Fundamental limits upon the measurement of state vectors

Abstract

Using the Shannon information theory and the Bayesian methodology for inverting quantum data [K. R. W. Jones, Ann. Phys. (N.Y.) 207, 140 (1991)] we prove a fundamental bound upon the measurability of finite-dimensional quantum states. To do so we imagine a thought experiment for the quantum communication of a pure state \ensuremath{\psi}, known to one experimenter, to his colleague via the transmission of N identical copies of it in the limit of zero temperature. Initial information available to the second experimenter is merely that of the allowed manifold of superpositions upon which the chosen \ensuremath{\psi} may lie. Her efforts to determine it, in an optimal way, subject to the fundamental constraints imposed by quantum noise, define a statistical uncertainty principle. This limits the accuracy with which \ensuremath{\psi} can be measured according to the number N of transmitted copies. The general result is illustrated in the physically realizable case of polarized photons.