Topic 3 of 7 · Statements 31–40

Uncertainty and quantum foundations

Uncertainty relations, Bell and Tsirelson bounds, contextuality, Gleason’s theorem and the Zeno effect.

Applies to the spread over many preparations, not to measurement disturbance.

Strengthens Robertson by adding the covariance term from the anticommutator.

Follows from $[x,p]=i\hbar$.

Noncommuting observables cannot all have sharp values at once, though particular states may be joint eigenstates.

No local hidden-variable theory satisfying Bell's assumptions reproduces all quantum correlations.

$S=E(a,b)+E(a,b')+E(a',b)-E(a',b')$.

Quantum mechanics reaches $2\sqrt2$ with a maximally entangled pair and suitable measurements, but no higher.

One cannot give every observable a predetermined value independent of which compatible observables are measured alongside it.

Any suitably additive probability on projectors must have the Born-rule form.

Frequent projective measurements can freeze evolution, since short-time decay is quadratic in $t$.

ψUncertainty and quantum foundations