| 词条 | Quantum probability |
| 释义 |
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 mechanicsOrthodox 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. 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. MotivationIn 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
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 definitionThe 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|date=March 2018}}. The idempotents p ∈ A are the events in A, and P(p) gives the probability of the event p. References1. ^{{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}} 2. ^{{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}} 3. ^{{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}} 4. ^{{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}} 5. ^{{cite journal | author = P.-A. Meyer | title = Quantum probability for probabilists | journal = Lecture Notes in Mathematics | volume = 1538 | year = 1993}} 6. ^{{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}} 7. ^{{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}} 8. ^{{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 }} 9. ^{{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}} 10. ^{{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 }} 11. ^{{Cite journal|last=Bouten|first=Luc|last2=Van Handel|first2=Ramon|last3=James|first3=Matthew R.|date=2007|title=An Introduction to Quantum Filtering|url=https://doi.org/10.1137/060651239|journal=SIAM Journal on Control and Optimization|language=en-US|volume=46|issue=6|pages=2199–2241|doi=10.1137/060651239|issn=0363-0129|via=}} 12. ^{{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 }} 13. ^{{cite journal|author=V. P. Belavkin|first=|date=|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=}} 14. ^{{cite journal | author = V. P. Belavkin | title = Optimal multiple quantum statistical hypothesis testing | journal = Stochastics | volume = 1 | pages = 315–345 | year = 1975 | doi = 10.1080/17442507508833114}} 15. ^{{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}} External links
2 : Quantum mechanics|Exotic probabilities |
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