Naimark Theorem Research Articles

Quantum formalism on the plane: POVM-Toeplitz quantization, Naimark theorem and linear polarization of the light

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.

On description of dual frames

One of key problems in signal reconstruction process with the use of frames is to find a dual frame. Typically, a canonical dual frame is used. However, there are many applications where this choice appears to be unfortunate. Due to that fact, it is necessary to develop a tool, which helps to find a suitable dual frame. In this paper we give a method to find every dual frames. The proposed method is based on Naimark's dilation theorem and the obtained description of dual frames involves parameters that characterize extension of a Parseval frame to an orthonormal basis. These formulas are simplified for frames in finite-dimensional spaces and for near-Riesz bases. In the latter case, the simplification is based on the extended and supplemented version of the Naimark theorem, which is proved in the last part of the paper.

On properties of real selfadjoint operators

In spite of the important applications of real selfadjoint operators and monotone operators, very few papers have dealt in depth with the properties of such operators. In the present paper, we follow A. Rhodius to define the spectrum $$\sigma _{\mathbb {F}}(T)$$ and the numerical range $$W_{\mathbb {F}}(T)$$ of a selfadjoint operator T acting on a Hilbert space H over the real/complex field $$\mathbb {F}$$ , and study their topological and geometrical properties which are well known in the complex case $${\mathbb {F}}={\mathbb {C}}$$ . If $$\mathbb F={\mathbb {R}}$$ , the results are new; if $${\mathbb {F}}={\mathbb {C}}$$ , the results constitute an expository body of results containing simple and short proofs of the known facts. The results are then applied to real selfadjoint operators and then to complex normal operators to sharpen their Borel functional calculi with new and shorter proofs avoiding the classical sophisticated Gelfand–Naimark theorem or the Berberian’s amalgamation theory. For such a real selfadjoint or complex normal operator N, a normed functional algebra $$L^\infty _{\mathbb {F}}(N)$$ consisting of certain Borel functions defined on $$\sigma _{\mathbb {F}}(N)$$ is constructed which inherits the isometric properties of the continuous functional calculus $$f\mapsto f(N):C_{\mathbb {F}}(\sigma _{\mathbb {F}}(N))\rightarrow B(H)$$ .

Naimark extension for the single-photon canonical phase measurement

We address the implementation of the positive operator-valued measure (POVM) describing the optimal M-outcomes discrimination of the polarization state of a single photon. Initially, the POVM elements are extended to projective operators by Naimark theorem, then the resulting projective measure is implemented by a Knill-Laflamme-Milburn scheme involving an optical network and photon counters. We find the analytical expression of the Naimark extension and the detection scheme that realise it for an arbitrary number of outcomes M = 2^N.

Implementing positive-operator-valued-measurement elements in photonic circuits for performing minimum quantum state tomography of path qudits

Manipulation of qudits in optical tables is a difficult and nonscalable task. The use of integrated optical circuits opens new possibilities for the generation, manipulation, and characterization of high dimensional states besides the ease of transmission of these states through an optical fiber. In this work we propose photonic circuits to perform minimum quantum state tomography of path qudits and show how to determine all the constituents parameters of these circuits (beam splitters and phase shifters). Our strategies were based on the symmetries of the involved POVMs (positive operator-valued measures) suggested for minimum tomography and allowed us to obtain interferometers smaller than those obtained by other already known methods. The calculations of the transmittances and reflectivities of the beam splitters were made using the definition of probability operators in extended Hilbert spaces and the application of Naimark's theorem. The employment of equidistant states for the definition of the POVM elements allowed us to develop a recipe applicable to the tomography of qudits of any dimension, generalizing our scheme.

Projections and dyadic Parseval frame MRA wavelets

A classical theorem attributed to Naimark states that, given a Parseval frame B in a Hilbert space H, one can embed H in a larger Hilbert space K so that the image of B is the projection of an orthonormal basis for K. In the present work, we revisit the notion of Parseval frame MRA wavelets from [12] and [13] and produce an analog of Naimark's theorem for these wavelets at the level of their scaling functions. We aim to make this discussion as self-contained as possible and provide a different point of view on Parseval frame MRA wavelets than that of [12] and [13].

On dense ideals of C*-algebras and generalizations of the Gelfand–Naimark Theorem

We develop the theory of Segal algebras of commutative $C^\ast $-algebras, with an emphasis on the functional representation. Our main results extend the Gelfand–Naimark Theorem. As an application, we describe faithful principal ideals of $C^\ast $-

The modern tools of quantum mechanics

We address the basic postulates of quantum mechanics and point out that they are formulated for a closed isolated system. Since we are mostly dealing with systems that interact or have interacted with the rest of the universe one may wonder whether a suitable modification is needed, or in order. This is indeed the case and this tutorial is devoted to review the modern tools of quantum mechanics, which are suitable to describe states, measurements, and operations of realistic, not isolated, systems. We underline the central role of the Born rule and and illustrate how the notion of density operator naturally emerges, together with the concept of purification of a mixed state. In reexamining the postulates of standard quantum measurement theory, we investigate how they may be formally generalized, going beyond the description in terms of selfadjoint operators and projective measurements, and how this leads to the introduction of generalized measurements, probability operator-valued measures (POVMs) and detection operators. We then state and prove the Naimark theorem, which elucidates the connections between generalized and standard measurements and illustrates how a generalized measurement may be physically implemented. The “impossibility” of a joint measurement of two non commuting observables is revisited and its canonical implementation as a generalized measurement is described in some details. The notion of generalized measurement is also used to point out the heuristic nature of the so-called Heisenberg principle. Finally, we address the basic properties, usually captured by the request of unitarity, that a map transforming quantum states into quantum states should satisfy to be physically admissible, and introduce the notion of complete positivity (CP). We then state and prove the Stinespring/Kraus-Choi-Sudarshan dilation theorem and elucidate the connections between the CP-maps description of quantum operations, together with their operator-sum representation, and the customary unitary description of quantum evolution. We also address transposition as an example of positive map which is not completely positive, and provide some examples of generalized measurements and quantum operations.

Approximately norm-unital products on [formula omitted]-algebras, and a non-associative Gelfand–Naimark theorem

We describe the non-associative products on a C ⁎ -algebra A which convert the Banach space of A into a Banach algebra having an approximate unit bounded by 1, and determine among them those which are associative. As a consequence, if such a product p satisfies p ( a , b ) □ = p ( b □ , a □ ) and ‖ p ( a □ , a ) ‖ = ‖ a ‖ 2 , for all a , b ∈ A and some conjugate-linear vector space involution □ on A, then p is associative. The proof of the above result involves also a new Gelfand–Naimark type theorem asserting that non-associative C ⁎ -algebras (defined verbatim as in the associative case, but removing associativity) are alternative if and only if they have an approximate unit bounded by 1.

Characterization of self-adjoint ordinary differential operators

Symmetric differential expressions ℓ of order n = 2 k with real valued coefficients give rise to self adjoint operators in the space of weighted square integrable functions. Characterization theorems exist in the literature that describe such self-adjoint operators. All such characterizations begin by constructing the maximal domain of definition of the expression ℓ . The Glazman–Krein–Naimark theorem constructs the maximal domain in terms of eigenfunctions corresponding to a nonreal parameter λ . Representations in terms of certain functions related to a real parameter λ can also be found in the literature. In this paper we construct the maximal domain from two complementary self-adjoint realizations of ℓ . One operator is assumed to be known and the other one is computed explicitly. From these two operators we explicitly give all other self-adjoint operators associated with ℓ . A special class of operators associated with ℓ is what we call Type I operators. They arise in connection with a certain bilinear form that results from the weak formulation of the expression ℓ . Depending on the deficiency index of ℓ and the properties of the bilinear form we can have two complementary self-adjoint operators (two Type I operators) and, as it turns out, one of them is the celebrated Friedrich Extension. The other operator appears to be new. As in the general case, using these two operators we give an explicit characterization of all other operators of the same Type I.

Unsharpness, Naimark Theorem and Informational Equivalence of Quantum Observables

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.

POVMs and Naimark's Theorem Without Sums

We provide a definition of POVM in terms of abstract tensor structure only. It is justified in two distinct manners. i. At this abstract level we are still able to prove Naimark's theorem, hence establishing a bijective correspondence between abstract POVMs and abstract projective measurements (cf. [B. Coecke and D. Pavlovic (2007) Quantum measurements without sums. In: Mathematics of Quantum Computing and Technology. Chapman & Hall, pp. 559–596. E-print available from arXiv:quant-ph/0608035]) on an extended system, and this proof is moreover purely graphical. ii. Our definition coincides with the usual one for the particular case of the Hilbert space tensor product. We also provide a very useful normal form result for the classical object structure introduced in [B. Coecke and D. Pavlovic (2007) Quantum measurements without sums. In: Mathematics of Quantum Computing and Technology. Chapman & Hall, pp. 559–596. E-print available from arXiv:quant-ph/0608035].

The Gelfand-Naimark theorem for C*-algebras over a ring of measurable functions

One of the most important results in the theory of Banach algebras is the Gelfand–Naimark theorem. It describes a commutative C∗-algebra U over the field of complex numbers C as an algebra of continuous complex-valued functions defined on the set of pure states of U endowed with an ∗-weak topology. The development of the general theory of Banach–Kantorovich C∗-algebras over a ring of measurable functions naturally leads to the question about an analog of the Gelfand–Naimark theorem for such C∗-modules. The structural theory of C∗-modules originates from the work of I. Kaplansky [1], who used these objects in the algebraic approach to the theory of W ∗-algebras. Considering C∗-algebras, AW ∗-algebras, andW ∗-algebras as modules over their centers, one can describe some properties of the mentioned classes of ∗-algebras with the help of Boolean analysis methods (e.g., [2–5]). In particular, in [4] for C∗-algebras, representing modules over the Stone algebra, one obtains vector analogs of Gelfand–Mazur and Gleason–Zhelyazko–Kahan theorems. C∗-modules serve as useful examples of Banach–Kantorovich modules, whose theory is being actively developed now (e.g., [5, 6]). An efficient tool for the study of these Banach–Kantorovich modules, together with the Boolean-valued analysis, is the theory of continuous and measurable Banach bundles [6]. In particular, this enables one to represent a C∗-module over a ring of measurable functions as a measurable bundle of classical C∗-algebras [7]. This, in turn, allows one to establish properties ofC∗-modules, “gluing” the corresponding properties of C∗-algebras over the field C. In this paper we apply this approach, proving one version of the Gelfand– Naimark theorem for C∗-modules. We use the terminology and the notation of the theory of Banach– Kantorovich spaces [5] and the theory of measurable bundles stated in [6].

On quaternionic functional analysis

AbstractIn this paper, we will show that the category of quaternion vector spaces, the category of (both one-sided and two sided) quaternion Hilbert spaces and the category of quaternion B*-algebras are equivalent to the category of real vector spaces, the category of real Hilbert spaces and the category of real C*-algebras respectively. We will also give a Riesz representation theorem for quaternion Hilbert spaces and will extend the main results in [12] (namely, we will give the full versions of the Gelfand–Naimark theorem and the Gelfand theorem for quaternion B*-algebras). On our way to these results, we compare, clarify and unify the term ‘quaternion Hilbert spaces’ in the literatures.

Entanglement cost of generalised measurements

Bipartite entanglement is one of the fundamental quantifiable resources of quantum information theory. We propose a new application of this resource to the theory of quantum measurements. According to Naimark's theorem any rank 1 generalised measurement (POVM) M may be represented as a von Neumann measurement in an extended (tensor product) space of the system plus ancilla. By considering a suitable average of the entanglements of these measurement directions and minimising over all Naimark extensions, we define a notion of entanglement cost E_{\min}(M) of M. We give a constructive means of characterising all Naimark extensions of a given POVM. We identify various classes of POVMs with zero and non-zero cost and explicitly characterise all POVMs in 2 dimensions having zero cost. We prove a constant upper bound on the entanglement cost of any POVM in any dimension. Hence the asymptotic entanglement cost (i.e. the large n limit of the cost of n applications of M, divided by n) is zero for all POVMs. The trine measurement is defined by three rank 1 elements, with directions symmetrically placed around a great circle on the Bloch sphere. We give an analytic expression for its entanglement cost. Defining a normalised cost of any $d$-dimensional POVM by E_{\min} (M)/\log_2 d, we show (using a combination of analytic and numerical techniques) that the trine measurement is more costly than any other POVM with d>2, or with d=2 and ancilla dimension 2. This strongly suggests that the trine measurement is the most costly of all POVMs.

A Gelfand–Naimark theorem for C*-algebras

A Gelfand–Naimark theorem for C*-algebras

Topological function-theoretic proofs in spectral theory

In this paper, the spectral theorem and related characterizations of the spectrum and the spectral projections for bounded self adjoint and normal operators on a Hilbert space, are proved in purely topological —function theoretic terms. The basis for such a development, is the Gelfand—Naimark theorem for commutativeC *-algebras and the fact that the structure space of the (abelian) von Neumann algebra generated by the operator is a Stonean space.

New proof of Naimark's theorem on the existence of nonpositive invariant subspaces for commuting families of unitary operators in Pontryagin spaces

In the present note we give a new and short proof of Naimark's theorem asserting that for every commuting family ℱ of unitary operators in a πk-space Πk there exists ak-dimensional, nonpositive subspace invariant under ℱ.

Compatibility of observables represented by positive operator-valued measures

The proof of a result analogous to that in Koelman and de Muynck [Phys. Lett. A 98, 1 (1983)] is given for the case of unbounded observables. If two, not necessarily bounded, observables are represented by a positive operator-valued measure, then the measurement of any of them is undisturbed if and only if they commute. The Naimark theorem on dilations of spectral functions is exploited. A stronger version of Wigner’s theorem is given.

A theorem of Naimark, linear systems, and scattering operators

Naimark's theorem on the dilation of Toeplitz kernels and its extension for generalized Toeplitz kernels provide a unified approach to some questions concerning couplings of isometries and the characterization of the response function of a linear system with a contractive propagator.