ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

32 2. Gaussian states: structural properties
Hilbert space H Phase space Γ
dimension
structure
∞ 2N
description ϱ χs, Ws
Table 2.I. Schematic comparison between Hilbert-space and phase-space
pictures for N-mode continuous variable systems.
2.2. Mathematical description of Gaussian states
The set of Gaussian states is, by definition, the set of states with Gaussian characteristic
functions and quasi-probability distributions on the multimode quantum
phase space. Such states are at the heart of information processing in CV systems
[GA22, GA23, 40] and are the main subject of this Dissertation.
2.2.1. Covariance matrix formalism
From the definition it follows that a Gaussian state ϱ is completely characterized
by the first and second statistical moments of the quadrature field operators,
which will be denoted, respectively, by the vector of first moments ¯
R =
〈 ˆ R1〉, 〈 ˆ R1〉, . . . , 〈 ˆ RN〉, 〈 ˆ
Rn〉 and by the covariance matrix (CM) σ of elements
σij = 1
2 〈 ˆ Ri ˆ Rj + ˆ Rj ˆ Ri〉 − 〈 ˆ Ri〉〈 ˆ Rj〉 . (2.17)
First moments can be arbitrarily adjusted by local unitary operations, namely
displacements in phase space, i.e. applications of the single-mode Weyl operator
Eq. (2.9) to locally re-center the reduced Gaussian corresponding to each single
mode 5 . Such operations leave any informationally relevant property, such as entropy
and entanglement, invariant: therefore, first moments are unimportant to the whole
scope of our analysis and from now on (unless explicitly stated) we will set them
to 0 without any loss of generality.
With this position, the Wigner function of a Gaussian state can be written as
follows in terms of phase-space quadrature variables
W (R) =
e− 1
2 Rσ−1 R T
π √ Det σ
, (2.18)
where R stands for the real phase-space vector (q1, p1, . . . , qN , pN) ∈ Γ. Despite the
infinite dimension of the Hilbert space in which it lives, a complete description of an
arbitrary Gaussian state (up to local unitary operations) is therefore encoded in the
2N × 2N CM σ, which in the following will be assumed indifferently to denote the
matrix of second moments of a Gaussian state, or the state itself. In the formalism
of statistical mechanics, the CM elements are the two-point truncated correlation
functions between the 2N canonical continuous variables. We notice also that the
entries of the CM can be expressed as energies by multiplying them by the quantity
ωk, where ωk is the frequency of each mode k, in such a way that Tr σ is related
to the mean energy of the state, i.e. the average of the non-interacting Hamiltonian
Eq. (2.2). This mean energy is generally unbounded in CV systems.
5 Recall that the reduced state obtained from a Gaussian state by partial tracing over a subset
of modes is still Gaussian.