Quantum Statistical Mechanics - Von Neumann Entropy

Von Neumann Entropy

Of particular significance for describing randomness of a state is the von Neumann entropy of S formally defined by

.

Actually, the operator S log₂ S is not necessarily trace-class. However, if S is a non-negative self-adjoint operator not of trace class we define Tr(S) = +∞. Also note that any density operator S can be diagonalized, that it can be represented in some orthonormal basis by a (possibly infinite) matrix of the form

and we define

The convention is that, since an event with probability zero should not contribute to the entropy. This value is an extended real number (that is in ) and this is clearly a unitary invariant of S.

Remark. It is indeed possible that H(S) = +∞ for some density operator S. In fact T be the diagonal matrix

T is non-negative trace class and one can show T log₂ T is not trace-class.

Theorem. Entropy is a unitary invariant.

In analogy with classical entropy (notice the similarity in the definitions), H(S) measures the amount of randomness in the state S. The more dispersed the eigenvalues are, the larger the system entropy. For a system in which the space H is finite-dimensional, entropy is maximized for the states S which in diagonal form have the representation

For such an S, H(S) = log₂ n. The state S is called the maximally mixed state.

Recall that a pure state is one of the form

for ψ a vector of norm 1.

Theorem. H(S) = 0 if and only if S is a pure state.

For S is a pure state if and only if its diagonal form has exactly one non-zero entry which is a 1.

Entropy can be used as a measure of quantum entanglement.