ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

1.1. Information contained in a quantum state 11
be singled out in the context of characterizing entanglement, as we will see in the
next Part.
1.1.2. Shannon–Von Neumann entropy
Let us go back to our physical system and to our ensemble of a priori known probabilities
p1, . . . , pN, associated to the possible outcomes of a particular measurement
we are going to perform on the system. If we imagine to repeat the measurement
on n copies of the system, all prepared in the same state, with n arbitrarily large,
we can expect the outcome “system in state j” be obtained ∼ nj = n pj times.
Based on our knowledge on the preparation of the system, we are in the position
to predict the statistical frequencies corresponding to the different outcomes, but
not the order in which the single outcomes will be obtained. Assuming that, on n
measurement runs, outcome 1 is obtained n1 times, outcome 2 n2 times, and so on,
the total number of permutations of the n outcomes is given by (n!/
k nk!) . For
n → ∞, also the individual frequencies will diverge, nj = n pj → ∞, so that by
using Stirling’s formula one finds
log
k
The expression
n!
n log n − n − (nk log nk − nk) = −n
nk!
pk log pk .
k
S = −
N
k=1
pk log pk
k
(1.3)
is named entropy associated to the probability distribution {p1, . . . , pN}: it is a
measure of our ignorance prior to the measurement.
The notion of entropy, originating from thermodynamics, has been reconsidered
in the context of classical information theory by Shannon [213]. In quantum
information theory the probabilities {pk} of Eq. (1.3) are simply the eigenvalues of
the density matrix ϱ, and Shannon entropy is substituted by Von Neumann entropy
[258]
SV = −Tr [ϱ log ϱ] = −
pk log pk . (1.4)
Purity µ, linear entropy SL and Von Neumann entropy SV of a quantum state
ϱ are all invariant quantities under unitary transformations, as they depend only
on the eigenvalues of ϱ. Moreover, Von Neumann entropy SV (ϱ) satisfies a series
of important mathematical properties, each reflecting a well-defined physical
requirement [260]. Some of them are listed as follows.
• Concavity.
SV (λ1ϱ1 + . . . + λnϱn) ≥ λ1SV (ϱ1) + . . . + λnSV (ϱn) , (1.5)
with λi ≥ 0,
i λi = 1. Eq. (1.5) means that Von Neumann entropy
increases by mixing states, i.e. is greater if we are more ignorant about
the preparation of the system. This property follows from the concavity
of the log function.
• Subadditivity. Consider a bipartite system S (described by the Hilbert
space H = H1 ⊗ H2) in the state ϱ. Then
k
SV (ϱ) ≤ SV (ϱ1) + SV (ϱ2) , (1.6)