Quantum Information Theory with Gaussian Systems

2.1 Phase space
yields as its expectation value the occupation number of mode k, i.e. the number of
quanta in this mode.
The commutation relation (2.2) requires that the Hilbert space for any representation
of the Rk is of infinite dimension and that the Rk are not bounded. In quantum
mechanics, the usual representation of the ccr for each degree of freedom is the
Schrödinger representation on the Hilbert space H = L 2 (Ê, dx) of square-integrable
functions, where Q and iP act by multiplication and differentiation with respect
to the variable x. 1 However, this representation leaves room for ambiguities, as is
discussed with a counterexample in [7, Ch. VIII.5]. This problem can be overcome
by building the theory upon suitable exponentials of the field operators instead. A
possible choice is to use the family of bounded, unitary Weyl operators
Wξ = e iξT ·σ· R for ξ ∈ Ξ; in particular W0 = . (2.5)
Hence for σ in standard form and ξ = (q1, p1, . . . , qf, pf), the Weyl operators can be
written explicitly as
Wξ = exp i f
k=1 (qk Pk − pk Qk) . (2.6)
By the ccr (2.2), the Weyl operators satisfy the Weyl relations
Wξ Wη = e −iσ(ξ,η)/2 Wξ+η and (2.7a)
Wξ Wη = e −iσ(ξ,η) Wη Wξ . (2.7b)
Note that by these relations and unitarity of Wξ, the inverse of a Weyl operator is
W ∗ ξ = W −ξ .
Remark on notation: Where appropriate, we expand the argument of Weyl operators,
i.e. we write equivalently to each other
Wξ ≡ Wξ1,ξ2,...,ξn ≡ Wq1,p1,...,qn,pn .
It is implicitly understood that ξ = (ξ1, ξ2, . . .,ξn) and ξi = (qi, pi). Occasionally, we
find it convenient to write the argument of Weyl operators in parentheses instead as
an index:
W(ξ) ≡ Wξ .
In reverse, the generators Rk of a family of unitary operators which satisfy the relations
(2.7) give rise to the ccr (2.2), cf. [7, Ch. VIII.5]. Moreover, for representations
of the Weyl relations in a finite-dimensional phase space, the Stone-von Neumann
theorem states conditions for unitary equivalence [5,7]:
1 That is, for ψ ∈ L 2 (Ê,dx): Q ψ(x) = x ψ(x) and iP ψ(x) = d
dx ψ(x).
11