quantum algorithms:
In probability theory, the concept of noncommutative probability space or quantum probability space is the generalization of that of probability space as the concept of “space” is generalized to non-commutative geometry.
The basic idea is to encode a would-be probability space dually in its algebra of functions , typically regarded as a star algebra, and encode the probability measure as a state on this star algebra
Hence this primarily axiomatizes the concept of expectation values (Segal 65, Whittle 92) while leaving the nature of the underlying probability measure implicit (in contrast to the classical formalization of probability theory by Andrey Kolmogorov).
Often is assumed/required to be a von Neumann algebra (e.g. Kuperberg 05, section 1.8). Often is taken to be the full algebra of bounded operators on some Hilbert space (e.g. Attal, def. 7.1).
In quantum physics, is an algebra of observables (or a local net thereof) and is a particular quantum state, for instance a vacuum state.
The formulation of non-perturbative quantum field theory from the algebraic perspective of quantum probability is known as algebraic quantum field theory (AQFT).
The formulation of perturbative quantum field theory from the algebraic perspective of quantum probability is known as perturbative algebraic quantum field theory (pAQFT).
The sentiment that quantum physics is quantum probability theory is also referred to as the Bayesian interpretation of quantum mechanics (“QBism”).
The idea that
quantum probability is “just as” classical probability theory but generalized to non-commutative probability spaces, hence, for quantum physics, to quantized phase spaces
may be made precise and fully manifest by understanding quantum probability theory as being classical probability theory internal to the Bohr topos of the given quantum mechanical system.
For details see at Bohr topos the section Kinematics in a Bohr topos.
For going deeper, see at order-theoretic structure in quantum mechanics.
Quantum probability theory shows that “wave function collapse” is just part of the formula for conditional expectation values in quantum probability theory (e.g. Kuperberg 05, section 1.2, Yuan 12):
Let be a quantum probability space, hence a complex star algebra of quantum observables, and a state on a star-algebra .
This means that for any observable, its expectation value in the given state is
More generally, if is a real idempotent/projector
thought of as an event, then for any observable the conditional expectation value of , conditioned on the observation of , is (e.g. Redei-Summers 06, section 7.3, see also Fröhlich-Schubnel 15, (5.49), Fröhlich 19 (45))
Now assume a star-representation of the algebra of observables by linear operators on a Hilbert space is given, and that the state is a pure state, hence given by an vector (“wave function”) via the Hilbert space inner product as
In this case the expression for the conditional expectation value (2) of an observable conditioned on an idempotent observable becomes (notationally suppressing the representation )
where in the last step we used (1).
This says that assuming that has been observed in the pure state , then the corresponding conditional expectation values are the same as actual expectation values but for the new pure state .
This is the statement of “wave function collapse”:
The original wave function is , and after observing it “collapses” to (up to normalization).
The understanding of quantum physics as a probabilistic theory originates with the formulation of the Born rule and was made fully explicit in:
Pascual Jordan, Über eine neue Begründung der Quantenmechanik, Zeitschrift für Physik 40 (1927) 809–838 [doi:10.1007/BF01390903]
John von Neumann, Die quantenmechanische Statistik, Part III in:
Mathematische Grundlagen der Quantenmechanik, Springer (1932, 1971) [doi:10.1007/978-3-642-96048-2]
Mathematical Foundations of Quantum Mechanics Princeton University Press (1955) [doi:10.1515/9781400889921, Wikipedia entry]
see also:
with emphasis on POVMs:
Paul Busch, Marian Grabowski, Pekka J. Lahti, Operational Quantum Physics, Lecture Notes in Physics Monographs 31, Springer (1995) [doi:10.1007/978-3-540-49239-9]
Paul Busch, Pekka J. Lahti, Juha-Pekka Pellonpää, Kari Ylinen, Quantum Measurement, Springer (2016) [doi:10.1007/978-3-319-43389-9]
The axiomatization of probability theory in terms of the concept of expectation values (instead of probability measures) is amplified in:
Irving Segal, Algebraic integration theory, Bull. Amer. Math. Soc. Volume 71, Number 3, Part 1 (1965), 419-489 (eculid:1183526903)
Peter Whittle, Probability via expectation, Springer 1992 (doi:10.1007/978-1-4612-0509-8)
Rolando Rebolledo, An algebraic view on Probability, Section 1.2 in: Complete Positivity and the Markov structure of Open Quantum Systems, chapter in: Stéphane Attal, Alain Joye, Claude-Alain Pillet (eds.), Open Quantum Systems II – The Markovian approach, Lecture Notes in Mathematics 1881, Springer (2006) 149-182 [doi:10.1007/b128451]
Dedicated discussion of quantum probability theory:
Itamar Pitowsky, Quantum Probability – Quantum Logic, Lecture Notes in Physics 321, Springer (1989) [doi:10.1007/BFb0021186]
(emphasis on the Bell's inequalities and quantum logic)
Paul-André Meyer, Quantum Probability for Probabilists, Lecture Notes in Mathematics 1538, Springer 1995 (doi:10.1007/BFb0084701)
Hans Maassen, Quantum Probability Theory, Lecture notes 1998 (pdf, pdf)
Greg Kuperberg, A concise introduction to quantum probability, quantum mechanics, and quantum computation (2005) [pdf, pdf]
Miklos Redei, Stephen Summers, Quantum Probability Theory, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics Volume 38, Issue 2, June 2007, Pages 390-417 (arXiv:quant-ph/0601158, doi:10.1016/j.shpsb.2006.05.006)
Stéphane Attal, Quantum Probability, Lecture 7 in: Lectures on Quantum Noises [pdf, webpage]
Franco Strocchi, Section 2.4 in: An introduction to the mathematical structure of quantum mechanics, Advanced Series in Mathematical Physics 28, World Scientific (2008) [doi:10.1142/7038]
Jonathan Gleason, The -algebraic formalism of quantum mechanics (2009) [pdf, pdf]
Jonathan Gleason, From Classical to Quantum: The -Algebraic Approach, contribution to VIGRE REU 2011, Chicago (2011) [pdf, pdf]
Qiaochu Yuan, Finite noncommutative probability, the Born rule, and wave function collapse, 2012
A. Ibort, V.I. Manko, G. Marmo, A. Simoni, F. Ventriglia, A pedagogical presentation of a -algebraic approach to quantum tomography, Phys. Scr., 84 (2011) 065006 (arXiv:1204.5231)
Jürg Fröhlich, B. Schubnel, Quantum Probability Theory and the Foundations of Quantum Mechanics. In: Blanchard P., Fröhlich J. (eds.) The Message of Quantum Science. Lecture Notes in Physics, vol 899. Springer 2015 (arXiv:1310.1484, doi:10.1007/978-3-662-46422-9_7)
Jürg Fröhlich, The structure of quantum theory, Chapter 6 in The quest for laws and structure, EMS 2016 (doi, doi:10.4171/164-1/8)
Jan Swart, Introduction to Quantum Probability, 2017 (pdf, pdf)
Klaas Landsman, Foundations of quantum theory – From classical concepts to Operator algebras, Springer Open 2017 (doi:10.1007/978-3-319-51777-3, pdf)
Further discussion of quantum probability:
Dror Bar-Natan, Two examples in noncommutative probability, Foundations of Physics, 19 (1989) 97–104 (doi:10.1007/BF00737769)
Nicolò Drago, Valter Moretti, The notion of observable and the moment problem for -algebras and their GNS representations (doi:1903.07496, spire:1725528)
Discussion of density matrices and entropy in quantum probability, via the GNS construction:
A. P. Balachandran, T. R. Govindarajan, Amilcar R. de Queiroz, A. F. Reyes-Lega, Algebraic approach to entanglement and entropy, Phys. Rev. A 88, 022301 (2013) (arXiv:1301.1300)
A. P. Balachandran, A. R. de Queiroz, S. Vaidya, Entropy of Quantum States: Ambiguities, Eur. Phys. J. Plus 128, 112 (2013) (arXiv:1212.1239, doi:10.1140/epjp/i2013-13112-3)
Paolo Facchi, Giovanni Gramegna, Arturo Konderak, Entropy of quantum states (arXiv:2104.12611)
Last revised on September 29, 2023 at 09:28:20. See the history of this page for a list of all contributions to it.