Quantum-mechanical uncertainty and the stability of incompatibility

In talking about the compatibility of quantum observables, discussions often center on the question of whether the corresponding operators commute - even though commutativity is a coarse-grained notion that largely fails to capture the salient "nonclassical" features of quantum theory. Often, too, such discussions involve the issue of whether the operators in question satisfy a Heisenberg-like inequality, of the form Delta A . Delta B greater than or equal to r > 0 - even though such inequalities are specific to unbounded operators and (for this and other reasons) are typically not a useful way to discuss joint uncertainty in quantum mechanics. In the present paper we emphasize a simpler dichotomy, in which operator pairs (A, B) are classified according to whether or not states can be found with arbitrarily small dispersions in both A and B. If A and B cannot be made arbitrarily dispersionless simultaneously, then we call A and B an uncertainty pair. Otherwise, we call A and B a certainty pair. An interesting feature of uncertainty pairs in particular is that they are stable, in the sense that if A and B form an uncertainty pair, then slight enough perturbations of A and B must also form an uncertainty pair. This stability, obvious in the finite-dimensional case, follows in general from an operator inequality derived herein. A consequence of this inequality is that perturbed position and momentum operators X+delta X and P+delta P cannot share an eigenvector unless \delta X\ . \delta P\ greater than or equal to h/2. (Here vertical bars denote the operator norm.) This, despite the fact that (as we show) arbitrarily slight perturbations of X and P can fail to satisfy a Heisenberg inequality - a fact which raises interesting measurement issues in its own right.