Quantum Logic - Pure States

Pure States

A convex combination of statistical states S₁ and S₂ is a state of the form S = p₁ S₁ +p₂ S₂ where p₁, p₂ are non-negative and p₁ + p₂ =1. Considering the statistical state of system as specified by lab conditions used for its preparation, the convex combination S can be regarded as the state formed in the following way: toss a biased coin with outcome probabilities p₁, p₂ and depending on outcome choose system prepared to S₁ or S₂

Density operators form a convex set. The convex set of density operators has extreme points; these are the density operators given by a projection onto a one-dimensional space. To see that any extreme point is such a projection, note that by the spectral theorem S can be represented by a diagonal matrix; since S is non-negative all the entries are non-negative and since S has trace 1, the diagonal entries must add up to 1. Now if it happens that the diagonal matrix has more than one non-zero entry it is clear that we can express it as a convex combination of other density operators.

The extreme points of the set of density operators are called pure states. If S is the projection on the 1-dimensional space generated by a vector ψ of norm 1 then

for any E in Q. In physics jargon, if

where ψ has norm 1, then

Thus pure states can be identified with rays in the Hilbert space H.