The second lesson addressed the question of purification of a statistical mixture and its link with the notions of complementarity and quantum gum. The density operator of a system A was described as the result of a partial trace taken over a pure state in an extended space A + B. This pure state is referred to as the purification of the density operator of A. This pure state is called the purification of the density operator of A. We established the relations between the different forms of the same density operator, associated with fictitious unread measurements in A's environment B. We established the GHJW theorem (Gisin, Horne, Josza and Wooters) showing that all forms of the density operator of a statistical mixture can be deduced from a unique purification. This result was applied to the description of the depolarized density matrix of a qubit. This formalism enabled us to recall the notions of decoherence, complementarity and quantum eraser that had been covered in previous years' lectures.
We then extended the notion of measurement in quantum physics by defining generalized measurements and POVM measurements (acronym for Positive Operator Valued Measure). The effect of such measurements on an open system A appears to result from a standard measurement performed on an extended space, obtained by adding an auxiliary system B to A. The actual realizations of these measures have been described. Examples of POVM measurements on a qubit have been presented, in particular measurements with three possible results, in a number greater than the dimension of the associated Hilbert space. The practical realization of such measurements using quantum gates was described, and their application to the partial recognition of two non-orthogonal states of a qubit was discussed.
