Robertson uncertainty relation

In plain terms

The uncertainty principle is quantum mechanics' most famous export and its most misquoted, and the precise statement is Robertson's. Take any two measurable quantities and any quantum state. Each quantity has a spread in that state, the amount by which repeated measurements would scatter around their average, and Robertson's inequality says the product of the two spreads can never fall below a certain floor. The floor is set by how much the order of the two quantities matters, as applying one and then the other needn't give the same result as the reverse order, and half the size of the average of that difference, called the commutator, is the bound. For quantities whose order doesn't matter the floor is zero, and both can be sharp at once.

No story about measurements disturbing each other is needed, as the inequality is a fact about statistics, and it constrains the spreads any state can have before anyone measures anything.

The corpus proves Robertson's inequality and then does the part that belongs to the CSD picture. Each spread is identified as an ordinary average over the underlying state space, the arena, of a quantity that genuinely varies from point to point, so the spread the inequality bounds is the spread of something really distributed and not just a number attached abstractly to a state.

Two concrete examples anchor the result, including one where the two sides are exactly equal, so the inequality is tight and the formalisation exhibits the tightness.

In CSD

This belongs to the observable-correspondence tier of the programme, the part that connects quantum observables to quantities on the arena. A quantum observable, one of the matrices that represents a measurable quantity in textbook quantum mechanics, acquires a representative on the arena, a real-valued function whose ordinary integral against the arena's measure reproduces the textbook expectation value. The correspondence is built from two sides, as on the quantum side the observable is broken into its possible values and the states carrying each value, and on the arena side the space is partitioned along its fibres, the sets of underlying points that share the same quantum state.

With that correspondence in place, Robertson's inequality holds with its left-hand side read on the arena, so the uncertainty bound becomes a statement about the spreads of arena quantities. The examples use pairs of spin observables of a qubit on explicit states, one example saturates the bound, and a parametric version proves the inequality for spin measured along arbitrary directions.

Mathematically

The main theorem, kahler_robertson_ontic_variance, states that the product of ontic- side variances dominates the commutator bound, with the variance identification running through the spectral-region integral identities, so the ontic centred second moment equals the Hilbert-space variance.

The companions are pauli_xy_robertson_saturation, with both sides equal to one, and the parametric pauliDot version over arbitrary axes.

The name

Howard Percy Robertson (1903 to 1961) was an American mathematical physicist at Caltech and Princeton, known jointly for the Robertson-Walker metric of cosmology and for the 1929 general uncertainty relation, a one-page Physical Review note that sharpened Heisenberg's heuristic into an inequality for arbitrary pairs of observables. Schrodinger later refined the relation by adding the covariance term.

Related entries

• Ozawa error-disturbance relationThe famous error-times-disturbance formula attributed to Heisenberg fails in the laboratory, and the relation that actually holds is Ozawa's, with three terms instead of one, and the corpus proves it.
• Hermitian operatorA single symmetry condition on an operator guarantees that every number it can produce in a measurement is real, and it delivers the whole machinery of outcomes and eigenstates at no extra cost.
• No exact finite-dimensional CCRIn a world with only finitely many distinguishable states, the founding equation of quantum mechanics can't hold exactly, and one line of arithmetic proves it while CSD says precisely where the equation gives way.
• Fisher informationThere's a floor under every error bar, classical or quantum, and Fisher information is what sets it.
• Pauli matricesThree small matrices carry the entire two-level world, and their refusal to commute is the uncertainty principle in its smallest packaging.