The information entropy of quantum mechanical states

– It is well known that a Shannon based definition of information entropy leads in the classical case to the Boltzmann entropy. It is tempting to regard the Von Neumann entropy as the corresponding quantummechanical definition. But the latter is problematic from quantum information point of view. Consequently we introduce a new definition of entropy that reflects the inherent uncertainty of quantum mechanical states. We derive for it an explicit expression, and discuss some of its general properties. We distinguish between the minimum uncertainty entropy of pure states, and the excess statistical entropy of mixtures. The statistical state of a system (ρ) is specified in classical mechanics using a probability function, while in the quantum mechanical case it is specified by a probability matrix. The information entropy S[ρ] is a measure for the amount of extra information which is required in order to predict the outcome of a measurement. If no extra information is needed we say that the system is in a definite statistical state with S = 0. A classical system can be in principle prepared in a definite state. But this is not true for a quantum mechanical system. Even if the system is prepared in a pure state, still there is an inherent uncertainty regarding the outcome of a general measurement. Therefore the minimum information entropy of a quantum mechanical state is larger than zero. It is clear that the common von Neumann definition of quantum mechanical entropy does not reflect the inherent uncertainty which is associated with quantum mechanical states [1,2]. For a pure state it gives S = 0. Let us assume that we prepare two spins in a (pure) singlet state. In such a case the von Neumann entropy of a single spin is S = ln(2), while the system as a whole has S = 0. If it were meaningful to give these results an information theoretic interpretation, it would be implied that the amount of information which is needed to determine the outcome of a measurement of a subsystem is larger than the amount of information which is required in order to determine the outcome of a measurement of the whole system. This does not make sense. Thus we are faced with the need to give a proper definition for the (information) entropy of a quantum mechanical state. As in the case of the von Neumann entropy it can be regarded as a measure for the lack of purity of a general (mixed) state. But unlike the von Neumann