ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

CHAPTER 2
Gaussian states: structural properties
In this Chapter we will recall the main definitions and set up our notation for the
mathematical treatment of Gaussian states of continuous variable systems. Some
of our results concerning the evaluation of entropic measures for Gaussian states
[GA3] and the reduction of Gaussian covariance matrices into standard forms under
local operations [GA18] will be included here as well.
2.1. Introduction to continuous variable systems
A continuous variable (CV) system [40, 77, 49] of N canonical bosonic modes is
described by a Hilbert space
N
H =
(2.1)
k=1
resulting from the tensor product structure of infinite-dimensional Fock spaces
Hk’s. One can think for instance to the quantized electromagnetic field, whose
Hamiltonian describes a system of N harmonic oscillators, the modes of the field,
N
ˆH = ωk â †
kâk + 1
. (2.2)
2
k=1
Here âk and â †
k are the annihilation and creation operators of a photon in mode k
(with frequency ωk), which satisfy the bosonic commutation relation
âk, â †
k ′
= δkk ′ , [â, kâk ′] = â †
k , â†
k ′
= 0 . (2.3)
From now on we will assume for convenience natural units with = 2. The corresponding
quadrature phase operators (position and momentum) for each mode are
defined as
Hk
ˆqk = (âk + â †
k ) , (2.4)
ˆpk = (âk − â †
k )/i (2.5)
We can group together the canonical operators in the vector
ˆR = (ˆq1, ˆp1, . . . , ˆqN, ˆpN ) T , (2.6)
which enables us to write in a compact form the bosonic commutation relations
between the quadrature phase operators,
[ ˆ Rk, ˆ Rl] = 2iΩkl , (2.7)
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