www.qutools.com quED Manual 29
Tomographic reconstruction of the two-photon density matrix
The output from real-life entanglement sources like the quED is not truly pure as described by
the Bell states in section 5.2. In reality we always have to deal with mixed polarization states,
which are completely characterized using density matrices. To determine the density matrix
of an unknown quantum state, the experimental procedure called state tomography has to be
accomplished.
In case of the entangled photon pairs the state tomography requires a set of 16 projective
measurements. These are given by all possible permutations of projecting the two photons to
horizontal
, vertical
, plus-diagonal
and right-circular
polarization states;
and
is defined in section 5.2. The tomography contains
measurements in the circular polarization base, which is accessible only with a quarter
waveplate (optionally delivered with the quED). Therefore, to project the photons into any of
the four mentioned polarization states, quarter-wave plates preceding the polarizing filters
have to be used. For each of the settings the coincidence count rate over a given integration
time is recorded and the accumulated experimental data are then evaluated. Since the
evaluation procedure of density matrix from the measured 16 coincidence counts is lengthy
and more involved, it is not explicitly described here, but can be found e.g. in
D. F. V. James et al.,
Phys. Rev A 64, 052312
. To quantitatively characterize the deviation of the experimental density
matrix
from the ideal, its overlap with the theoretical density matrix
, ( is
one of the Bell states
), is calculated:
is usually called the fidelity and it takes the values between 0 and 1. The maximum value of 1
is obtained if
, so that the two states are completely indistinguishable. Furthermore,
many useful measures to characterize the experimental quantum state can be derived from
the reconstructed density matrix, e.g. concurrence or tangle can be computed to quantify the
degree of entanglement and entropy to quantify the degree of mixture in quantum state. The
definitions of the various measures and their interpretation can be found in a number of
quantum mechanics textbooks.
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