Information Exclusion Principle for Complementary Observables

The sum of the information gains corresponding to measurements of position and momentum is bounded by $\log 2\Delta X\Delta P/\hslash$ for a quantum ensemble with position and momentum uncertainties $\Delta X$ and $\Delta P$. The bound implies the Heisenberg uncertainty principle and that the gain of position information can be maximized only at the expense of momentum information, and vice versa. This exclusion principle for the information contents of complementary observables is extended to finite Hilbert spaces, and to the quadrature, number, and phase observables of bosonic fields degraded by Gaussian noise.