Von Neumann Entropy and Nonorthogonal States

The von Neumann entropy: Trace( -density ln(density)), which is applied to a set of quantum mechanical states, is similar in form to Shannon's entropy, - Sum over i P(i) ln(P(i)). In fact, the two are equal if the density matrix is diagonal with P(i) as the eigenvalues and Sum over i P(i)=1. In the literature (1), it is noted that the value of the von Neumann entropy may change if one uses different basis states. This is considered a problem. If the states are created using an unitary operator acting on an original orthonormal set, then (2) shows that the von Neumann entropy is unchanged. In general (1), however, this is not the case. (1) suggests that one should find the set of states which yield the lowest Shannon entropy.

We try to argue that the origins of the form: P(i) ln(P(i)) in classical statistical mechanics follow from the independence of probabilities and the nonoverlap of their states and try to extend this idea to quantum states. For a set of states which yield a diagonal density matrix, unitary transformations yield a set of basis states which are still orthonormal, i.e. independent. If one uses states which are not orthogonal as a basis, the states are no longer independent even though one still has a set of P(i) which sum to 1. We consider a 2 state orthogonal system (eigenfunctions of a Hamiltonian) and simultaneously of an operator A (i.e [H.A]=0). We define density operators as: d = Sum over i p(i) I i><iI whether or not the states i are orthogonal and insist that <A> be the same regardless of the basis used. In the two-state case, this may lead to different von Neumann entropy values for states which are not created by a unitary transform of the inital two energy eigenstates. We argue that the p(1)=p(2) = .5 eigenstate represents an extremization of entropy and that a shift to nonorthogonal states leads to a higher negative entropy. We also argue that nonorthogonal states have an overlap and so are not independent whereas physically the probabilities used classically are assocated with states which do not overlap, i.e. a heads up or down coin represent nonoverlapping states.