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];
Garola a1806 [epistemic interpretation];
> 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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