aa r X i v : . [ m a t h - ph ] A ug Filtering of Wide Sense StationaryQuantum Stochastic Processes
John GoughInstitute for Mathematical and Physical Sciences,University of Wales, Aberystwyth,Ceredigion, SY23 3BZ, Wales
Abstract
We introduce a concept of a quantum wide sense stationary processtaking values in a C*-algebra and expected in a sub-algebra. The powerspectrum of such a process is defined, in analogy to classical theory, as apositive measure on frequency space taking values in the expected algebra.The notion of linear quantum filters is introduced as some simple examplesmentioned.
In the following we consider processes depending on a continuous time variable t ∈ R : the frequency domain will be denoted as ˆ R for distinction. Thereforewe have the Fourier transform conventions ˆ f ( ν ) = R R e πi tν f ( t ) dt , f ( t ) = R ˆ R e − πi tν ˆ f ( ν ) dν . Let h = L (Ω , F , P ) be the Hilbert space of second-order random variableson a probability space (Ω , F , P ). A second order process X = ( X t ) t is astochastic process with each X t ∈ h . (That is, X corresponds to a parame-terized curve in the Hilbert space h . The linear manifold in h spanned by thefamily { X t : t ∈ R } is denote as l X and its closure h X is a Hilbert subspaceof h . The mean function is µ X ( t ) := E [ X t ] and the covariance function is C X ( s, t ) := E [ X ∗ s X t ] − µ X ( t ) µ X ( s ) ≡ h X s , X t i h − µ X ( t ) µ X ( s ). If µ X is in-dependent of t and C X ( s, t ) depends only on t − s then the process is termedwide sense stationary (WSS).If we are given a mean-zero WSS process X then we write its covariancefunction as C X ( τ ) ≡ C X ( t, t + τ ). The covariance is semi-positive definite ( P nj,k =1 z ∗ j C X ( t j , t k ) z k ≥ n , for arbitrary times t , . . . , t n ∈ and arbitrary complex numbers z , . . . , z n ) and by Bochner’s theorem (Her-glotz’ theorem for discrete time parameter) it can therefore be written as aFourier-Stieltjes transform [1][2] C X ( t ) = Z ˆ R e πi tν dS X ( ν )where S X ( ν ) is a non-decreasing right-continuous on ˆ R with lim ν →−∞ S X ( ν ) =0 and lim ν → + ∞ S X ( ν ) = C X (0). S X is called the spectral function for X .Let E be the linear manifold spanned by the harmonic functions e t ( ν ) =exp (2 πi tν ) on ˆ R . We set f X = L (cid:16) ˆ R , dS X (cid:17) . Evidently E is a subset of f X which is dense in the k . k f X -topology. Theorem 1
Let X be a mean-zero WSS process then there exists a linear iso-metric isomorphism ˆ X : f X h X . The proof is constructive: define ˆ X : E h X first of all by setting ˆ X ( e t ) ≡ X t and extending by linearity. It follows that ˆ X is an isometry, since D ˆ X ( e s ) | ˆ X ( e t ) E h = C X ( t − s )= Z ˆ R e s ( ν ) ∗ e t ( ν ) dS X ( ν )= h e s | e t i f X . Next given any Z ∈ h X there will be a sequence ( Z n ) n converging to Z in normsuch that each Z n ≡ ˆ X ( f n ) for some f n ∈ E . The sequence of function ( f n ) n will have a well-defined limit f in f X depending only on Z . We therefore set Z = ˆ X ( f ). Existence and uniqueness follows from the completeness of theHilbert spaces f X and h X .The map ˆ X : f X h X induces a h X -valued measure on the Borel setsof ˆ R , again denoted as ˆ X [ . ], according to ˆ X [ F ] ≡ ˆ X (1 F ) where 1 F is thecharacteristic function for the subset F . In this way we can writeˆ X ( f ) ≡ Z ˆ R f ( ν ) ˆ X [ dν ]and in particular X t = ˆ X ( e t ) ≡ R ˆ R e πi tν ˆ X [ dν ]. (Typically a Fourier transformˆ X ν may only exist in some more singular sense.) Remark (1)
The set E in fact forms a commutative *-algebra with unit e . Remark (2) If X and Y are orthogonal (independent!) mean-zero WSS pro-cesses then dS αX + βY = | α | dS X + | β | dS Y .2 emark (3) For each mean-zero WSS process X we construct a kernel K X : f X × f X C according to K X ( f | g ) = D ˆ X ( f ) | ˆ X ( g ) E h = Z ˆ R f ( ν ) ∗ g ( ν ) dS X ( ν ) . A quantum probability space ( A , E ) consists of a *-algebra A of operators anda state E on A . A quantum stochastic process is then a family X = ( X t ) t ofelements of A parameterized by time. Following the standard interpretation ofquantum mechanics, the self-adjoint elements of A correspond to the physicalobservables. More generally we consider conditional expectations [3][4] wherethe partially-averaged variables will live in some C*-subalgebra B of A .Let B be a C*-algebra and Φ : E B be a completely positive map. Akernel K : E × E B is defined by K ( f | g ) = Φ ( f ∗ g ). Then C ( τ ) = Φ ( e τ ) ispositive semi-definite on B in the sense that n X j,k =1 Z ∗ j C X ( t k − t j ) Z k ≥ n , for arbitrary times t , . . . , t n ∈ R and arbitrary operators Z , . . . , Z n ∈ B . This suggests the following definition. Definition 2
Let A be a C*-algebra, B a C*-subalgebra of A and E [ . | B ] aconditional expectation from A to B . A mean-zero WSS quantum stochasticprocess X = ( X t ) t taking values in A and expected in B is a family of operators X t ∈ A such that E [ X t | B ] = 0 and C X ( τ ) = E [ X ∗ t X t + τ | B ] is semi-positivedefinite on B . In the situations where we are interested in a quantum probability space( A , E ), we might typically ask that the conditional expectation from A to B becompatible with the state (i.e. E [ E [ Z | B ]] = E [ Z ]).A natural example is where H and K are fixed Hilbert spaces and the C*-algebras are concretely realized as A = B ( H ⊗ K ), B = B ( H ). The conditionalexpectation then being partial expectation wrt. a fixed density matrix ρ K on K : that is, tr H { T E [ Z | B ] } ≡ tr H ⊗ K { T ⊗ ρ K Z } for all trace-class operators T on H . The expectation E will be wrt. a productstate ρ = ρ H ⊗ ρ K .Let R be a Hilbert space and let B ( H, R ) denote the set of linear operatorsfrom H to R . Suppose that we are given a map V : E
7→ B ( H, R ) , then asemi-positive definite kernel K : E × E
7→ B ( H ) is obtained by setting K ( f | g ) = V ( f ) ∗ V ( g ). This is known as a Kolmogorov decomposition of the kernel [5]. It3s known that a kernel K : E × E
7→ B ( H ) admits a Kolmogorov decompositionif and only if it is semi-positive definite. A canonical choice is given by taking R to be the closed linear manifold of H -valued functions on E of the type g K ( f | g ) ξ , where f ∈ E and ξ ∈ H , and taking V ( g ) ∈ B ( H, R ) to be V ( f ) : ξ K ( f | g ) ξ . The decomposition is minimal in the sense that R = h V ( f ) ξ : f ∈ E , ξ ∈ H i and is unique up to unitary equivalence. Theorem 3
Let X = ( X t ) t be a mean-zero WSS quantum stochastic processon ( A , E ) modelled on B . Let B + denote the cone of positive elements of B .There exists a non-decreasing right-continuous B + -valued function S X on ˆ R with lim ν →−∞ S X ( ν ) = 0 and lim ν → + ∞ S X ( ν ) = C X (0) such that C X ( t ) = Z ˆ R e πi tν d S X ( ν ) . S X is called the spectral operator for X . Now let H = L ( A , E ) be the vector space of all operators X ∈ A suchthat E [ X ∗ X ] < ∞ . Then H is a Hilbert space and if X = ( X t ) t is a quantumstochastic process on ( A , E ), then H X = h X t : t ∈ R i will be a Hilbert subspace.We shall denote by F X = L (cid:16) ˆ R , d S X (cid:17) the space of all measurable functions f on ˆ R such that R ˆ R | f ( ν ) | d S X ( ν ) defines a bounded element of B + . Theorem 4
Let X = ( X t ) t be a mean-zero WSS quantum stochastic processon ( A , E ) modelled on B and having spectral operator S X . Then there exists alinear map ˆ X : F X H X such that E h ˆ X ( f ) ∗ X ( g ) | B i = Z ˆ R f ( ν ) ∗ g ( ν ) d S X ( ν ) . In analogy to the classical case, we denote by ˆ X [ dν ] the corresponding B + -valued measure on ˆ R . The notion of a linear filtering is of considerable importance in stochastic mod-elling [6][7]. It is possible to extend this to the quantum case.
Definition 5
Let ψ L : ˆ R B be measurable. A linear filter L with character-istic function ψ L ( ν ) acting on the A -valued mean-zero WSS quantum stochasticprocesses expected on B is the linear transformation defined by X LX where ( LX ) t := Z ˆ R e πi νt ˆ X [ dν ] ψ L ( ν )4 nd the domain of L is the set D L of all such processes X for which the integral R ˆ R ψ L ( ν ) ∗ d S X ( ν ) ψ L ( ν ) converges in B . Given X ∈ D L , the process LX will again be mean-zero WSS process andits spectral measure will be d S LX ( ν ) = ψ L ( ν ) ∗ d S X ( ν ) ψ L ( ν ). We begin with some simple “classical” filters having c -number characteristicfunctions: • Time shifts ( LX ) t = X t + s ; ψ L = e s . • Time derivatives ( LX ) t = ˙ X t ; ψ L ( ν ) = 2 πiν • Scalar convolutions ( LX ) t = ( X ∗ h ) t ; ψ L ( ν ) = ˆ h ( ν )However, filters with operator-valued characteristic functions are possible.Let A ∈ B and Γ ∈ B + ; then set h ( t ) = e − Γ t Aθ ( t ). We can consider a pro-cess X passed through the linear filter corresponding to convolution with h . Theoutput process Y will have spectral density d S Y ( ν ) = A ∗ πi − Γ d S X ( ν ) πi +Γ A .As a specific case, we could consider X to be a white noise input with spec-trum d S X ( ν ) = Sdν where S ∈ B + . (For instance, X might be combinationsof creator or annihilator white noise [9][10][11].) Then provided the dampingoperator Γ commutes with S , the output will have a continuous Lorentzian-typespectral density, no singular or pure point component, and its covariance willbe the Ornstein-Uhenbeck type C X ( t ) = A ∗ Se − Γ | t | A . If [Γ , S ] = 0 then thespectrum will be more complicated. References [1] Bochner, S.
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