ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

16 1. Characterizing entanglement
From the Schmidt decomposition it follows that the reduced density matrices
of |ψ〉,
ϱ1 =
ϱ2 =
d
λ 2 k|uk〉〈uk| ,
k=1
d
λ 2 k|vk〉〈vk| . (1.23)
k=1
have the same nonzero eigenvalues (equal to the squared Schmidt coefficients) and
their total number is the Schmidt number d. One can then observe that product
states |ψ〉 = |ϕ, χ〉 are automatically written in Schmidt form with d = 1, i.e. the
reduced density matrices correspond to pure states (ϱ1 = |ϕ〉〈ϕ| , ϱ2 = |χ〉〈χ|). On
the other hand, if a state admits a Schmidt decomposition with only one coefficient,
then it is necessarily a product state. We can then formulate an entanglement
criterion for pure quantum states. Namely, a state |ψ〉 of a bipartite system is
entangled if and only if the reduced density matrices describe mixed states,
|ψ〉 entangled ⇔ d > 1 , (1.24)
with d defined by Eq. (1.21).
We thus retrieve that bipartite entanglement of pure quantum states is qualitatively
equivalent to the presence of local mixedness, as intuitively expected. This
connection is in fact also quantitative. The entropy of entanglement EV (|ψ〉) of a
pure bipartite state |ψ〉 is defined as the Von Neumann entropy, Eq. (1.4), of its
reduced density matrices [21],
EV (|ψ〉) = SV (ϱ1) = SV (ϱ2) = −
d
λ 2 k log λ 2 k . (1.25)
The entropy of entanglement is the canonical measure of bipartite entanglement in
pure states. It depends only on the Schmidt coefficients λk, not on the corresponding
bases; as a consequence it is invariant under local unitary operations
( Û1 ⊗ Û2)|ψ〉
|ψ〉 . (1.26)
EV
= EV
It can be shown [190] that EV (|ψ〉) cannot increase under LOCC performed on the
state |ψ〉: this is a fundamental physical requirement as it reflects the fact that
entanglement cannot be created via LOCC only [245, 249]. It can be formalized as
follows. Let us suppose, starting with a state |ψ〉 of the global system S, to perform
local measurements on S1 and S2, and to obtain, after the measurement, the state
|ϕ1〉 with probability p1, the state |ϕ2〉 with probability p2, and so on. Then
EV (|ψ〉) ≥
pkEV (|ϕk〉) . (1.27)
k
Note that entanglement cannot increase on average, i.e. nothing prevents, for a
given k, that EV (|ϕk〉) > EV (|ψ〉). On this the concept of entanglement distillation
is based [23, 24, 97]: with a probability pk, it is possible to increase entanglement
via LOCC manipulations.
k=1