For technical reasons, one needs to consider separately the positive and negative parts of A defined by the Borel functional calculus for unbounded operators.
, then: 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.
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
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.
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 every positive r. The Gibbs canonical ensemble is described by the state Where β is such that the ensemble average of energy satisfies and This is called the partition function; it is the quantum mechanical version of the canonical partition function of classical statistical mechanics.
is Under certain conditions, the Gibbs canonical ensemble maximizes the von Neumann entropy of the state subject to the energy conservation requirement.
Note that this is a density matrix including many more states (of varying N) compared to the canonical ensemble.