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.
Read more about this topic: Quantum Logic
Famous quotes containing the words pure and/or states:
“What if all ponds were shallow? Would it not react on the minds of men? I am thankful that this pond was made deep and pure for a symbol. While men believe in the infinite some ponds will be thought to be bottomless.”
—Henry David Thoreau (1817–1862)
“It may be said that the elegant Swann’s simplicity was but another, more refined form of vanity and that, like other Israelites, my parents’ old friend could present, one by one, the succession of states through which had passed his race, from the most naive snobbishness to the worst coarseness to the finest politeness.”
—Marcel Proust (1871–1922)