Illustrated Physics
physics, drawn
Concept

Shannon entropy — where it appears

The average information needed to specify an outcome drawn from a probability distribution, measured in bits: one for a fair coin, zero for a certain result. Uncertainty relations written in it bound how predictable incompatible measurements can be together, where variances can fail.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Two measures of how uncertain a split state is. For the two-peaked state, against half the separation of the peaks in units of their width: Heisenberg's product σₓσₚ divided by its minimum ħ/2 (solid), and the entropic sum Hₓ + Hₚ minus its minimum ln(πeħ) (dashed), both computed from the state's distributions. As the peaks separate the product grows without limit, 8.1 times the minimum at the right of the figure, because the variance counts the empty space between the peaks as uncertainty. The entropic excess levels off at 2 ln 2 − 1 = 0.386 (0.386 at the right): the position's entropy gains ln 2, one bit, for not knowing which peak, and the momentum's gives back 1 − ln 2 because its fringes pack it into narrow bands. The entropic relation says what is true of the state: it is a fraction of a coin toss away from a minimum-uncertainty packet, however far apart the two places are.

The sharpness a variance cannot see

Heisenberg's inequality is a statement about standard deviations, and a standard deviation is a poor measure of how sharply something is known. A particle that is certainly in one of two narrow places a metre apart has a position spread of half a metre, and satisfies the inequality with room to spare while being sharply localised. Measure uncertainty by information instead — by the Shannon entropy of each distribution — and the relation tightens: the sum of the two entropies has a floor that the split state cannot get under either, and the floor survives in places where the variance form says nothing at all.

quantum · Uncertainty Every pure spin owes at least two bits. The total uncertainty, in bits, of the answers to three questions about a spin-½ — is it up along x, along y, along z — for pure states round two great circles of directions. A spin along z answers the z question with certainty and the other two as coin tosses: 0 + 1 + 1 = 2 bits, the least any pure state can have. A spin tilted equally towards all three axes is uncertain about every question and sums to 2.232 bits, the most. Every pure state lies between, a narrow band: the three questions together cannot be answered with less than two bits of doubt.

The third question pairwise bounds miss

Ask a spin-½ whether it points up along x, along y and along z, and quantum mechanics guarantees that the three answers together carry at least two bits of uncertainty. No pairwise uncertainty relation says so: added up across the three pairs they promise only one and a half bits, and the best general formula for many questions promises 1.755. The true floor is higher than any bound built from its parts, the familiar variance relation cannot see it at all, and an entangled partner can remove it entirely.

quantum · Uncertainty

Named alongside it

The objects these essays reach for when they reach for this one.

Uncertainty principleAngular momentumBloch sphereEntanglementEntropic uncertaintyFourier transformMutually unbiased basesQubitSuperpositionVarianceWave packet

All concepts