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].

@ __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 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].

@ __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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send feedback and suggestions to bombelli at olemiss.edu – modified 3 aug 2018