Spatial Characterization Of Two-Photon States - GAP-Optique

2. Correlations and entanglement
(a)
Space Frequency Signal
q
q
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qs
qi
Idler
Figure 2.1: The frequency part, gray in (a), is traced out in section 2.2, to calculate
the state of the subsystem composed by the two-photon state in the spatial degree
of freedom. Calculating the signal photon state in space and frequency, section 2.3,
requires to trace out the idler photon, gray in (b).
2.1 The purity as a correlation indicator
When a physical system is used to implement a task, its state should be characterized.
For quantum states, the reconstruction of the state density matrix
demands a tomography [1]. This process requires an infinite amount of copies
of the state in order to eliminate statistical errors [42]. In an experimental
implementation, the number of copies of the state available is finite, but even
under this condition an experimental tomography is a complex process. Instead
when just a certain feature of the system is relevant for the task, it is preferable
to use a figure of merit to characterize that part of its quantum state.
For a system in a quantum state given by the density operator ˆρ, the purity
P = T r[ˆρ 2 ] is a figure of merit that describes its ability to be correlated with
others. That is, the purity indicates if correlations can or cannot exist. Since
the purity is a second order function of the density operator, measuring it does
not require a full tomography, but just a simultaneous measurement over two
copies of the system [43, 44].
To illustrate other characteristics of the purity, consider as an example
a complete orthonormal basis for the electromagnetic field, where each basis
vector |n〉 describes a mode of the field [36]. A pure state |R〉 is any state
written as a superposition of basis vectors:
|R〉 =
Cn|n〉, (2.1)
n
where the mode amplitude and the relative phases between modes, given by
the coefficients Cn, are fixed. This pure state can be described by the density
operator ˆρ = |R〉〈R|, which has the maximum possible value for the purity
T r[ˆρ 2 ] = 1.
A mixed state of the electromagnetic field is a state that can not be written
as in equation 2.1. For instance, in the case of a statistical mixture of field
modes, where there are no fixed amplitude or fixed relative phases associated
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(b)