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Asymptotic Spectral Measures: Between Quantum Theory and E

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Asymptotic Spectral Measures: Between Quantum Theory and E Asymptotic Spectral Measures: Between Quantum Theory and E-theory Jody Trout Department of Mathematics Dartmouth College Hanover, NH 03755 Email: [email protected] Abstract— We review the relationship between positive of classical observables to the C∗-algebra of quantum operator-valued measures (POVMs) in quantum measurement observables. See the papers [12]–[14] and theB books [15], C∗ E theory and asymptotic morphisms in the -algebra -theory of [16] for more on the connections between operator algebra Connes and Higson. The theory of asymptotic spectral measures, as introduced by Martinez and Trout [1], is integrally related K-theory, E-theory, and quantization. to positive asymptotic morphisms on locally compact spaces In [1], Martinez and Trout showed that there is a fundamen- via an asymptotic Riesz Representation Theorem. Examples tal quantum-E-theory relationship by introducing the concept and applications to quantum physics, including quantum noise of an asymptotic spectral measure (ASM or asymptotic PVM) models, semiclassical limits, pure spin one-half systems and quantum information processing will also be discussed. A~ ~ :Σ ( ) { } ∈(0,1] →B H I. INTRODUCTION associated to a measurable space (X, Σ). (See Definition 4.1.) In the von Neumann Hilbert space model [2] of quantum Roughly, this is a continuous family of POV-measures which mechanics, quantum observables are modeled as self-adjoint are “asymptotically” projective (or quasiprojective) as ~ 0: → operators on the Hilbert space of states of the quantum system. ′ ′ A~(∆ ∆ ) A~(∆)A~(∆ ) 0 as ~ 0 The Spectral Theorem relates this theoretical view of a quan- ∩ − → → tum observable to the more operational one of a projection- for certain measurable sets ∆, ∆′ Σ. ∈ valued measure (PVM or spectral measure) which determines Let X be the locally compact phase space of a classical ∗ the probability distribution of the experimentally measurable system. Let C0(X) denote the C -algebra of continuous values of the observable. In order to solve foundational prob- functions vanishing at infinity on X. Let be the Hilbert state H lems with the concept of measurement and to better analyze space of the quantum version of the system. Let denote a B unsharp results in experiments, this view was generalized to hereditary C∗-subalgebra of ( ). One of their main results B H include positive operator-valued measures (POVMs). Since the is an “asymptotic” Riesz representation theorem (Theorem initial work of Jauch and Piron [3], POV-measures have played 5.2) which gives a bijective correspondence between certain an ever increasing role in both the foundations and operational positive asymptotic morphisms (i.e. “quantizations”) aspects of quantum physics [4], [5]. Q~ ~ : C (X) In this paper, we review the relationship between POVMs { } ∈(0,1] 0 →B arXiv:math-ph/0309050v1 19 Sep 2003 and asymptotic morphisms in the operator algebra E-theory and certain Borel asymptotic spectral measures of Connes and Higson [6], which has already found many A~ ~ : (ΣX , X ) ( ( ), ) applications in mathematics [7]–[10], most notably to classifi- { } ∈(0,1] C → B H B cation problems in operator K-theory, index theory, represen- where ΣX denote the Borel subsets of X and X denotes the tation theory, geometry, and topology. The basic ingredients open subsets of X with compact closure. This correspondenceC of E-theory are asymptotic morphisms, which are given by is given by an expectation: For all φ in and f in C0(X), continuous families of functions | i H φ Q~(f) φ = f(x) d φ A~(x) φ . Q~ ~ : h | | i Z h | | i { } ∈(0,1] A→B X ∗ ∗ from a C -algebra to a C -algebra that satisfy the axioms The associated asymptotic morphism Q~ : C0(X) then { } →B of a -homomorphismA in the limit asB the parameter ~ 0. allows one to define an E-theory invariant for the asymptotic Asymptotic∗ multiplicativity is a modern version of the Bohr→ - spectral measure A~ (and hence the quantum system), { } von Neumann correspondence principle [11] from quantization [[A~]] =def [[Q~]] E(C0(X), ) = E0(X; ), theory: For all f,g in , ∈ B ∼ B A in the E-homology group of X with coefficients in . Q~(fg) Q~(f)Q~(g) 0 as ~ 0. B − → → However, we will not discuss such E-theory invariants here, It is then no surprise that quantization schemes may naturally preferring to focus more on the relationship between ASMs define asymptotic morphisms, say, from the C∗-algebra and operational quantum physics. A The outline of this paper is as follows. In Section II we where (S T ) φ = S(T φ ) denotes the composition of review the basic Hilbert space model of quantum measure- operators S◦ and| Ti. It is very| i closely related to the conjugate ment theory and its relationship with POVMs and C∗-algebra transpose of a complex matrix [17]. A very important result quantization theory. Positive asymptotic morphisms associated is that the C∗-identity holds: For all T in ( ), to hereditary and nuclear C∗-algebras are discussed in Sec- B H T 2 = T † 2 = T † T 2. (2) tion III. The basic definitions and properties of asymptotic k k k k k ◦ k spectral measures are reviewed in Section IV. Asymptotic It follows that ( )has the structure of a C∗-algebra. (See Riesz representation theorems and some of their consequences subsection II-DB belowH for a discussion of C∗-algebras and are contained in Section V. Examples and applications of quantum theory.) ASMs associated to various aspects of quantum physics are An operator T is called self-adjoint (Hermitian) if T = T † discussed in Section VI, e.g., constructing ASMs from PVMs or equivalently, if for all φ and ψ in , | i | i H by quantum noise models, quasiprojectors and semiclassical ∗ 1 φ T ψ = ψ T φ . limits, and unsharp spin measurements of spin- 2 particles h | | i h | | i (including an example from quantum cryptography). For example, the identity operator I on given by I φ = φ is self-adjoint I = I† and has operatorH norm one I| i=1.| Ifi II. POV-MEASURES AND QUANTUM THEORY T is self-adjoint, then T has real spectrum k k A. Review of Hilbert Space Theory spec(T )=def λ C : T λI not invertible in ( ) R We begin by reviewing basic Hilbert space theory, mainly { ∈ − B H }⊂ to fix terminology and notation, especially the Dirac bra-ket An operator T ( ) is called positive, denoted T 0, if notation, which is more common in physics than pure math. ∈B H ≥ A Hilbert space [17], [18] is a complex vector space φ T φ 0 H h | | i≥ equipped with an inner product for all φ in . It follows that T = T † and has nonnegative spectrum| ispec(HT ) [0, ). A density operator is a positive : C ⊂ ∞ h·|·i H×H→ operator T 0 in ( ) of unit trace: trace(T )=1. Note that ≥ B H such that is complete with respect to the norm defined by if dim( )= , then there are operators without finite trace H [18]. ThereH is∞ a partial order on ( ) defined by S T on φ = φ φ ( ) iff T S 0. There≤ is alsoB anH orthogonality relation≤ k k ph | i B H − ≥ for all vectors (kets) φ in . The inner product has the given by S T if and only if range(S) range(T ). This ⊥ ⊥ ⊥ † † following properties: | i H is equivalent to the algebraic condition that S T = T S =0. Positivity: φ φ 0 and φ φ =0 iff φ =0, B. The von Neumann Model of Quantum Mechanics • h | i≥ h | i | i Linearity: φ αψ1 + βψ2 = α φ ψ1 + β φ ψ2 , A quantum-mechanical system [2], [19], [20] is modeled by • Skew-symmetry:h | φ ψ ∗ =i ψ hφ |, i h | i a Hilbert space which forms the state space of the system. • h | i h | i A(pure) state ofH the system is given by a unit vector φ in , for all φ , ψ , ψi in and complex numbers α, β in C, | i H | i | i | i H i.e., a vector of norm one φ =1. In general, a state of the where α∗ = a bi denotes the conjugate of the complex k k − system is given by a density operator ρ on . For example, a number α = a + bi where i2 = 1. H − pure state φ can be represented by the density operator An operator on is a linear transformation | i H ρ = φ φ , T : | ih | H → H φ T φ . where ρ ψ = φ φ ψ = φ ψ φ . (Recall that the ’bra’ | i 7→ | i φ : | iC is| theih dual| i of theh vector| i| i (ket) φ and is defined The operator T is called bounded (= continuous) if the byh | ψH → φ ψ .) More generally, if a quantum| i system can operator norm of T is finite, i.e., be in| i pure 7→ statesh | i T = sup T φ : φ , φ =1 < . (1) φ , φ , , φn , k k {k k | i ∈ H k k } ∞ | 1i | 2i ··· | i The set of all bounded linear operators T on is denoted by (which need not be orthogonal) each with respective (classical) H ( ). The adjoint of an operator T in ( ) is the (unique) probabilities B H † B H operator T : in ( ) such that 0 p ,p , ,pn 1, H → H B H ≤ 1 2 ··· ≤ † ∗ φ T ψ = ψ T φ where p1 + p2 + + pn = 1, then the density operator h | | i h | | i representing this mixed··· state is given by for all φ and ψ in . The adjoint operation T T † | i | i H 7→ satisfies the axioms of an involution: For all T ( ), ρ = p1 φ1 φ1 + p2 φ2 φ2 + + pn φn φn .
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