On the missing axiom of Quantum Mechanicss

Ariano, G. M. D

doi:10.1063/1.2158715

Cited by 16 publications

(17 citation statements)

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“…In the present paper I show how it is possible to derive the Hilbert space formulation of Quantum Mechanics from a comprehensive definition of physical experiment and assuming experimental accessibility and simplicity as specified by five simple Postulates. This accomplishes the program presented in form of conjectures in the previous paper [1]. Pivotal roles are played by the local observability principle, which reconciles the holism of nonlocality with the reductionism of local observation, and by the postulated existence of informationally complete observables and of a symmetric faithful state.…”

mentioning

confidence: 98%

“…We also emphasize that acausality of local actions is not logically entailed by system independence (for a discussion about acausality see Ref. [1]). For the following it is convenient to extend the notion of state to that of weight, namely nonnegative bounded functionalsω over the set of transformations with 0 ≤ω(A ) ≤ω(I ) < +∞ for all transformations A .…”

Section: Definition 5 (Local State)mentioning

confidence: 99%

“…As already noticed in Ref. [1], in order to include also non-disturbing experiments, one must conceive situations in which all states are left invariant by each transformation.…”

Section: Introductionmentioning

confidence: 99%

“…As I have shown in Ref. [1], the adoption of such a general definition of experiment constitutes a very seminal point for axiomatization, entailing a thorough series of notions that are usually considered of quantum nature-such as the same probabilistic notion of state, and the notions of conditional state, local state, pure state, faithful state, instru-Thus the action/experiment is just a complete set of possible transformations that can occur in an experiment. As we can see now, in a general probabilistic framework the action A is the "cause", whereas the outcome j labeling the transformation A j that actually occurred is the "effect".…”

Section: Introductionmentioning

confidence: 99%

“…It follows that the only possible definition of state is the following Definition 2 (States) A state ω for a physical system is a rule that provides the probability for any possible transformation, namely ω : state, ω(A ) : probability that the transformation A occurs. (1) We assume that the identical transformation I occurs with probability one, namely…”

Section: Introductionmentioning

confidence: 99%

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How to Derive the Hilbert-Space Formulation of Quantum Mechanics From Purely Operational Axioms

D’Ariano

2006

AIP Conference Proceedings

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Abstract. In the present paper I show how it is possible to derive the Hilbert space formulation of Quantum Mechanics from a comprehensive definition of physical experiment and assuming experimental accessibility and simplicity as specified by five simple Postulates. This accomplishes the program presented in form of conjectures in the previous paper [1]. Pivotal roles are played by the local observability principle, which reconciles the holism of nonlocality with the reductionism of local observation, and by the postulated existence of informationally complete observables and of a symmetric faithful state. This last notion allows one to introduce an operational definition for the real version of the "adjoint"-i. e. the transpositionfrom which one can derive a real Hilbert-space structure via either the Mackey-Kakutani or the Gelfand-Naimark-Segal constructions. Here I analyze in detail only the Gelfand-Naimark-Segal construction, which leads to a real Hilbert space structure analogous to that of (classes of generally unbounded) selfadjoint operators in Quantum Mechanics. For finite dimensions, general dimensionality theorems that can be derived from a local observability principle, allow us to represent the elements of the real Hilbert space as operators over an underlying complex Hilbert space (see, however, a still open problem at the end of the paper). The route for the present operational axiomatization was suggested by novel ideas originated from Quantum Tomography.

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mentioning

confidence: 98%

Section: Definition 5 (Local State)mentioning

confidence: 99%

“…As already noticed in Ref. [1], in order to include also non-disturbing experiments, one must conceive situations in which all states are left invariant by each transformation.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

How to Derive the Hilbert-Space Formulation of Quantum Mechanics From Purely Operational Axioms

D’Ariano

2006

AIP Conference Proceedings

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Universal Gaps for XOR Games from Estimates on Tensor Norm Ratios

Aubrun¹,

Lami²,

Palazuelos³

et al. 2020

Commun. Math. Phys.

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We define and study XOR games in the framework of general probabilistic theories, which encompasses all physical models whose predictive power obeys minimal requirements. The bias of an XOR game under local or global strategies is shown to be given by a certain injective or projective tensor norm, respectively. The intrinsic (i.e. model-independent) advantage of global over local strategies is thus connected to a universal function r(n, m) called 'projective-injective ratio'. This is defined as the minimal constant ρ such that · X⊗π Y ρ · X⊗εY holds for all Banach spaces of dimensions dim X = n and dim Y = m, where X ⊗π Y and X ⊗ε Y are the projective and injective tensor products. By requiring that X = Y , one obtains a symmetrised version of the above ratio, denoted by rs(n). We prove that r(n, m) 19/18 for all n, m 2, implying that injective and projective tensor products are never isometric. We then study the asymptotic behaviour of r(n, m) and rs(n), showing that, up to log factors: rs(n) is of the order √ n (which is sharp); r(n, n) is at least of the order n 1/6 ; and r(n, m) grows at least as min{n, m} 1/8 . These results constitute our main contribution to the theory of tensor norms. In our proof, a crucial role is played by an ' 1/ 2/ ∞ trichotomy theorem' based on ideas by Pisier, Rudelson, Szarek, and Tomczak-Jaegermann. The main operational consequence we draw is that there is a universal gap between local and global strategies in general XOR games, and that this grows as a power of the minimal local dimension. In the quantum case, we are able to determine this gap up to universal constants. As a corollary, we obtain an improved bound on the scaling of the maximal quantum data hiding efficiency against local measurements. arXiv:1809.10616v2 [quant-ph] 7 Dec 2018 2

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