One-mode quantum Gaussian channels
نویسنده
چکیده
A classification of one-mode Gaussian channels is given up to canonical unitary equivalence. A complementary to the quantum channel with additive classical Gaussian noise is described providing an example of onemode Gaussian channel which is neither degradable nor anti-degradable. 1 The canonical form For a mathematical framework of linear Bosonic system used in this note the reader is referred to [1]. Consider Bosonic system with one degree of freedom described by the canonical observables Q,P satisfying the Heisenberg canonical commutation relations (CCR) [Q,P ] = iI. (1) Let Z be the two-dimensional symplectic space, i. e. the linear space of vectors z = [x, y] with the symplectic form ∆(z, z′) = x′y − xy′. (2) A basis e, h in Z is symplectic if ∆(e, h) = 1, i.e. if the area of the oriented parallelogram based on e, h is equal to 1. A linear transformation T in Z is symplectic if it maps a symplectic basis into symplectic basis, or equivalently ∆(Tz, T z′) = ∆(z, z′); z, z′ ∈ Z. Let V (z) = exp i(xQ+yP ) be the unitary Weyl operators in a Hilbert space H satisfying the CCR V (z)V (z′) = exp[ i 2 ∆(z, z′)]V (z + z′) formally equivalent to (1). For any symplectic transformation T in Z there is a canonical unitary transformation UT in H such that U∗ TV (z)UT = V (Tz). 1 An arbitrary Gaussian channel Φ in B (H) has the following action on the Weyl operators (we use the dual channel Φ∗ in Heisenberg picture) Φ∗(V (z)) = V (Kz)f(z), (3) where K is a linear transformation in Z while f(z) is a Gaussian characteristic function satisfying the condition that for arbitrary finite collection {zr} ⊂ Z the matrix with the elements f(zr − zs) exp ( − i 2 ∆(zr, zs) + i 2 ∆(Kzr,Kzs) ) (4) is positive definite. Considering the function f(z), we can always eliminate the linear terms by a canonical transformation and assume that f(z) = exp [ − 1 2 α(z, z) ] , where α is a quadratic form. Then (4) is equivalent to positive definiteness of the matrix with the elements α(zr, zs)− i 2 ∆(zr, zs) + i 2 ∆(Kzr,Kzs). (5) Motivated by [2], [3], we are interested in the simplest form of one-mode Gaussian channel which can be obtained by applying suitable canonical unitary transformations to the input and the output of the channel: Φ∗′ [V (z)] = U∗ T1Φ ∗ [U∗ T2V (z)UT2 ] UT1 i.e. Φ∗′ [V (z)] = V (T1KT2z)f(T2z). Theorem. Let e, h be a symplectic basis; depending on the value A) ∆(Ke,Kh) = 0; B) ∆(Ke,Kh) = 1; C) ∆(Ke,Kh) = k > 0, k 6= 1; D) ∆(Ke,Kh) = −k < 0
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تاریخ انتشار 2006