Geometry of Quantum States" SAMUEL L. BRAUNSTEINb AND CARLTON M. CAVESC Centerfor Advanced Studies Department of Physics and Astronomy University of New Mexico Albuquerque, New Mexico zyxw 87131-1156 zyxw INTRODUCTION There have been two threads running through John A. Wheeler's distinguished career: geometry as central to physics; the puzzle of the quantum. What could be more appropriate than that, in this volume dedicated to Wheeler, we pursue these threads in our own small way by formulating a natural Riemannian geometry on the space of quantum states-a geometry built on a concept, statistical distinguishability, that can be traced to Wheeler, who encouraged Bill Wootters' investigation of it. Another Wheeler theme is also important: communicate ideas clearly, as in his elegant summary of the key ideas of general relativity-space-time tells matter how to move; matter tells space-time how to curve. We can do no better than to characterize statistical distinguishability as Wootters did-a distance between quan- tum states, a statistical distance, quantified by how well measurements distinguish the states. It is a pleasure and an honor for us to dedicate this report to John A. Wheeler-a consummate researcher and extraordinary teacher and always a gentle- man. GEOMETRY ON THE PROBABILITY SIMPLEX In this section, we review Wootters' derivation' of the distinguishability metric on the space of probability distributions, the probability simplex. Let the n-tuple 6 z = (PI, . . . zyxwvuts ,pn) denote a probability distribution for discrete alternativesj = 1, . . . , z n. After N samplings from this distribution, we have a frequency f) for each alternative. These frequencies are distributed according to the multinomial distribution, N! prob(fl,. . . , fN) = aThis work was supported in part by the Office of Naval Research (Grant No. N00014-93-1- bCurrent address: Department of Chemical Physics, Weizmann Institute of Science, 76100 'To whom all correspondence should be addressed. 0116). Rehovot, Israel. zyxwvut 786