# Quantum probability

**Quantum probability** was developed in the 1980s as a noncommutative analog of the Kolmogorovian theory of stochastic processes.^{[1]}^{[2]}^{[3]}^{[4]}^{[5]} One of its aims is to clarify the mathematical foundations of quantum theory and its statistical interpretation.^{[6]}^{[7]}

A significant recent application to physics is the dynamical solution of the quantum measurement problem,^{[8]}^{[9]} by giving constructive models of quantum observation processes which resolve many famous paradoxes of quantum mechanics.

Some recent advances are based on quantum filtering^{[10]} and feedback control theory as applications of quantum stochastic calculus.

## Orthodox quantum mechanics

Orthodox quantum mechanics has two seemingly contradictory mathematical descriptions:

- deterministic unitary time evolution (governed by the Schrödinger equation) and
- 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.^{[11]}^{[12]} for introduction or Belavkin, 1970s^{[13]}^{[14]}^{[15]}) 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 *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 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.

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

## References

- ↑ L. Accardi; A. Frigerio & J.T. Lewis (1982). "Quantum stochastic processes".
*Publ. Res. Inst. Math. Sci*.**18**(1): 97–133. doi:10.2977/prims/1195184017. - ↑ R.L. Hudson, K.R. Parthasarathy; Parthasarathy (1984). "Quantum Ito's formula and stochastic evolutions".
*Comm. Math. Phys*.**93**(3): 301–323. Bibcode:1984CMaPh..93..301H. doi:10.1007/BF01258530. - ↑ K.R. Parthasarathy (1992).
*An introduction to quantum stochastic calculus*. Monographs in Mathematics.**85**. Basel: Birkhäuser Verlag. - ↑ D. Voiculescu; K. Dykema; A. Nica (1992).
*Free random variables. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups*. CRM Monograph Series.**1**. Providence, RI: American Mathematical Society. - ↑ P.-A. Meyer (1993). "Quantum probability for probabilists".
*Lecture Notes in Mathematics*.**1538**. - ↑ John von Neumann (1929). "Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren".
*Mathematische Annalen*.**102**: 49–131. doi:10.1007/BF01782338. - ↑ John von Neumann (1932).
*Mathematische Grundlagen der Quantenmechanik*. Die Grundlehren der Mathematischen Wissenschaften, Band 38. Berlin: Springer. - ↑ V. P. Belavkin (1995). "A Dynamical Theory of Quantum Measurement and Spontaneous Localization".
*Russian Journal of Mathematical Physics*.**3**(1): 3–24. arXiv:math-ph/0512069. Bibcode:2005math.ph..12069B. - ↑ V. P. Belavkin (2000). "Dynamical Solution to the Quantum Measurement Problem, Causality, and Paradoxes of the Quantum Century".
*Open Systems and Information Dynamics*.**7**(2): 101–129. arXiv:quant-ph/0512187. doi:10.1023/A:1009663822827. - ↑ V. P. Belavkin (1999). "Measurement, filtering and control in quantum open dynamical systems".
*Reports on Mathematical Physics*.**43**(3): A405–A425. arXiv:quant-ph/0208108. Bibcode:1999RpMP...43A.405B. CiteSeerX 10.1.1.252.701. doi:10.1016/S0034-4877(00)86386-7. - ↑ Bouten, Luc; Van Handel, Ramon; James, Matthew R. (2007). "An Introduction to Quantum Filtering".
*SIAM Journal on Control and Optimization*.**46**(6): 2199–2241. doi:10.1137/060651239. ISSN 0363-0129. - ↑ Luc Bouten; Ramon van Handel; Matthew R. James (2009). "A discrete invitation to quantum filtering and feedback control".
*SIAM Review*.**51**(2): 239–316. arXiv:math/0606118v4. Bibcode:2009SIAMR..51..239B. doi:10.1137/060671504. - ↑ V. P. Belavkin (1972–1974). "Optimal linear randomized filtration of quantum boson signals".
*Problems of Control and Information Theory*.**3**(1): 47–62. - ↑ V. P. Belavkin (1975). "Optimal multiple quantum statistical hypothesis testing".
*Stochastics*.**1**: 315–345. doi:10.1080/17442507508833114. - ↑ V. P. Belavkin (1978). "Optimal quantum filtration of Makovian signals [In Russian]".
*Problems of Control and Information Theory*.**7**(5): 345–360.