Superspace geometry: the exact uncertainty relationship between complementary aspects

Abstract

After reviewing the standard uncertainty relations due to Heisenberg (1927), Robertson (1929), and Schrodinger (1930), as well as the relations of Deutsch, and of Maassen and Uffink-including the so-called entropic relations-the author presents a complete account of the uncertainty relationship between complementary aspects in terms of superspace geometry, an approach not hitherto employed. Two incompatible properties A= Sigma alpha A alpha mod alpha )( alpha mod and N= Sigma n N n mod n)(n mod belong to a pair of complementary aspects defined by two orthonormal bases ( mod alpha )) and ( mod n)) in the Hilbert space H. If the state is mod psi >, then P( alpha )= mod ( alpha mod psi ) mod 2 is the probability of obtaining the value A alpha in a measurement of A, and P(n)= mod (n mod psi ) mod 2 is the probability of obtaining the value N n in a measurement of N. The two aspects are characterised, relative to mod psi ), by the numbers (so-called purities): pi = Sigma alpha P( alpha ) 2 and pi = Sigma n P(n) 2 , both <or=1. He gives a complete characterisation of the uncertainty relationship between A and N (more precisely: between their aspects) in terms of the range of joint values of ( pi , pi ) for arbitrary initial states (pure as well as mixed). A theorem of Lenard is given an alternative proof, employing only elementary (superspace) geometry. The results depend on two angles, phi m =minimal angle, and phi M =maximal angle between the two aspects. Exact expressions for phi m and phi M are obtained in terms of the overlap matrix Lambda =( Lambda alpha n )=( mod ( alpha mod n) mod 2 ). As a corollary he finds the uncertainty relation for a pure state mod psi ) pi + pi <or= 1+1/g+1-1/g cos phi m (where g=dim H), and a sharper one for mixed states. pi + pi = 2 is obtainable if and only if the intersection of the aspects holds a pure state. If phi m = pi /2 (maximal incompatibility), then pi + pi <or= 1+1/g is a special case of a stronger relation: Sigma mu g =o pi ( mu ) =2, which one obtains for g+1 maximally incompatible aspects by means of a thereom of Ivanovic.