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 a1711 [minimal formulation]; Bertram a1711 [geometric setting].

@ __Necessity of axioms__: Kotúlek JMP(09); Jeknić-Dugić et al a1711.

@ __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 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].

@ __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].

@ __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.

@ __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].

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