Quantum Information Theory with Gaussian Systems

2 Basics of Gaussian systems
In phase space, transposition of Hermitian density operators is the same as complex
conjugation, which in turn can be identified with inversion of sign for the momenta
[15], i.e. P ↦→ −P while Q ↦→ Q under Θ. This corresponds to areversal of
timeor rather a reversal of time evolution. As a density operator ρ of a bipartite
Gaussian state is positive if its covariance matrix obeys the state condition (2.22),
γ + iσA ⊕ σB ≥ 0 (where σA, σB are the symplectic forms for systems A and B),
the partial transpose ρ T A is positive if the covariance matrix obeys
γ + i(−σA) ⊕ σB ≥ 0 ,
where the sign on σA reflects the change of sign for momenta in the ccr (2.2).
While the ppt criterion is necessary for separability, it is sufficient only forsmall
systems:2 ⊗2 and2 ⊗3 in finite dimensions [17], Gaussian states with 1×n
modes for continuous-variable systems [18]. In particular, the criterion fails if both
parties A and B of a Gaussian states have more than one mode (an explicit example
is presented in [18]). Since the entanglement of entangled states with positive partial
transpose cannot be freely converted into other forms, the entanglement isbound
and the sates are calledppt-bound entangled.
2.2.4 Singular states
In a general sense, a quantum state ω is a normalized positive linear functional 11 on
the algebra of observables [19], i.e. here on ccr(Ξ, σ) for f degrees of freedom:
ω: ccr(Ξ, σ) →, where ω(X ∗ X) ≥ 0 for all X ∈ ccr(Ξ, σ) and ω( ) = 1 .
Note that = W0 ∈ ccr(Ξ, σ). A state is normal if it can be described by a density
operator, i.e. a positive trace class operator ρ on the representation Hilbert space
H ⊗f :
ω(X) = tr[ρ X] .
Otherwise, the state ω is singular and can be decomposed into a normal part ωn given
by a density operator and a purely singular contribution ωs, which has expectation
value zero for all compact operators12 : ω = ωn + ωs (cf. Section 3.3.1). Singular
states have a characteristic function by
χ(ξ) = ω
Wξ
and can thus be Gaussian if χ(ξ) is a Gaussian (2.21).
If the ccr algebra is represented on the Hilbert space H⊗f , then the normalized
positive linear functionals ω on ccr(Ξ, σ) form the space B∗H⊗f
. Similarly, the
⊗f linear space generated by the density operators is denoted by B∗ H , whose closure
in the weak topology is B∗H⊗f .
11 Normalized positive linear functionals are automatically bounded and continuous.
12 Compact operators on a Hilbert space are those which can be approximated in norm by finite
rank operators, i.e. operators represented as a finite sum of terms |φ〉〈ψ|, cf. e.g. [7, Vol.I].
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