Search

We give a necessary and sufficient condition for the incompatibility of a pair of effects in a general probabilistic model in which the state space is a total convex space, which can be obtained by minimising a real parameter. This has an interpretation in terms of the least noise that must be included to make the given pair compatible.

We investigate two aspects of the elementary example of POVMs on the Euclidean plane, namely their status as quantum observables and their role as quantizers in the integral quantization procedure. The compatibility of POVMs in the ensuing quantum formalism is discussed, and a Naimark dilation is found for the quantum operators. The relation with Toeplitz quantization is explained. A physical situation is discussed, where we describe the linear polarization of the light with the use of Stokes parameters. In particular, the case of sequential measurements in a real bidimensional Hilbert space is addressed. An interpretation of the Stokes parameters in the framework of unsharp or fuzzy observables is given. Finally, a necessary condition for the compatibility of two dichotomic fuzzy observables which provides a condition for the approximate joint measurement of two incompatible sharp observables is found., 43 pages, 2 figures. Title changed. Matches the version published in Annals of Physics

Starting from the phase space representation of quantum mechanics we provide an Euclidean system of covariance for the photon. In particular, we consider systems with the Poincare group as the symmetry group and use a standard procedure in order to build a phase space and a localization observable on the phase space. Then we focus on the massless representations of the Poincare group that we use to build a space localization observable for the photon.

Commutative POVMs are smearing of spectral measures with the smearing realized by a Markov kernel. In the case of POVMs defined on the Borel σ-algebra of the reals, the existence of a universal Markov kernel has been established. Here we show that every weak Markov kernel is functionally subordinated to the universal Markov kernel. Then, we analyze the implications of that result to the characterization of a Markov kernel as a Choquet’s integral on the space of deterministic Markov kernels.

The real plane with its set of orientations or angles in $[0,\pi)$ is the simplest non trivial example of a (projective) Hilbert space and provides nice illustrations of quantum formalism. We present some of them, namely covariant integral quantization, linear polarisation of light as a quantum measurement, interpretation of entanglement leading to the violation of Bell inequalities, and spin one-half coherent states viewed as two entangled angles., Comment: 8 pages. Matches the published version in the AAMP Special Issue of the diamond Open-Access journal Acta Polytechnica

We use Naimark’s dilation theorem in order to characterize the joint measurability of two POVMs. Then, we analyze the joint measurability of two commutative POVMs F1 and F2 which are the smearing of two self-adjoint operators A1 and A2, respectively. We prove that the compatibility of F1 and F2 is connected to the existence of two compatible self-adjoint dilations A1+ and A2+ of A1 and A2, respectively. As a corollary we prove that each couple of self-adjoint operators can be dilated to a couple of compatible self-adjoint operators. Next, we analyze the joint measurability of the unsharp position and momentum observables and show that it provides a master example of the scheme we propose. In other words, the scheme we have in the case of the position and momentum operators can be generalized to the case of an arbitrary couple of self-adjoint operators. Finally, we give a sufficient condition for the compatibility of two effects.

We prove the existence of a universal Markov kernel, i.e., a Markov kernel µ such that every commutative POVM F is the smearing of a selfadjoint operator AF with the smearing realized through µ. The relevance of the smearing is illustrated in connection with the problem of the joint measurability of two quantum observables. Also the connections with phase space quantum mechanics is outlined.

A Positive Operator Valued Measure (POVM) is a map $F:\mathcal{B}(X)\to\mathcal{L}_s^+(\mathcal{H})$ from the Borel $\sigma$-algebra of a topological space $X$ to the space of positive self-adjoint operators on a Hilbert space $\mathcal{H}$. 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 $\nu$ 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 \cite{Busch}, they are important for a range of quantum information processing protocols, where classical post-processing plays a role \cite{Heinosaari}. 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., Comment: 26 pages. arXiv admin note: substantial text overlap with arXiv:1207.0086, arXiv:1307.5733

After a short review of the main properties of unsharp observables we provide examples of commutative unsharp observables obtained as the projection of sharp observables. In particular, we consider a family of PVMs defined on the tensor product Hilbert space H ⊗ K which include the one introduced by Ozawa in his modification of the von Neumann position measurement model and show that their projections onto H are commutative POVMs. Then, we make some observations on the concept of Naimark dilation and its connections with the integral representation of commutative POVMs. Finally, we introduce a partial order relation in the set of Naimark dilations, show that there is a minimal element and that it is unique. Minimality is a consequence of the partial ordered structure and coincides with the usual definition of minimal Naimark dilation. In passing, we use Naimark’s theorem in order to prove some well known properties of positive operator valued measures.

We investigate the algebraic properties of the operation a ∘ b = a b a on the set of all positive invertible elements of a C ⁎ -algebra A . We show that its commutativity, associativity and distributivity are each equivalent to the commutativity of A . We present abstract characterizations of the operation ∘ and a few related ones, too.

We propose, for a finite crystal, that the entire system be expressed in terms of the phase space, including momentum, configuration position and spin. This is to be done both classically and quantum mechanically. We illustrate this with a silicon crystal. Then, quantum mechanically measuring with a wire containing electrons, we obtain a theoretically good approximation for an electron in one semiconductor. As well, we outline what has to be done for any crystal.

In the present paper, we review some recent results about commutative positive operator valued measures (POVMs) and single out some open problems. We introduce a conjecture about the extension of some recent results and prove some important consequences of such conjecture. In particular, we prove that it implies the universal character of some of the mathematical objects we introduce, i.e., the fact that they do not depend on the POV measure we are considering. We analyze the relevance of this result. Finally, we point out that some of the results we present admit a constructive proof and we show the relevance of this fact to the theory of commutative POV measures.

Let F : ℬ(ℝ)→ℱ(ℋ) be a commutative positive operator-valued measure from the Borel σ-algebra of the reals to the space of bounded positive operators on the Hilbert space ℋ. It is well known that there exist a self-adjoint operator A and a Markov kernel μ (·) (λ) such that F(Δ) = μ Δ (A), where Δ ∈ ℬ(ℝ). We prove that, for any Δ ∈ ℬ(ℝ), if F(Δ) is a discrete operator, then A is a linear combination of the moments of F. This result allows us to characterize the quantum observable represented by A by means of the Naimark dilation of F. The result is connected with the following properties of the sequences of linear functionals. Consider an infinite sequence of linear functionals {T i } i ∈ ℕ , such that T i f = ∫ f(t) dμ t (i), corresponding to an infinite sequence of probability measures {μ (·) (i)} i ∈ ℕ on the Borel σ-algebra ℬ([0, 1]) such that μ (·) (i) ≠ μ (·) (j), where i, j ∈ ℕ and i≠j. There exists a real bounded one-to-one continuous function f such that

For each commutative POV measure F there exists (Beneduci, J. Math. Phys. 47:062104-1, 2006; Int. J. Geom. Methods Mod. Phys. 3:1559, 2006) a PV measure E such that F can be interpreted as a random diffusion of E. In its turn, the self-adjoint operator A=∫λ dEλ corresponding to E, can be interpreted (Beneduci, J. Math. Phys. 48:022102-1, 2007; Nuovo Cimento B 123:43–62, 2008) as the projection of a Naimark operator corresponding to the Naimark dilation E+ of F. Moreover E can be algorithmically reconstructed by F. All that suggests that, in some sense, the observables represented by E and F should have the same informational content. We introduce an equivalence relation on the set of observables which we compare with other well known equivalence relations and prove that it is the only one for which E is always equivalent to F.

Given a commutative POV measure [Formula: see text] on the Borel σ-algebra of the reals it is possible to construct a PV measure [Formula: see text] (the sharp reconstruction of F) such that the outcomes of the measurement of F can be interpreted as the random diffusion of the outcomes of the measurement of E. On the other hand Neumark's theorem ensures the existence of an extended Hilbert space [Formula: see text] and of a PV measure [Formula: see text] such that [Formula: see text] for every [Formula: see text]. We consider the projection [Formula: see text] of the self-adjoint operator [Formula: see text] corresponding to E+ and the self-adjoint operator A corresponding to the sharp reconstruction E and show the existence of a function G: [0,1] → [0,1] such that [Formula: see text]. An example for which the last relation can be inverted is shown.

We introduce a meaningful consistency criterion, called self-decoherence, for families of 2-event quantum histories, which implies weak decoherence. We show that a condition of heredity, stemming from Ludwig's interpretation of density matrix, is satisfied by self-decoherence. On the contrary, usual consistency criteria are not hereditary.

In this work, we study a copolymer-micellar system characterized by clustering processes due to a short-range attractive interaction. This originates a percolation process and a new type of kinetic glass transition. We have studied these intriguing dynamical situations by means of an extensive set of light scattering and viscoelasticity experiments. Obtained data, in both the phenomena, are accounted for by considering in a proper way fractal clustering processes and the related scaling concepts. Near the percolation line the main role in the system structure and dynamics is played by the cluster's partial screening of hydrodynamic interaction, that behaves, on approaching the percolation threshold, dramatic effects on the rheological properties and on the density decay relaxations. The ergodic–nonergodic transition line (glass transition) is studied in terms of the intermediate scattering functions (ISF) in the frame of the mode coupling theory. The measured ISF gives evidence of a logarithmic decay on the density fluctuation followed by a power law behavior. This latter phenomenon is the signature of a high-order glass transition of the A3 type (cusp-like singularity).

Recently a characterization of uniformly continuous POVMs and a necessary condition for a uniformly continuous POVM $F$ to have the norm-1 property have been provided. Moreover it was proved that in the commutative case, uniform continuity corresponds to the existence of a Feller Markov kernel. We apply such results to the analysis of some relevant physical examples; i.e., the phase space localization observables, the unsharp phase observable and the unsharp number observable of which we study the uniform continuity, the norm-1 property and the existence of a Feller Markov kernel., Comment: 16 pages. arXiv admin note: substantial text overlap with arXiv:1207.0086

Recently, the presence in the literature (also recent) of voluminous discussions about the interpretation of quantum mechanics has been labelled as scandalous. We claim that a weak point of the debate which hosted such a conclusion is a lack of a definition of the term "interpretation". In the present note, we would "like to make precise" that meaning and to show how such a clarification is necessary in order to avoid misunderstandings., Accepted by the American Journal of Physics

Our starting point is the proof of the following property of a particular class of matrices. Let T = { T i , j } be a n × m non-negative matrix such that ∑ j T i , j = 1 for each i . Suppose that for every pair of indices ( i , j ) , there exists an index l such that T i , l ≠ T j , l . Then, there exists a real vector k = ( k 1 , k 2 , … , k m ) T , k i ≠ k j , i ≠ j ; 0 k i ⩽ 1 , such that, ( T k ) i ≠ ( T k ) j if i ≠ j . Then, we apply that property of matrices to probability theory. Let us consider an infinite sequence of linear functionals { T i } i ∈ N , T i f = ∫ f ( t ) d μ t ( i ) , corresponding to an infinite sequence of probability measures { μ ( · ) ( i ) } i ∈ N , on the Borel σ -algebra B ( [ 0 , 1 ] ) such that, μ ( · ) ( i ) ≠ μ ( · ) ( j ) , i , j ∈ N , i ≠ j . The property of matrices described above allows us to construct a real bounded one-to-one piecewise continuous and continuous from the left function f such that T i f = ∫ f ( t ) d μ t ( i ) ≠ ∫ f ( t ) d μ t ( j ) = T j f , i , j ∈ N , i ≠ j . The relevance to quantum mechanics is showed.

In this paper we present a hierarchy of extended hydrodynamical models for electron transport in Silicon, which differ from each other for the number of scalar and vector moments of the electron distribution function used as state variables. The closure of the moment equations is achieved by means of the Maximum Entropy Principle. The main scattering mechanisms between electrons and phonons are taken into account. An application to the case of bulk Silicon is presented.

We show that a POV measure F on the Borel σ-algebra of the reals B(R) is commutative if and only if there exists a PV measure E on B(R) and, for every λ in the spectrum of E, a probability measure γ(∙)(λ) on B(R) such that the effect F(Δ) coincides with γΔ(A), where A is the self-adjoint operator associated to E. The relevance of this result to the theory of the sharp reconstruction is analyzed.

This paper focuses on the problem of localization in quantum mechanics. It is well known that it is not possible to define a localization observable for the photon by means of projection-valued measures, or conversely, that it is possible by using positive operator-valued measures. On the other hand, projection-valued measures imply a kind of localization which is stronger than that implied by positive operator-valued measures. It has been claimed that the norm-1 property would in some sense reduce the gap between the two kinds of localizations. We give a necessary condition for the norm-1 property and show that it is not satisfied by several important localization observables.

A commutative POV measure $F$ with real spectrum is characterized by the existence of a PV measure $E$ (the sharp reconstruction of $F$) with real spectrum such that $F$ can be interpreted as a randomization of $E$. This paper focuses on the relationships between this characterization of commutative POV measures and Neumark's extension theorem. In particular, we show that in the finite dimensional case there exists a relation between the Neumark operator corresponding to the extension of $F$ and the sharp reconstruction of $F$. The relevance of this result to the theory of non-ideal quantum measurement and to the definition of unsharpness is analyzed., 37 pages

A well defined procedure exists which allows us to “reconstruct” a sharp, i.e., standard, quantum observable A starting from a given commutative unsharp observable F. In this work we prove that the outcomes of measurements of F can be consistently interpreted as the result of a stochastic diffusion of outcomes of its sharp reconstruction A. Furthermore, for every sharp observable B, such that F is unsharp realization of B, we explicitly construct a real mapping g such that A=g(B).