Quantum Measurement Bounds beyond the Uncertainty Relations

In quantum mechanics, the Heisenberg uncertainty relations and the Cramér-Rao inequalities typically limit the precision in the estimation of a parameter through the standard deviation of a conjugate observable. Here we extend these relations by giving a bound to the precision of a parameter in terms of the expectation value of the conjugate observable. This has both foundational and practical consequences: in quantum optics it resolves a controversy over which is the ultimate precision in interferometry.

Figure 1

Lower bounds to the precision estimation $\Delta X$ as a function of the experimental repetitions $\nu$. The gray area in the graph represents the forbidden values due to our bound (2). The hatched (dashed-line) area represents the forbidden values due to the Cramér-Rao bound, or the Heisenberg uncertainty, (1). Possible estimation strategies have precision $\Delta X$ that cannot penetrate in the colored regions. For large $\nu$ the Cramér-Rao bound (which scales as $1/\sqrt{\nu }$) is stronger, as expected since in this regime it is achievable. Our bound is not achievable in general, so that the gray area may be expanded when considering specific estimation strategies. [Here we used $⟨H⟩-E_0=0.1$ (a.u.) and $\Delta H=4$ (a.u.).]