Quantum Information Theory with Gaussian Systems

2 Basics of Gaussian systems
As pure states, coherent states correspond to Hilbert space vectors |α〉, which we
label by the phase space vector α determining the displacement in (2.26). This allows
to write
|α〉 = Wα |0〉. (2.28)
The state |0〉〈0| has expectation value zero in the field operators and is a minimum
uncertainty state (see above). Moreover, among all such states it has the smallest
possible expectation value of the operator Q2 +P 2 , since its variances with respect to
the field operators are equal, 9 〈Q2 〉 = 〈P 2 〉 = 1
2 . Considering that Q2 +P 2 = 2 ˆ N +
and hence tr[ρ0 ˆ N] = 0, |0〉〈0| necessarily is the vacuum state.
The relation (2.28) allows to compute the overlap between two coherent states:
〈α|β〉 = 〈0| W ∗ α W β |0〉 = 〈0| W β−α |0〉eiσ(α,β)/2 = e −(β−α)2 /4+iσ(α,β)/2 . (2.29)
This overlap is strictly nonzero, hence coherent states are not orthogonal to each
other.
Coherent states are eigenstates of the annihilation operator a = (Q + iP)/ √ 2:
denoting α = (αq, αp), one has 10
a |α〉 = W α W ∗ α aW α |0〉
= 1
√ 2 Wα
(Q − αq ) + i(P − αp ) |0〉
= Wα a |0〉 − 1
√ 2 (αq + iαp)Wα |0〉
= − 1 √ 2 (αq + iαp)|α〉. (2.30)
The expectation value of the occupation number operator ˆ N = a∗a in a coherent
state is thus tr |α〉〈α| ˆ N = 〈α| a∗a |α〉 = |α| 2 /2. This can be interpreted as the
mean energy of a system in the coherent state |α〉〈α| if the result is scaled by the
characteristic energy ω of the mode.
A thermal state of the Hamiltonian H = (Q2 + P 2 )/2 with covariance matrix
γ = τ , τ > 1 is by (2.24) and (2.25) a classical mixture of coherent states, where
γpure = and the noise is described by γnoise = (τ − 1) :
ρτ = dξ exp −1 4 ξ2 (τ − 1) −1 Wξ |α〉〈α|W ∗ ξ . (2.31)
The displacement of ρτ is the same as of |α〉〈α|.
9 Recall that for a, b, c ∈Êand a, b, c > 0, the quantity a + b under the restriction a b = c is
minimized for a = b.
10 Note that this differs from the convention where coherent states are labeled by their eigenvalue
with respect to the annihilation operator a:
20
a ˛ ˛ αq+iαp
√ 2
¸ = αq+iαp
√ 2
˛ αq+iαp ¸
√ .
2
Defining a complex number α = (αq + iαp)/ √ 2, this reads a |α〉 = α |α〉. Consequentially,
relations between coherent states look different, e.g. the overlap (2.29) is given by 〈α|β〉 =
exp(−|α| 2 /2 − |β| 2 /2 + α β ´ .