Axioms for Quantum Theory  

In General
* Conventional approach: One possible set (adapted from slides by W Zurek) is
(1) The universe is made of systems.
(2) Each system has a Hilbert space; The Hilbert space of a composite system is the tensor product of the constituent ones.
(3) A pure state for a system is a vector in the Hilbert space of that system.
(4) The evolution of an isolated system is a unitary transformation on Hilbert space.
(5) Observables are associated with Hermitian operators on Hilbert space.
(6) The only possible outcome of a measurement is an eigenstate (and eigenvalue) of the operator.
(7) Probabilities of various outcomes are given by Born's rule.
* Information-theory approach: Hardy's 2001 work recast the issue in information-theory terms, based on "reasonable" operational axioms rather than formalism.
* Yakir Aharonov's idea: The conjecture that the two axioms of relativistic causality ("no superluminal signalling") and non-locality so nearly contradict each other that a unique theory – quantum mechanics – reconciles them; D Rohrlich adds a third axiom, the existence of a classical limit in which macroscopic observables commute.

References > s.a. causality in quantum theory; logic; origin of quantum theory.
@ General: Margenau PR(36) [removal of one postulate]; Piron HPA(64); Fivel pr(73);
Guz IJTP(77); Bugajski & Lahti IJTP(80); Landsman qp/96; Landsman 98; Ghaboussi qp/98; Hardy qp/00-wd; Aerts qp/01-in; Khrennikov in(02)qp; Volovich qp/02 [new set]; Minic & Tze PLB(04)ht/03; Sakai qp/04/JMP; Slavnov TMP(05)qp [based on algebra of observables and functionals]; Mehrafarin IJTP(05); Parwani IJTP(06); Held FP(08)-a0705; Diego a0801; Wada a0909 [minimal set]; Dakić & Brukner a0911-ch [and entanglement]; Santos a0912 [with realist interpretation]; Wilce a0912; Dutailly a1301; Friedberg & Hohenberg FP(18)-a1711 [minimal formulation]; Bertram a1711 [geometric setting]; Aravinda et al EPJD(19)-a1809 [hierarchical]; Jeandel a1810 [diagrammatic language]; D'Ariano FP-a2011 [spurious postulates and purification ontology].
@ Necessity of axioms: Kotúlek JMP(09); Jeknić-Dugić et al a1711; De Raedt et al a1805 [no axioms needed, separation of preparation and measurement].
@ Based on probability theory: Hardy qp/01 [from classical probability, comment Schack FP(03)qp/02, Duck qp/03, Johnson qp/06]; Masanes & Müller NJP(11)-a1004, Müller & Masanes in(16)-a1203 [four requirements compatible only with classical probability and quantum theory]; Cassinelli & Lahti FP(16)-a1508 [using the framework of generalized probabilistic theories].
@ Based on information: Grinbaum PhD(04)qp; Fivel FP(12)-a1010; Chiribella et al PRA(11)-a1011 + Brukner Phy(11); Fields a1102; Masanes et al PNAS(13)-a1208; Chiribella & Scandolo EPJWC(15)-a1411; > s.a. quantum information.
@ Other physical requirements: Rohrlich a1011-ch [two postulates, based on non-locality and causality]; Moldoveanu JPCS(15)-a1303 [from invariance laws], equivalent to Kapustin JMP(13)-a1303 [categorical language]; Rohrlich in(13)-a1407 [derivation of Tsirelson's bound from three axioms]; Jia a1808 [with indefinite causal structure].
@ Operational: D'Ariano AIP-qp/05, qp/06-conf, qp/06-conf, qp/06-conf; D'Ariano & Tosini QIP(10) [testing with toy theories]; Hardy a1104 [reformulation of finite-dimensional quantum theory in the circuit framework], a1303; Fuchs & Stacey a1401-conf [comments on operational approaches]; Benavoli et al a1902 [quantum theory as a computationally tractable theory]; Oeckl a1903-proc [from notions of measurement and composition, positive formalism]; Hardy a2104 [time-symmetric framework].
@ Quantum-gravity-motivated: Giddings PRD(08)-a0711.
@ From cryptography: Clifton et al FP(03)qp/02; Smolin qp/03; Halvorson & Bub qp/03.

Related Topics
@ Use of Hilbert space: de Ronde & Massri a1412 [inconsistency?].
@ Projection postulate: Ballentine FP(90); Kronz PhSc(92)mar; > s.a. wave-function collapse.
@ Linearity: Caticha PLA(98)qp, PRA(98)qp [need]; Aerts & Valckenborgh qp/02-in, qp/02-in [questioning]; Jordan PRA(06)qp/05 [assumptions]; > s.a. non-linear quantum mechanics.
@ For relativistic quantum theory: Kent PTRS(15)-a1411, PRA(17)-a1608 [realist, one-world, relativistic framework].
@ Other topics: Lahti IJTP(80) [uncertainty and complementarity]; Mould qp/05, qp/05, AIP(06)qp [auxiliary rules, nRules], qp/06 [covariance of nRules]; Nottale & Célérier JPA(07)-a0711 [from principles of scale relativity]; Stachow IJTP(80) [logic foundations]; de la Torre et al PRL(12)-a1110 [quantum theory from local structure and reversibility]; Budiyono & Rohrlich nComm(17)-a1711 [common framework with classical statistical mechanics]; Carcassi et al PRL-a2003 [deriving the tensor product postulate from the others].

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