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1 Expectation

From classical probability theory we know that the expectation of a random variable X is completely determined by its distribution D_X by

assuming, of course that the random variable is integrable or the random variable is non-negative. Similarly, let A be an observable of a quantum mechanical system. A is given by a densely defined self-adjoint operator on H. The spectral measure of A defined by

uniquely determines A and converesely, is uniquely determined by A. E_A is a boolean homomorphism from the Borel subsets of R into the lattice Q of self-adjoint projections of H. In analogy with probability theory, given a state S, we introduce the distribution of A under S which is the probability measure defined on the Borel subsets of R by

Similarly, the expected value of A is defined in terms of the probability distribution D_A by

Note that this expectation is relative to the mixed state S which is used in the definition of D_A.

Remark. For technical reasons, one needs to consider separately the positive and negative parts of A defined by the Borel functional calculus for unbounded operators.

One can easily show:

Note that if S is a pure state corresponding to the vector ψ,

2 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 necessarilly 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 operaror S can be diagonalized, that it can be represented in some orthonormal basis by a (possibly infinite) matrix of the form

and we define

This value is an extended real number (that is in [0, ∞] 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, H(S) measures the amount of randomness in the state S. The more disperse the eigenvlaues 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.

Recall that a pure state is one the form

for ψ a vector of norm 1.

Theorem. H(S) = 0 iff 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.

This incidentally is one justification for the use of entropy as a measure of quantum entanglement.

3 Gibbs canonical ensemble

Consider an ensemble of systems described by a Hamiltonian H with average energy E. If H has pure-point spectrum and the eigenvalues of H go to + ∞ sufficiently fast, e^{-r H} will be a non-negative trace-class operator for ever positive r.

The Gibbs canonical ensemble is the state

where β is such that the ensemble average of energy satisfies

Under certain conditions the Gibbs canonical ensemble maximizes the von Neumann entropy of the state subject to the energy conservation requirement.