Quantum Information Theory with Gaussian Systems

4.2 Reversible Gaussian qca
Similarly, Weyl operators 4 W(ξ) ∈ A() on the whole system are defined as tensor
products of single-site Weyl operators wx(ξx) ∈ Ax,
W(ξ) =
x∈wx(ξx).
Both definitions are well-formed even on the infinite lattice, since the ξx are zero
except for finitely many sites.
As we work in the Heisenberg picture, states are positive, normalized, linear
functionals ω: A() →on the observable algebra, yielding a positive expectation
value ω(A) for positive observables A. Alternatively, they can as usual be
described by their characteristic function χ, the expectation value of all Weyl operators,
χ(ξ) = ω W(ξ) . For Gaussian states this is Gaussian and in strict analogy
of Eq.(2.21)
χ(ξ) = exp −1
4 γ(ξ, ξ) + i
x∈ξ T
x · dx .
Similar to the symplectic form, the covariances are contained in a bilinear correlation
function γ(ξ, η) =
x,z∈ξ T
x · γx,z · ηz defined as an effectively finite sum of
terms involving (2×2 blocks of) covariance matrices for finitely many modes. The
covariance matrix of a finite restriction of the chain is obtained as a block matrix of
the respective γx,z. For example, the covariance matrix γ| {x,z} of two modes x and z
is the 2×2 block matrix
γ| {x,z} =
γx,x γx,z
γz,x γz,z
For translationally invariant states, the displacement dx has to be independent of
the position in the chain, dx ≡ d, and can be interpreted as a globalamplitude.
Likewise, the real 2×2 matrices γx,z depend only on the distance between the two
sites x and z, i.e. γx,z = γ(x − z). Since the correlation function γ(x) takes the role
of the covariance matrix, it has to be symmetric, so we require γ(−x) = γ(x) T
.
A translationally invariant Gaussian state thus has a characteristic function of the
form
χ(ξ) = exp −1
4
x,y∈ξ T
.
x · γ(x − y) · ξy + i
x∈ξ T
x
· d . (4.4)
In order to describe an admissible Gaussian quantum state, the correlation function
γ has to obey the state condition (2.22). The positivity condition γ + iσ ≥ 0
on matrices is in the present case replaced by the respective condition on bilinear
functions, where complex-valued analogs to the phase space functions take the place
of complex phase space vectors:
γ(µ, µ) + iσ(µ, µ) ≥ 0 , (4.5)
for all µ = µre + iµim with µre, µim ∈ Ξ and µ as the complex conjugate. This condition
stems from a direct generalization of the argument leading to (2.22). Writing
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