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The problem I want to solve is this:

There is an ensemble in a state corresponding to vector (i, 2)

A measurement of Sy (with the operator represented by the 2x2 pauli spin matrix) is made but the values measured are not kept. What is the density matrix that must be used in subsequent measurements of any observable?

I know the denisty matrix is defined as the sum of the classical probability that the vector is in a particular state times by the projection operator of that particular state.

However I am having problems seeing how this is helpful to me.

Any help would be very much appreciated

Many thanks

Kian