ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

2.2. Mathematical description of Gaussian states 33
As the real σ contains the complete locally-invariant information on a Gaussian
state, we can expect some constraints to exist to be obeyed by any bona fide CM, reflecting
in particular the requirements of positive-semidefiniteness of the associated
density matrix ϱ. Indeed, such condition together with the canonical commutation
relations imply
σ + iΩ ≥ 0 , (2.19)
Ineq. (2.19) is the only necessary and sufficient constraint the matrix σ has to fulfill
to be the CM corresponding to a physical Gaussian state [220, 219]. More in general,
the previous condition is necessary for the CM of any, generally non-Gaussian, CV
state (characterized in principle by the moments of any order). We note that
such a constraint implies σ ≥ 0. Ineq. (2.19) is the expression of the uncertainty
principle on the canonical operators in its strong, Robertson–Schrödinger form
[197, 200, 208].
For future convenience, let us define and write down the CM σ1...N of an Nmode
Gaussian state in terms of two by two submatrices as
⎛
⎞
⎜
σ1...N = ⎜
⎝
σ1 ε1,2 · · · ε1,N
ε T 1,2
.
. ..
. ..
. ..
.
. .. εN−1,N
ε T 1,N · · · ε T N−1,N σN
⎟ . (2.20)
⎟
⎠
Each diagonal block σk is respectively the local CM corresponding to the reduced
state of mode k, for all k = 1, . . . , N. On the other hand, the off-diagonal matrices
εi,j encode the intermodal correlations (quantum and classical) between subsystems
i and j. The matrices εi,j all vanish for a product state.
In this preliminary overview, let us just mention an important instance of twomode
Gaussian state, the two-mode squeezed state |ψ sq 〉i,j = Ûi,j(r) (|0〉i⊗ |0〉j)
with squeezing factor r ∈ , where the (phase-free) two-mode squeezing operator
is given by
Ûi,j(r) = exp − r
2 (â†
i ↠j
− âiâj) , (2.21)
In the limit of infinite squeezing (r → ∞), the state approaches the ideal Einstein-
Podolsky-Rosen (EPR) state [73], simultaneous eigenstate of total momentum and
relative position of the two subsystems, which thus share infinite entanglement.
The EPR state is unnormalizable and unphysical: two-mode squeezed states, being
arbitrarily good approximations of it with increasing squeezing, are therefore key
resources for practical implementations of CV quantum information protocols [40]
and play a central role in the subsequent study of the entanglement properties of
general Gaussian states. A two-mode squeezed state with squeezing degree r (also
known in optics as twin-beam state [259]) will be described by a CM
σ sq
i,j (r) =
⎛
⎜
⎝
cosh(2r) 0 sinh(2r) 0
0 cosh(2r) 0 − sinh(2r)
sinh(2r) 0 cosh(2r) 0
0 − sinh(2r) 0 cosh(2r)
⎞
⎟
⎠ . (2.22)