M. Takesaki [[#References|[a2]]], and others.

==Classical probability.==
The fundamental object of Kolmogorov's probability theory is a triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q0763201.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q0763202.png" /> is the set of possible outcomes of some experiment, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q0763203.png" /> a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q0763204.png" />-algebra of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q0763205.png" /> called "events" , and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q0763206.png" /> a probability measure on the measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q0763207.png" />. A [[Random variable|random variable]] is a measurable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q0763208.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q0763209.png" /> taking values in some measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632010.png" />, typically the real line with its Borel sets: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632011.png" />. An event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632012.png" /> becomes an event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632013.png" /> when viewed as a statement "about" <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632014.png" />, namely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632015.png" />. Thus, the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632016.png" /> corresponds to an imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632018.png" />.

One may now shift one's point of view somewhat, and replace the event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632019.png" /> by its characteristic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632020.png" />. This is a bounded measurable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632021.png" />. The set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632022.png" /> bounded measurable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632023.png" /> is the [[Von Neumann algebra|von Neumann algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632024.png" />. The probability measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632025.png" /> extends to a positive linear functional on this algebra, the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632026.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632027.png" /> is the probability distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632028.png" />, then the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632029.png" /> mentioned above extends to the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632030.png" /> of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632031.png" /> into the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632032.png" />, given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632033.png" />.

Then one may go one step further, and consider a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632034.png" /> as a linear operator on the Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632035.png" /> by the multiplication <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632037.png" />. The bounded linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632038.png" /> comes from an event if and only if it is an orthogonal projection.

==Generalization.==
By a quantum probability space one means any pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632039.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632040.png" /> a (possibly non-commutative) von Neumann algebra of operators on some Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632042.png" /> a weak<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632043.png" />-continuous positive linear functional (a state) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632044.png" />. The events of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632045.png" /> are the orthogonal projections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632046.png" />. The probability that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632047.png" /> occurs is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632048.png" />. A quantum random variable is an imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632049.png" /> of one quantum probability space into another. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632050.png" /> is of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632051.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632052.png" /> stands for a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632053.png" /> taking values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632054.png" />. In particular, the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632055.png" /> stands for the event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632056.png" />. In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632057.png" />, the projection-valued measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632058.png" /> determines a self-adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632059.png" /> (usually identified with the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632060.png" />) by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632061.png" /></td> </tr></table>

Therefore, real-valued quantum random variables correspond to self-adjoint operators affiliated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632062.png" />, as postulated in quantum mechanics.

A quantum probability space carries a structure which is absent in the classical situation [[#References|[a2]]]: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632063.png" /> is faithful (i.e. if for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632065.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632066.png" />), a one-parameter group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632067.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632068.png" />, of automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632069.png" /> is canonically determined by the mismatch between the quadratic forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632070.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632071.png" />. This group, which is trivial in the commutative case, is called the modular group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632072.png" /> and plays a central role in its probabilistic properties; for instance, it decides the existence question of conditional expectations. A random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632073.png" /> is said te be conditioned if there exists a completely positive mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632074.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632075.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632076.png" />. If this is the case, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632077.png" /> is a projection of norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632078.png" /> (a conditional expectation) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632079.png" />. Such a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632080.png" /> exists if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632081.png" /> is left globally invariant by the modular group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632082.png" />.

==Quantum mechanics.==
Quantum mechanics provides many examples of the above structure. For instance, the situation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632083.png" /> (distinguishable) particles in space corresponds to the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632084.png" /> consisting of all bounded operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632085.png" />. The state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632086.png" /> is of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632087.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632088.png" /> ranges over all positive operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632089.png" /> with trace 1. Some interesting random variables are the positions and momenta of the particles, and their total energy.

Other examples can be found in [[Quantum field theory|quantum field theory]] and quantum statistical mechanics.

However, application of probabilistic ideas becomes most fruitful when the number of relevant degrees of freedom becomes infinite, and some [[Central limit theorem|central limit theorem]] is active. In particular, if a quantum system is studied which interacts weakly with a large number of objects in the outside world, this influence from the outside can be described by a quantum "noise" . Examples of such noises are: thermal radiation, thermal collisions with heat bath particles, laser fields and atomic beams. The evolution in time of the system under consideration is then described by a quantum stochastic process.

==Quantum stochastic processes.==
A quantum stochastic process [[#References|[a3]]] is a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632090.png" /> of quantum random variables indexed by time. (The time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632091.png" /> may be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632092.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632093.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632094.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632095.png" />.) If the process is conditioned, it determines a family of transition probabilities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632096.png" />, i.e. completely positive mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632097.png" /> given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632098.png" />. A quantum Markov process is a quantum stochastic process in which the conditional expectation of the future, given past and present, is entirely determined by the present. In these processes one has an analogue of the classical Chapman–Kolmogorov equation: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q07632099.png" />. The quantum stochastic process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q076320100.png" /> is called stationary if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q076320101.png" /> is a group and their exists a group of automorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q076320102.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q076320103.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q076320104.png" />. The process is said to be in thermal equilibrium if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q076320105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q076320106.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076320/q076320107.png" />).

The theory of stationary quantum Markov processes was developed under the name of dilation theory [[#References|[a4]]]. It is intimately connected with stochastic differential equations (cf. [[Langevin equation|Langevin equation]]). A large part of the literature on quantum probability theory is concentrated in a series of proceedings volumes [[#References|[a5]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R.V. Kadison, J.R. Ringrose, "Fundamentals of the theory of operator algebras" , '''I-II''' , Acad. Press (1983–1986) {{MR|1468230}} {{MR|1468229}} {{MR|1170351}} {{MR|1134132}} {{MR|0859186}} {{MR|0719020}} {{ZBL|0991.46031}} {{ZBL|0888.46039}} {{ZBL|0831.46060}} {{ZBL|0869.46029}} {{ZBL|0869.46028}} {{ZBL|0601.46054}} {{ZBL|0518.46046}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Takesaki, "Tomita's theory of modular Hilbert algebras and its applications" , ''Lect. notes in math.'' , '''128''' , Springer (1970) {{MR|270168}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> L. Accardi, A. Frigerio, J.T. Lewis, "Quantum stochastic processes" ''Publ. RIMS Kyoto'' , '''18''' (1982) pp. 97–133 {{MR|0660823}} {{ZBL|0498.60099}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> B. Kümmerer, "Markov dilations on W*-algebras" ''J. Funct. Anal.'' , '''63''' (1985) pp. 139–177</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> L. Accardi (ed.) W. von Waldenfels (ed.) , ''Quantum probability and applications I-IV'' , ''Lect. notes in math.'' , '''1055, 1136, 1303, 1396''' , Springer (1984, 1985, 1988, 1989)</TD></TR></table>