Quantum Logic - Statistical Structure

Statistical Structure

Imagine a forensics lab which has some apparatus to measure the speed of a bullet fired from a gun. Under carefully controlled conditions of temperature, humidity, pressure and so on the same gun is fired repeatedly and speed measurements taken. This produces some distribution of speeds. Though we will not get exactly the same value for each individual measurement, for each cluster of measurements, we would expect the experiment to lead to the same distribution of speeds. In particular, we can expect to assign probability distributions to propositions such as {a ≤ speed ≤ b}. This leads naturally to propose that under controlled conditions of preparation, the measurement of a classical system can be described by a probability measure on the state space. This same statistical structure is also present in quantum mechanics.

A quantum probability measure is a function P defined on Q with values in such that P(0)=0, P(I)=1 and if {E_i}_i is a sequence of pairwise orthogonal elements of Q then

The following highly non-trivial theorem is due to Andrew Gleason:

Theorem. Suppose H is a separable Hilbert space of complex dimension at least 3. Then for any quantum probability measure on Q there exists a unique trace class operator S such that

for any self-adjoint projection E.

The operator S is necessarily non-negative (that is all eigenvalues are non-negative) and of trace 1. Such an operator is often called a density operator.

Physicists commonly regard a density operator as being represented by a (possibly infinite) density matrix relative to some orthonormal basis.

For more information on statistics of quantum systems, see quantum statistical mechanics.