Reconstructing Quantum Theory

Hardy, Lucién

doi:10.1007/978-94-017-7303-4_7

Cited by 94 publications

(195 citation statements)

References 46 publications

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“…In recent years, a wide range of reconstruction theorems with a large variety of choices for the axioms have been derived, as pioneered by Hardy [11,12]. In these theorems, 'quantum mechanics' refers to the Hilbert space formulation in finite dimensions, and the reconstruction theorems recover the Hilbert space structure within the framework of general probabilistic theories.…”

Section: Cecilia Flori and Tobias Fritzmentioning

confidence: 99%

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(Almost) C*-algebras as sheaves with self-action

Flori¹,

Fritz²

2017

J. Noncommut. Geom.

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Abstract. Via Gelfand duality, a unital C*-algebra A induces a functor from compact Hausdorff spaces to sets, CHaus → Set. We show how this functor encodes standard functional calculus in A as well as its multivariate generalization. Certain sheaf conditions satisfied by this functor provide a further generalization of functional calculus. Considering such sheaves CHaus → Set abstractly, we prove that the piecewise C*-algebras of van den Berg and Heunen are equivalent to a full subcategory of the category of sheaves, where a simple additional constraint characterizes the objects in the subcategory. It is open whether this additional constraint holds automatically, in which case piecewise C*-algebras would be the same as sheaves CHaus → Set.Intuitively, these structures capture the commutative aspects of C*-algebra theory. In order to find a complete reaxiomatization of unital C*-algebras within this language, we introduce almost C*-algebras as piecewise C*-algebras equipped with a notion of inner automorphisms in terms of a self-action. We provide some evidence for the conjecture that the forgetful functor from unital C*-algebras to almost C*-algebras is fully faithful, and ask whether it is an equivalence of categories. We also develop an analogous notion of almost group, and prove that the forgetful functor from groups to almost groups is not full.In terms of quantum physics, our work can be seen as an attempt at a reconstruction of quantum theory from physically meaningful axioms, as realized by Hardy and others in a different framework. Our ideas are inspired by and also provide new input for the topos-theoretic approach to quantum theory. Contents 1.Introduction 2 2. C*-algebras as functors CHaus → Set 5 3. C*-algebras as sheaves CHaus → Set 9 4. Piecewise C*-algebras as sheaves CHaus → Set 26 5. Almost C*-algebras as piecewise C*-algebras with self-action 30 6. Groups as piecewise groups with self-action 34 References 362010 Mathematics Subject Classification. Primary: 46L05 (General theory of C*-algebras), 46L60 (Applications of selfadjoint operator algebras to physics). Secondary: 18F20 (Presheaves and sheaves), 20A05 (Group theory, axiomatics and elementary properties).Key words and phrases. Axiomatics of C*-algebras; sheaf theory; algebraic quantum mechanics; topos quantum theory.Acknowledgements. We thank Benno van den Berg, Chris Heunen and Manuel Reyes for discussion and crucial comments on a draft; Ryszard Kostecki, Klaas Landsman, Markus Müller, Sam Staton, Andreas Thom and Bas Westerbaan for further discussion; and Tom Leinster for pointing out Isbell's results on codensity. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development and Innovation. During his time at Perimeter Institute, the second author has been supported by the John Templeton Foundation. 1. Introduction C*-algebra theory is a blend of algebra and analysis which turns out to be much more than the sum of its parts, a...

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Section: Cecilia Flori and Tobias Fritzmentioning

confidence: 99%

“…is already an equivalence relation thanks to (12). As a finite union of closed sets, it is also closed, and hence two points in Y ∐ Z get identified in Y ∐ f g Z if and only if they satisfy this relation.…”

Section: C*-algebras As Sheaves Chaus → Setmentioning

confidence: 99%

(Almost) C*-algebras as sheaves with self-action

Flori¹,

Fritz²

2017

J. Noncommut. Geom.

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“…Let me just highlight amongst the most recent attempts those of Hardy [Har01,Har11] and those of Chiribella, D'Ariano & Perinotti [CDP10,CDP11] ( see [Bru11] for a short presentation of this last formulation). See also [MM11].…”

Section: Quantum Correlations 5-3mentioning

confidence: 99%

The Formalisms of Quantum Mechanics

David

2015

Lecture Notes in Physics

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These notes are an elaboration on: (i) a short course that I gave at the IPhT-Saclay in MayJune 2012; (ii) a previous letter[Dav11] on reversibility in quantum mechanics. They present an introductory, but hopefully coherent, view of the main formalizations of quantum mechanics, of their interrelations and of their common physical underpinnings: causality, reversibility and locality/separability. The approaches covered are mainly: (ii) the canonical formalism; (ii) the algebraic formalism; (iii) the quantum logic formulation. Other subjects: quantum information approaches, quantum correlations, contextuality and non-locality issues, quantum measurements, interpretations and alternate theories, quantum gravity, are only very briefly and superficially discussed.Most of the material is not new, but is presented in an original, homogeneous and hopefully not technical or abstract way. I try to define simply all the mathematical concepts used and to justify them physically. These notes should be accessible to young physicists (graduate level) with a good knowledge of the standard formalism of quantum mechanics, and some interest for theoretical physics (and mathematics).These notes do not cover the historical and philosophical aspects of quantum physics.

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“…In fact, there actually exist a number of successful reconstructions of finite dimensional quantum theory from operational axioms, most of which have been performed within the frameworks of generalized or operational probability theories [9,[23][24][25][26][27][28][29][30][31][32] (see also [33] which is adapted to a space-time language and the more mathematical reconstructions [34,35]). Despite the beauty and great technical achievements of some of these reconstructions, they arguably come short of providing a fully satisfactory physical and conceptual picture of quantum theory.…”

Section: Introductionmentioning

confidence: 99%

Quantum theory from questions

2017

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We reconstruct the explicit formalism of qubit quantum theory from elementary rules on an observer's information acquisition. Our approach is purely operational: we consider an observer O interrogating a system S with binary questions and define S's state as O's 'catalogue of knowledge' about S. From the rules we derive the state spaces for N elementary systems and show that (a) they coincide with the set of density matrices over an N -qubit Hilbert space C 2 N ; (b) states evolve unitarily under the group PSU(2 N ) according to the von Neumann evolution equation; and (c) O's binary questions correspond to projective Pauli operator measurements with outcome probabilities given by the Born rule. As a by-product, this results in a propositional formulation of quantum theory. Aside from offering an informational explanation for the theory's architecture, the reconstruction also unravels new structural insights. We show that, in a derived quadratic information measure, (d) qubits satisfy inequalities which bound the information content in any set of mutually complementary questions to 1 bit; and (e) maximal sets of mutually complementary questions for one and two qubits must carry precisely 1 bit of information in pure states. The latter relations constitute conserved informational charges which define the unitary groups and, together with their conservation conditions, the sets of pure quantum states. These results highlight information as a 'charge of quantum theory' and the benefits of this informational approach. This work emphasizes the sufficiency of restricting to an observer's information to reconstruct the theory and completes the quantum reconstruction initiated in [1].

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Reconstructing Quantum Theory

Cited by 94 publications

References 46 publications

(Almost) C*-algebras as sheaves with self-action

(Almost) C*-algebras as sheaves with self-action

The Formalisms of Quantum Mechanics

Quantum theory from questions

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