ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

10 1. Characterizing entanglement
associated to the N possible outcomes. After the measurement, one of these outcomes
will have occurred and we will possess a complete information (certainty)
about the state of our system.
The degree of information contained in a state corresponds to how much certainty
we possess a priori on predicting the outcome of any test performed on the
state [177].
1.1.1. Purity and linear entropy
The quantification of information will in general depend not only on the state
preparation procedure, but also on the choice of the measurement with its associated
probabilities {pk}. If for any test one has a complete ignorance (uncertainty), i.e. for
a system in a N-dimensional Hilbert space one finds pk = 1/N ∀ k, then the state is
maximally mixed, in other words prepared in a totally random mixture, with density
matrix proportional to the identity, ϱm = N/N. For instance, a photon emitted
by a thermal source is called ‘unpolarized’, reflecting the fact that, with respect
to any unbiased polarization measurement, the two outcomes (horizontal/vertical)
have the same probability. The opposite case is represented by pure quantum states,
whose density matrix is a projector ϱp = |ψ〉〈ψ|, such that ϱ 2 p = ϱp . A pure state of
a quantum system contains the maximum information one has at disposal on the
preparation of the system. All the intermediate instances correspond to a partial
information encoded in the state of the system under consideration.
A hint on how to quantify this information comes from the general properties
of a quantum density operator. We recall that
Tr ϱ 2
= 1 ⇔ ϱ pure state ;
< 1 ⇔ ϱ mixed state .
It is thus natural to address the trace of ϱ 2 as purity µ of a state ϱ,
µ(ϱ) = Tr ϱ 2 . (1.1)
The purity is a measure of information. For states of a Hilbert space H with
dim H = N, the purity varies in the range
1
≤ µ ≤ 1 ,
N
reaching its minimum on the totally random mixture, and equating unity of course
on pure states. In the limit of continuous variable systems (N → ∞), the minimum
purity tends asymptotically to zero.
Accordingly, the “impurity” or degree of mixedness of a quantum state ϱ, which
characterizes our ignorance before performing any quantum test on ϱ, can be quantified
via the functional
SL(ϱ) = N
N 2
(1 − µ) = 1 − Tr ϱ
N − 1 N − 1
. (1.2)
The quantity SL (ranging between 0 and 1) defined by Eq. (1.2) is known as linear
entropy and it is a very useful measure of mixedness in quantum information theory
due to its direct connection with the purity and the effective simplicity in its
computation. Actually, the name ‘linear entropy’ follows from the observation that
SL can be interpreted as a first-order approximation of the canonical measure of
lack-of-information in quantum theory, that is Von Neumann entropy. In practice,
the two quantities are not exactly equivalent, and differences between the two will