Joint measurability through Naimark’s dilation theorem

Beneduci, Roberto

doi:10.1016/s0034-4877(17)30035-6

Cited by 16 publications

(17 citation statements)

References 36 publications

(53 reference statements)

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“…Consider the unital * -algebra A of differential operators acting on the function of the invariant space S , made of all finite complex linear combinations of finite compositions in arbitrary order of (i) the operator P := −i d dx representing the momentum algebraic observable for a particle confined in the box [0, 1], (ii) smoothed position algebraic observables represented by multiplicative operators f • induced by real-valued functions f ∈ S , and (iii) the constantly 1 function again acting multiplicatively and also defining the unit of the algebra. The involution is A * := A † ↾ S ( † being the adjoint in L 2 ([0, 1], dx) ⊃ S ) 2 The converse does not hold, since ker(πω) is a two-sided * -ideal and thus ker(πω) G (A,ω) in the general case.…”

Section: Examplementioning

confidence: 99%

“…, a n with associated joint measures on R n accounting for the expectation-value interpretation. The many-variables moment problem is not a straightforward generalization of the one-variable moment problem [21] and also the notion of joint POVM presents some non-trivial technical difficulties [2]. Already at the level of selfadjoint observables, commutativity of symmetric operators (say π ω (a 1 ) and π ω (a 2 )) on a dense invariant domain of essential selfadjointness (D ω ) does not imply the much more physically meaningful commutativity of their respective PVMs (as proved by Nelson [24]) and the existence of a joint PVM.…”

Section: Open Issuesmentioning

confidence: 99%

See 1 more Smart Citation

The notion of observable and the moment problem for $$*$$-algebras and their GNS representations

Drago

Moretti

2020

Lett Math Phys

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We address some usually overlooked issues concerning the use of * -algebras in quantum theory and their physical interpretation. If A is a * -algebra describing a quantum system and ω : A → C a state, we focus in particular on the interpretation of ω(a) as expectation value for an algebraic observable a = a * ∈ A, studying the problem of finding a probability measure reproducing the moments {ω(a n )} n∈N . This problem enjoys a close relation with the selfadjointness of the (in general only symmetric) operator π ω (a) in the GNS representation of ω and thus it has important consequences for the interpretation of a as an observable. We provide physical examples (also from QFT) where the moment problem for {ω(a n )} n∈N does not admit a unique solution. To reduce this ambiguity, we consider the moment problem for the sequences {ω b (a n )} n∈N , being b ∈ A and ω b (·) := ω(b * · b). Letting µ

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Section: Examplementioning

confidence: 99%

Section: Open Issuesmentioning

confidence: 99%

The notion of observable and the moment problem for $$*$$-algebras and their GNS representations

Drago

Moretti

2020

Lett Math Phys

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“…Recall that a POVM is a set of positive semi-definite observables that sum to the identity. The original result of Naimark is the scheme by which any POVM, or, generalized resolution of the identity, is realized as a set of commuting projectors on a larger space; see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Shortly after the derivation by Naimark, an extension was presented by Sz.-Nagy [3] such that:…”

Section: Naimark's Theorem and The Sz-nagy Extensionmentioning

confidence: 99%

“…For these reasons POVMs are often assigned the coveted status of most general type of quantum measurement possible, and are often a starting point for foundational discussions in quantum information theory. Background POVM details are given in [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and proofs of Naimark's result appear in [3][4][5][6][7][8].…”

Section: Introductionmentioning

confidence: 99%

POVMs and the Two Theorems of Naimark and Sz.-Nagy

Malley¹,

Fletcher²

2015

Axioms

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In 1940 Naimark showed that if a set of quantum observables are positive semi-definite and sum to the identity then, on a larger space, they have a joint resolution as commuting projectors. In 1955 Sz.-Nagy showed that any set of observables could be so resolved, with the resolution respecting all linear sums. Crucially, both resolutions return the correct Born probabilities for the original observables. Here, an alternative proof of the Sz.-Nagy result is given using elementary inner product spaces. A version of the resolution is then shown to respect all products of observables on the base space. Practical and theoretical consequences are indicated. For example, quantum statistical inference problems that involve any algebraic functionals can now be studied using classical statistical methods over commuting observables. The estimation of quantum states is a problem of this type. Further, as theoretical objects, classical and quantum systems are now distinguished by only more or less degrees of freedom.Comment: Accepted at Axioms, 1 September 201

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“…Moreover, they are important for a range of quantum information processing protocols, where classical post-processing plays a role [30]. Another relevant application of commutative POVMs is the smearing of incompatible observables in order to get compatible observables (see [22,16]). All that explains the relevance of commutative POVMs both form the mathematical and the physical viewpoint.…”

Section: Introductionmentioning

confidence: 99%

Positive Operator Valued Measures and Feller Markov kernels

Beneduci

2016

Journal of Mathematical Analysis and Applications

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A Positive Operator Valued Measure (POVM) is a map F : B(X ) → L + s (H ) from the Borel σ -algebra of a topological space X to the space of positive selfadjoint operators on a Hilbert space H . We assume X to be Hausdorff, locally compact and second countable and prove that a POVM F is commutative if and only if it is the smearing of a spectral measure E by means of a Feller Markov kernel. Moreover, we prove that the smearing can be realized by means of a strong Feller Markov kernel if and only if F is uniformly continuous. Finally, we prove that a POVM which is norm bounded by a finite measure ν admits a strong Feller Markov kernel.That provides a characterization of the smearing which connects a commutative POVM F to a spectral measure E and is relevant both from the mathematical and the physical viewpoint since smearings of spectral measures form a large and very relevant subclass of POVMs: they are paradigmatic for the modeling of certain standard forms of noise in quantum measurements, they provide optimal approximators as marginals in joint measurements of incompatible observables [21], they are important for a range of quantum information processing protocols, where classical post-processing plays a role [30].The mathematical and physical relevance of the results is discussed and particular emphasis is given to the connections between the Markov kernel and the imprecision of the measurement process.

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Joint measurability through Naimark’s dilation theorem

Cited by 16 publications

References 36 publications

The notion of observable and the moment problem for $$*$$-algebras and their GNS representations

The notion of observable and the moment problem for $$*$$-algebras and their GNS representations

POVMs and the Two Theorems of Naimark and Sz.-Nagy

Positive Operator Valued Measures and Feller Markov kernels

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