Dual Measurements and Information in Quantum Optics

Abstract

Quantum systems defined in a finite dimensional Hilbert space H _D spanned by complete, finite Fourier transform states |λ; n〉, (resp. |

$$\tilde \lambda$$

; n〉) with n ∈ [1, D], (mod. D), are examined. The main properties of dual observables are briefly reviewed. The information I(Ω, A) associated with a measurement A of a system described by a state Ω, is introduced and its properties are analyzed. The inequality$I\left( {\Omega ,A} \right) + I\left( {\Omega ,\tilde A} \right) \leqslant \log D$ is demonstrated for dual measurements (A, Ã). The results are then applied to some operators of interest in quantum optics. For instance, it is shown that under some conditions, the number and “phase” observables for a harmonic oscillator are complementary and the inequality is “equivalent” to the uncertainty relation. Quantum correlations and the case D → ∞ are also considered.