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.

Read more about this topic:  Quantum Logic

Famous quotes containing the words pure and/or states:

    I
    Am a pure acetylene
    Virgin
    Attended by roses,
    By kisses, by cherubim,
    By whatever these pink things mean.
    Sylvia Plath (1932–1963)

    The admission of the States of Wyoming and Idaho to the Union are events full of interest and congratulation, not only to the people of those States now happily endowed with a full participation in our privileges and responsibilities, but to all our people. Another belt of States stretches from the Atlantic to the Pacific.
    Benjamin Harrison (1833–1901)