Spatial Characterization Of Two-Photon States - GAP-Optique

2.1. The purity as a correlation indicator
to each vector of the base. There is, however, a fixed probability pR of finding
the electromagnetic field in a state |R〉. Using the density operator formalism,
the probability distribution is given by
ˆρ =
pR|R〉〈R|. (2.2)
R
The purity of the state described by ˆρ quantifies how close it is to a pure
state. If the state is pure then T r[ˆρ 2 ] = 1, and if it is maximally mixed then
T r[ˆρ 2 ] = 1/n, where n is de dimension of the Hilbert space in which the state is
expanded. Maximally mixed states of the electromagnetic field, in the infinite
dimensional spatial or temporal degrees of freedom have a purity T r[ˆρ 2 ] = 0.
To extend the discussion about the purity to the kind of states generated
by spdc, consider the pure bipartite system described by |ψ〉 and composed by
the fields A and B. The Schmidt decomposition guarantees that orthonormal
states |RA〉 and |RB〉 exist for the fields A and B, so that
|ψ〉 =
λR|RA〉|RB〉 (2.3)
R
where λR are non-negative real numbers that satisfy
R λR = 1, and are
known as Schmidt coefficients [1, 17]. The average of the nonzero Schmidt
coefficients is a common entanglement quantifier [11] known as the Schmidt
number and given by
K =
1
R λ2 R
. (2.4)
While equation 2.3 describes the state of the whole bipartite system, the state
of each part is calculated by tracing out the other part. Thus, the states of the
fields A and B are given by the density operators
ˆρA =
λR|RA〉〈RA| (2.5)
and
R
ˆρB =
λR|RB〉〈RB|. (2.6)
R
Since both operators have equal eigenvalues λ2 R , the purity of both states is
equal, and given by the inverse of the Schmidt number K
T r[(ˆρ A ) 2 ] = T r[(ˆρ B ) 2 ] = 1
K
= λ 2 R. (2.7)
Thus, when the field A is in a pure state, the field B is in a pure state too,
and
R λ2
R = 1. Since R λR = 1, the Schmidt coefficients satisfy the new
condition only if all except one of them are equal to zero. If that is the case,
the bipartite system in equation 2.3 is a product state, and no correlations
exist between A and B. In other words, if there are correlations, the purity of
A (and B) is smaller than 1.
In conclusion, when considering a composed global system, the purity of
each subsystem measures the strength of the correlation between such subsystem
and the rest. This is always true independently of how the subsystems are
R
17