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Spectral Theorem Approach to the Characteristic Function of Quantum Communications on Stochastic Analysis Volume 13 | Number 2 Article 3 6-2019 Spectral Theorem Approach to the Characteristic Function of Quantum Observables Andreas Boukas Universitá di Roma Tor Vergata, Via di Torvergata, Roma, Italy, [email protected] Philip J. Feinsilver Department of Mathematics, Southern Illinois University, Carbondale, Illinois, USA, [email protected] Follow this and additional works at: https://digitalcommons.lsu.edu/cosa Part of the Analysis Commons, and the Other Mathematics Commons Recommended Citation Boukas, Andreas and Feinsilver, Philip J. (2019) "Spectral Theorem Approach to the Characteristic Function of Quantum Observables," Communications on Stochastic Analysis: Vol. 13 : No. 2 , Article 3. DOI: 10.31390/cosa.13.2.03 Available at: https://digitalcommons.lsu.edu/cosa/vol13/iss2/3 Communications on Stochastic Analysis Vol. 13, No. 2 (2019) Article 3 (27 pages) DOI: 10.31390/cosa.13.2.03 SPECTRAL THEOREM APPROACH TO THE CHARACTERISTIC FUNCTION OF QUANTUM OBSERVABLES ANDREAS BOUKAS* AND PHILIP FEINSILVER Abstract. Using the spectral theorem we compute the Quantum Fourier Transform or Vacuum Characteristic Function hΦ; eitH Φi of an observable H defined as a self-adjoint sum of the generators of a finite-dimensional Lie algebra, where Φ is a unit vector in a Hilbert space H. We show how Stone's formula for computing the spectral resolution of a Hilbert space self-adjoint operator, can serve as an alternative to the traditional reliance on splitting or disentanglement formulas for the operator exponential. 1. Introduction The simplest quantum analogue of a classical probability space (Ω; σ; µ) where Ω is a set, σ is a sigma-algebra of subsets of Ω and µ is a measure defined on σ with µ(Ω) = 1, is a finite dimensional quantum probability space [14] defined as a triple (H; P(H); ρ) where H is a finite dimensional Hilbert space, P(H) is the set of projections (called events) E : H!H and ρ : H!H is a state on H, i.e., a positive operator of unit trace. We call trρE the probability of the event E in the state ρ. For a quantum observable H, i.e. for a symmetric or Hermitian matrix H, the characteristic function or Fourier transform of H in the state ρ is defined as trρeitH. If ρ is a pure state defined in terms of a unit vector u, i.e., if ρ = juihuj itH then the characteristic functionP of H in the state defined by u is hu; e ui. By the spectral theorem, if H = λnEn then for every continuous function ϕ : R ! C, n X ϕ(H) = ϕ(λn)En: n Therefore, for ϕ(H) = eitH we have X X itH itλn itλn hu; e ui = hu; e Enui = e hu; Enui; n n where we have assumed that the inner product is linear in the second and conjugate linear in the first argument. If the Hilbert space H is infinite dimensional then the Received 2019-7-1; Accepted 2019-8-23; Communicated by the editors. 2010 Mathematics Subject Classification. Primary 81Q10, 47B25, 47B15, 47A10; Secondary 47B40, 47B47, 80M22. Key words and phrases. Quantum Fourier transform, vacuum characteristic function, quan- tum observable, Stone's formula, spectral resolution, spectral integral, unbounded self-adjoint operator, multiplication and differentiation operator, operator exponential, Lie algebra, Heisen- berg algebra, splitting formula, disentanglement. * Corresponding author. 1 2 ANDREAS BOUKAS AND PHILIP FEINSILVER above sums are replaced by spectral integrals with respect to a resolution of the identity fEλ : λ 2 Rg and we have the corresponding formulas Z Z H = λ dEλ ; ϕ(H) = ϕ(λ) dEλ R R and Z itH itλ hu; e ui = e dhu; Eλui : R For compact self-adjoint operators H, the above spectral integrals are reduced to finite or infinite sums over the nonzero eigenvalues of H, see e.g. [17], Theorem 4.2. The above probabilistic interpretation is based on Bochner's theorem (see [18] p. 346) which states that a positive definite continuous function f : R 7! C , i.e., a continuous function f such that Z Z f(t − s)ϕ(t)ϕ¯(s) dt ds ≥ 0; R R for every continuous function ϕ : R 7! C with compact support, can be represented as Z f(t) = eitλ dv(λ); R where v is a non-decreasing right-continuous bounded function. If f(0) = 1 then such a function v defines a probability measure on R and Bochner's theorem says that f is the Fourier transform of a probability measure, i.e., the characteristic function of a random variable that follows the probability distribution defined by v. Moreover, the condition of positive definiteness of f is necessary and sufficient for such a representation. The function f(t) = hu; eitH ui, where u is a unit vector and H is a self-adjoint operator as described above, is an example of such a positive definite function. In this paper we use this spectral theorem based approach to compute the characteristic function of several quantum random variables H defined as self- adjoint sums of the generators of some finite dimensional Lie algebras of interest in quantum mechanics (the only reason why a Lie structure is assumed is because splitting the exponential of a sum of operators is usually done through a Campbell- Baker-Hausdorff type formula that relies on commutation relations). n n If H = R or H = C for vectors u = (u1; :::; un) and v = (v1; :::; vn) in H we will use the standard inner products hu; vi = uvT and hu; vi =uv ¯ T respectively. The identity matrix/operator is denoted by I, while δ denotes Dirac's delta function defined, for a test function ϕ, by Z Z δ(x − a)ϕ(x) dx = δ(a − x)ϕ(x) dx = ϕ(a): R R We define the Fourier transform of f by Z 1 f^(t) = (Uf)(t) = (2π)−1=2 eiλtf(λ)dλ, −∞ SPECTRAL THEOREM APPROACH TO QUANTUM OBSERVABLES 3 and the inverse Fourier transform of f^ by ( ) Z 1 f(λ) = U −1f^ (λ) = (2π)−1=2 e−iλtf^(t)dt; −∞ so the inverse Fourier transform of (2π)−1=2 hu; eitH ui gives the probability density function p(λ) of H. For a 2 C, let R(a; H) = (a − H)−1 denote the resolvent of an operator H. The spectral resolution fEλ j λ 2 Rg of a bounded or unbounded self-adjoint operator H in a complex separable Hilbert Space H, is given by Stone's formula (see [5], Theorems X.6.1 and XII.2.10)) Z 1 b−δ E ((a; b)) = lim lim (R(t − ϵi; H) − R(t + ϵi; H)) dt; + + δ!0 ϵ!0 2πi a+δ − where (a; b) is the open interval a < λ < b, R(t ϵi; H) = (t ϵi − H) 1, and the limit is in the strong operator topology. For a ! −∞ and b = λ we have Eλ = E ((−∞; λ]) = lim E ((−∞; λ + ρ)) ρ!0+ Z 1 λ+ρ−δ = lim lim lim (R(t − ϵi; H) − R(t + ϵi; H)) dt: ρ!0+ δ!0+ ϵ!0+ 2πi −∞ In particular (see [16], Theorem 4.31), for f; g 2 H, Z 1 λ+ρ−δ hf; Eλgi = lim lim lim hf; (R(t − ϵi; H) − R(t + ϵi; H)) gi dt: ρ!0+ δ!0+ ϵ!0+ 2πi −∞ Thus, for a unit vector u, the vacuum resolution of the identity (terminology coming from the case when u = Φ, the vacuum vector in a Fock space) of the operator H is given by Z 1 λ+ρ−δ hu; Eλui = lim lim lim hu; (R(t − ϵi; H) − R(t + ϵi; H)) ui dt: ρ!0+ δ!0+ ϵ!0+ 2πi −∞ (1.1) In Section 9 we will show how, using formula (1.1), we can avoid the reliance on splitting or disentanglement lemmas, such as Lemma 8.3 of Section 8 for the splitting of operator exponentials, in order to compute the characteristic function of a quantum random variable. In particular, Stone's formula frees us from any dependence on Lie algebraic structures. However, the difficulty of obtaining a splitting lemma, is replaced by that of computing the resolvent and the resulting spectral integrals. 2. Quantum Observables in sl(2; R) The Lie algebra sl(2; R) of real (2 × 2) matrices of zero trace, is generated [7] by the matrices ( ) ( ) ( ) 0 0 0 1 1 0 ∆ = ;R = ; ρ = ; −1 0 0 0 0 −1 4 ANDREAS BOUKAS AND PHILIP FEINSILVER with commutation relations [∆;R] = ρ, [ρ, R] = 2R; [ρ, ∆] = −2∆: We notice that the matrix ( ) 1 1 H = R − ∆ + ρ = 1 −1 is real symmetric, thus it is a quantum observable. Its eigenvalues are p p λ1 = − 2; λ2 = 2; with corresponding eigenspaces ( p ) 1 − 2 V = fxv : v = ; x 2 Rg; 1 1 1 1 ( p ) 1 + 2 V = fxv : v = ; x 2 Rg; 2 1 2 1 corresponding normalized basic eigenvectors 0 p 1 0 p 1 1− 2 1+ 2 p p p p B 4−2 2 C B 4+2 2 C v1 B C v2 B C u1 = jj jj = @ A ; u2 = jj jj = @ A ; v1 1 v2 1 p p p p 4−2 2 4+2 2 and eigen-projections 0 p 1 1 (2 − 2) − p1 B 4 2 2 C h i T E1 = u1; u1 = u1 u1 = @ A ; − p1 1p 2 2 4−2 2 0 p 1 1 (2 + 2) p1 B 4 2 2 C h i T E2 = u2; u2 = u2 u2 = @ A : p1 1p 2 2 4+2 2 We notice that E1 and E2 are a resolution of the identity, i.e., I = E1 + E2 and H = λ1E1 + λ2E2: Moreover 0 p p p 1 cos( 2t) + i p1 sin( 2t) i p1 sin( 2t) 2 2 itH B C e = @ p p p A : i p1 sin( 2t) cos( 2t) − i p1 sin( 2t) 2 2 SPECTRAL THEOREM APPROACH TO QUANTUM OBSERVABLES 5 If u = (a; b) is a unit vector in R2 then for t 2 R, itH itλ1 itλ2 itλ1 T itλ2 T hu; e ui = e hu; E1ui + e hu; E2ui = e u E1u + e u E2u p a2 − b2 + 2ab p = (a2 + b2) cos( 2t) + i p sin( 2t) 2 p a2 − b2 + 2ab p = cos( 2t) + i p sin( 2t); 2 so H follows a Bernoulli distribution with probability density function (( p ) p 1 2 2 pa;b(λ) = 2 + 2(a + 2ab − b ) δ( 2 − λ) 4 ( p ) p ) + 2 − 2(a2 + 2ab − b2) δ( 2 + λ) ; p p that is, H takes the values λ1 = − 2 and λ2 = 2 with probabilities ( ) p 1 p P (H = − 2) = 2 − 2(a2 + 2ab − b2) 4 and ( ) p 1 p P (H = 2) = 2 + 2(a2 + 2ab − b2) 4 respectively.
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