Probability
in Quantum Physics |

**In General** > s.a. foundations; interpretations;
hidden variables; many-worlds interpretation;
pilot-wave theory; quantum mechanics.

* __Role__: Probabilities are an essential part of the interpretation,
obtained from inner products (0 ≤ cos^{2}*θ* ≤ 1).

* __Idea__: Probabilities
do not behave like in classical physics; The basic objects for questions *Q* are
probability amplitudes *A*(*Q*), from which the probabilities are calculated
as *P*(*Q*) = |*A*(*Q*)|^{2};
Quantum probability is a variant of contextual probability.

* __Calculation__: In general, when an event can occur
in several different ways, its probability amplitude is the sum of those for the individual ways (interference),

*A*(*Q*) = ∑_{i}
*A*(*Q*, *i*) ;

However, if an experiment is capable of determining which alternative is
followed, then interference is lost; For example, if *Q* is a question whose
answer depends on what the system does up to a time *t*, then

*P*(*Q*) = ∑_{x} |*A*(*x* at *t*_{f}, *a*)|^{2} = |*A*(*Q*)|^{2} , where *t*_{f} > *t* .

* __Remark__: Independence
on *t* in the above calculation is equivalent to unitarity; One point of
view is that probability doesn't change in time; We often ignore the fact that
when we talk about time dependence we're talking about *different* experiments.

* __QBism__: Quantum Bayesianism
or the quantum-Bayesian approach to quantum theory, the personalist Bayesian view of probability in quantum theory;
This view is widely held in general but not by many physicists; Some physicists who have argued for it are Caves,
Fuchs, Schack, and Mermin (Don Page thinks that there are both frequentist and personalist probabilities);
It has profound implications for the meaning of quantum mechanics.

> __Related topics__:
see
Probability Current; Wigner's Friend.

**References** > s.a. measurement in quantum theory; probabilities in physics [negative probabilities, general probabilistic theories]; QBism.

@ __Intros, reviews__:
Cufaro Petroni FP(92);
Sudarshan qp/01;
Rédei & Summers SHPMP(07)qp/06 [and von Neumann algebras]; Sontz a0902 [simple introduction]; Janotta & Hinrichsen JPA(14)-a1402; Khrennikov 16 [and classical]; Svozil a1707 [and correlations].

@ __General references__: Accardi PRP(81);
Accardi & von
Waldenfels ed-85; Van den Berg et al PhSc(90)mar;
Halpin PhSc(91)mar;
Youssef MPLA(91);
Farina AJP(93)may;
Gudder IJTP(93);
Ismael BJPS(96)
[conceptual]; Velleman AJP(98)nov;
Noyes & Etter PE(99)qp/98;
Adler qp/00-proc
[postulated vs emergent]; Barnum et al PRS(00)qp/99;
Khrennikov qp/01 [context-dependent];
Rylov qp/01 [dynamically
based]; Belavkin IDAQP(00)m.PR/05 [history];
Dreyer qp/06 [emergent
probabilities]; Lehrer & Shmaya PRS(06)
[qualitative approach]; Tipler qp/06;
de la Torre EJP(08);
Rave a0806 [interpretation
with closed loops and phases]; Page PLB(09)
[insufficiency of quantum state]; Bub a1005-in [and information theory]; Janssens PhD(10)-a1011; Leifer & Spekkens PRA(13)-a1107 [quantum theory as a causally neutral theory of Bayesian inference]; Blackman & Hsiang PE-a1110 [from large number of degrees of freedom]; Pfister MS-a1203; Aerts & Sassoli de Bianchi a1401, a1401; Hiley LNCS(14)-a1408-conf [structure processes, and non-commutative probability theory]; Yukalov & Sornette PTRS-a1601 [general definition].

@ __And quantum foundations__: Khrennikov qp/01-conf; Wilce FP(10); Holik et al AP(14)-a1211 [origin]; Fröhlich & Schubnel a1310; de Ronde in-a1506 [probabilities as objective knowledge]; > s.a. origin of quantum theory.

@ __From classical probability__:
Slavnov TMP(06)qp/07;
Grigorescu PhyA(08)-a0711 [classical
Fokker-Planck equation and quantum Brownian motion]; Groessing et al a1403.

@ __And classical probabilities__: Khrennikov & Loubenets FP(04)qp/02;
Khrennikov JMP(02), AIP(05)qp/03;
Nyman IJTP(10)-a0906 ["quantum-like
representation algorithm"]; Farenick et al JMP(11)-a1110 [classical and non-classical randomness, in terms of operator-valued measures]; Hardy a1303-ch; Dzhafarov & Kujala FP(14)-a1305, a1312-PLoS; Khrennikov a1406-conf; Hofmann a1606-proc.

@ __And decoherence__: Bacciagaluppi SHPMP(07)qp [time-directed
probability]; Jordan & Chisolm PLA(09)-a0801.

@ __Objective vs subjective probabilities__: Mohrhoff AJP(01)aug-qp/00;
Ballentine AIP(07)-a0710.

@ __Non-Kolmogorov__:
Szabó IJTP(94), IJTP(95)gq/94, FPL(95)gq/94, qp/96;
Khrennikov JMP(00)qp, a0709 [and
Bell
inequalities].

@ __Bayesian approach__: Caves et al PRA(02)qp/01;
Pitowsky SHPMP(03)qp/02;
Schack qp/02 [and
Hardy's axioms for quantum mechanics]; Appleby FP(05)qp/04, O&S(05)qp/04;
Marlow AP(06)qp/05 [histories], gq/06, JMP(06)qp;
Caves
et al SHPMP(07)qp/06 [concept
of certainty]; Bub SHPMP(07);
Rau AP(09)-a0710 [quantum
vs classical]; Fuchs & Schack RMP(13).

@ __Quasi-probability representation__: Ferrie RPP(11)-a1001; Ferrie et al PRA(10) [necessity of negative probabilities]; Ryu et al PRA(13)-a1206 [operationally defined, for qudits].

@ __Related topics__: Page qp/95, IJMPD(96)gq/95 ["sensible
quantum mechanics", without probabilities]; Shirokov qp/06 [on
set of states]; Kupczynski a0810-conf
[statistical predictions]; Döring & Isham JMP(12)-a1102 [as truth values in suitable sheaf topoi]; Loeliger & Vontobel a1201-conf [factor-graph representation]; Stairs SHPMP(11); Henson a1210 [Consistent Exclusivity]; Sokolovski PRD(13)-a1301 [for classes of Feynman paths in spacetime]; Nenashev PS(14)-a1406 [from Zurek's envariance and Gleason's theorem];
> s.a. bell inequalities; Born Rule;
experiments on quantum mechanics; Gleason's Theorem;
histories formulations [including extended probabilities];
measure theory; mixed states;
quantum collapse [GRW]; representations [tomographic].

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