Two objects from Part 1: the spin along x (top view, needle along x) and the dot, the fair coin. A run of z measurements on each gives the same histogram, half up and half down. Their tables differ: the spin has one half in every cell, red corners included; the coin has black corners. The Meter could not tell which way the corners pointed, but it tells a pure table from a coin: one full eigenvalue bar and the Meter at zero for the spin; two equal bars and the Meter at log 2 for the coin.
The move this part is about: a friend measures your spin along z, writes the result down and never tells you. You know it is up or down, not which: that is the coin. The red corners are gone, the bars are flat and the Meter has climbed from 0 to log 2; the secret cost log 2 (gold). The same ruler over the coin changes nothing: it was already a coin; the secret cost 0.
That cost is the meter we build today. New: defined in 2023 (Ares, Murciano, Calabrese); in 2024 it explained the quantum Mpemba effect (Rylands and co-workers), seen with trapped ions the same year (Joshi and co-workers); a 2025 review. Black box: the definition is taken from those papers; the Mpemba effect is named, never derived; the paper reads how far a window sits from a symmetry the ground state breaks. Our question: what did the secret cost, and why does it grow with the window?
The rule, seen from the top: rotate every spin about z by the same angle alpha; needles in the plane turn together, arrows along z do not move. Its charge Q_A = ups minus downs, halved = 1/2 sum sigma^z: up up down up gives 1, and the rule cannot change it. The kets of the window sorted into drawers, one per charge from ell/2 down to -ell/2 in the sector colours; a drawer holds binom(ell, ell/2 + q) kets, so the middle drawers are fullest. The rotation multiplies a drawer state by e^{i alpha q}: a needle turning q times per turn of alpha, none for q = 0. The legend docks for the rest of the series; odd windows have half-integer charges but integer differences.
The Grid of the window ell of the small ring magnet (a ring of L spins, exactly solved, at the home couplings Delta_z, gamma; no value is quoted), rows and columns reordered by drawer q = 2, 1, 0, -1, -2: the diagonal blocks in the sector colours, one per drawer, every off-diagonal block red and hatched. Each red block has a weight, the total squared size of its cells, shown as a bar: one drawer apart carries most of the red weight, two apart a little, three apart almost nothing (the two corners four apart get no row), so the neighbouring drawers talk most. rho_A = sum_{q,q'} Pi_q rho_A Pi_{q'}, Pi_q keeps what sits in drawer q; every block shrunk to one cell gives the block cartoon (brightness = square root of the weight), the Grid for the rest of the part.
One tilted spin in the top view with its Grid: two drawers q = +1/2, -1/2, two red corners, mirror images. Rotating by alpha grows a purple needle on each corner, the two turning opposite ways; the colours never change. The block cartoon of the window: every red cell grows a needle, one drawer from the diagonal turns once per turn of alpha, two drawers twice, three three times, the outermost corners fastest; the diagonal cells do not turn. The formula rho_qq' -> e^{i alpha (q - q')} rho_qq': speed = charge difference. The Bell pair sorted by charge: its two corners pair up-down with down-up, the same charge, inside the q = 0 block; nothing turns. Its red corners are not the kind this rule can see. If asked why the spin's needle and the cell needles turn opposite ways: the Bloch needle turns by -alpha and the (q, q') cell by +alpha (q - q'); both are what e^{i alpha Q_A} rho e^{-i alpha Q_A} does.
Vote. The Pause card alone: a friend measures the charge of the window (the magnet's window of 2.3) and never tells you the result; is the window now more certain, less certain, or the same? Hands up.
The block cartoon and the charge-sorted Grid of 2.3; the grey z-ruler passes over both (a friend measures Q_A and does not tell): every off-block cell fades to black, every cell inside a diagonal block stays, the former corners inside the q = 0 block included. The erased table is rho_{A,Q} = sum_q Pi_q rho_A Pi_q. Its eigenvalue bars are flatter; the white Meter (Dial at n = 2) rises, never down (Black box: Delta S >= 0, asserted; the circle average of 2.8 is the reason). The rise, painted gold, is the asymmetry Delta S^(2)_A of the small ring's window, a difference of two Meter readings: the table after the secret measurement minus the table before; no red blocks, nothing to erase, zero. Beside it the paper's infinite-chain value for the same window as a thinner mark, a hair shorter: the ring is a toy, and a close one; no value is quoted. The answer to the vote: less certain.
Two toys on a napkin. One tilted spin: after the secret z measurement its table is the diagonal cos^2, sin^2 of theta/2, and the gold meter at n = 2 reads -log(cos^4 + sin^4): nothing at theta = 0, a little at pi/6, more at pi/3, log 2 at pi/2 (the spin along x); the Meter's ticks name the angles, no value is read out. One spin, no partner, and the meter moves. Second toy: one half on the diagonal, c in the corners; the secret measurement wipes out c and the answer is log(1 + 4c^2): 0 at c = 0, log 2 at c = 1/2; a mix of the spin along x and the fair coin, c sets the proportions.
The scoreboard: one spin of the Bell pair, Meter log 2, asymmetry 0; one spin along x, Meter 0, asymmetry log 2; the two meters disagree completely, and that is the point. In the papers this quantity is called the entanglement asymmetry; the name is historical: the spin along x has nothing to be entangled with and reads log 2, the Bell pair is as entangled as two spins get and reads 0. From here on: the asymmetry, or the Meter for the corners.
Vote again. Octo, every arrow along x, a product state: the window's Meter reads zero whatever its size, and its Grid is dense, every cell the same grey, one over two to the ell. As the window grows, what does the asymmetry do? Saturate, grow like log ell, or grow like ell? Hands up.
The shortcut for a pure window: after the secret measurement only the charge histogram P(q) is left (the diagonal blocks are rank one, their traces P(q) are the eigenvalue bars), and the asymmetry is the Renyi entropy of that histogram; binomial for coins along x, the drawer sizes over two to the ell. The first points climb, each doubling adding about the same amount, a slope just under one half per log ell; doubling on and on, the slope settles at one half: half the log of ell, the answer in the middle (gold line, 'It grows like log ell' highlighted). Why one half: the binomial's bell widens like the square root of ell and the asymmetry is the log of that width; the next three scenes turn this sentence into a formula.
Three clocks, each hand e^{i m alpha}, alpha once round the circle: the m = 0 hand stands still, m = 1 goes round once, m = 2 twice. Average each hand over a full turn (the gold dot): for m = 0 the average is the hand itself; for every other m the running average spirals in to the centre, zero. Averaging over all rotations keeps m = 0 and kills the rest. As a formula, Pi_q = int dalpha/2pi e^{i alpha (Q_A - q)}: only drawer q survives, the hand that stands still; that is the drawer projector of 2.3.
Apply the clocks to the Grid: the block cartoon with its needles and, beside it, its running average; as alpha runs once round, every red block averages to black, the diagonal blocks stay: the average over all rotations is exactly the erased table, the friend's measurement is that average (z-ruler stamped). n = 2 was the overlap of two copies: overlap the Grid with its rotated copy, call it Z_2(alpha) (the purple bump), average over alpha (gold area) and you get the erased table's overlap with itself: Tr rho_{A,Q}^2 / Tr rho_A^2 = int dalpha/2pi Z_2(alpha); n = 2 is minus the log of an overlap. On the napkin toy: Tr[rho rho(alpha)] = 1/2 + 2c^2 cos alpha, its running average 1/2 + 2c^2 sin alpha / (alpha + pi) lands on the circle average 1/2 = Tr rho_{A,Q}^2, and the ratio, minus the log, is log(1 + 4c^2), the 2.6 formula; nothing erased by hand. The same identity holds for any table (it is asserted on a random two-spin table in the code; no number is quoted). General n: one angle per copy on a ring of copies (2.11; App. A of the paper), on the Caveats.
Two coins. The charge histogram is 1/4, 1/2, 1/4. For a pure window the overlap with the rotated copy is the squared size of one complex number, the overlap of the state with its rotated self, the average of e^{i alpha q} over the histogram: three purple needles tip to tail, a quarter turning backwards, a half standing still, a quarter turning forwards; their sum stays on the real axis and traces cos^2(alpha/2) as the alpha arc sweeps. Squared: Z_2 = cos^4(alpha/2) (pure window; the mixed case waits for 4.2). The gold area under it over 2 pi is 3/8, and -log 3/8 is the first point of 2.7, with no big table touched.
Four panels, ell doubling and doubling again (ell, 2 ell, 8 ell, 32 ell): the bump Z_2 = cos^{2l}(alpha/2) (one factor cos(alpha/2) per spin along x, squared) narrows from panel to panel; the gold dashed Gaussian e^{-l alpha^2/4} laid over each is indistinguishable from it by the last panel; the white brace at half height shrinks like 1/sqrt(l). Z_2 ~ e^{-V alpha^2} with V = l/4: not a fit, the variance of the coins' charge histogram. Remember the letter V; it is tonight's open hole.
Left: the purple bump Z_2(alpha) ~ e^{-V alpha^2} on the circle [-pi, pi]. Right: a Galton board, rows of pegs (each row one coin), white dots fall along the stored walks and pile into the bins: the charge histogram of a few spins (no counts printed). The pile fades into the gold P(q) bars of the window, then the window doubled, a gold Gaussian of variance V = l/4 through the bar tops: the Bell. The two panels step l -> 2l -> 4l together: the bump narrows as the Bell widens, the two 1/e widths (1/sqrt(l), sqrt(l)) multiply to a constant: Fourier partners. The average: the area under the bump on [-pi, pi] shaded gold; the axis zooms out to [-3pi, 3pi], the circle's edges recede, the shading runs along the whole line (the tails we add are nothing). int d alpha/2pi e^{-V alpha^2} = 1/(2 sqrt(pi V)); minus the log: Delta S^(2)_A = 1/2 log(4 pi V) = 1/2 log(pi l) for coins.
How good is it? The exact binomial value -log[binom(2l, l) 4^{-l}] sits a hair above the formula 1/2 log(pi l), by 1/(8l) and smaller terms (Stirling): already small for a few dozen coins, gone to the eye for a few hundred; no value is quoted. Half the log of the variance: a bell curve is known by its variance alone, and so is this meter.
Turn the Dial: the coins' asymmetry at n -> 1, 2, 3, 5 against log ell is four parallel lines of slope one half. Only the offsets move, pure numbers log n/(2(n-1)): one half, half log 2, a quarter log 3, an eighth log 5, with the common 1/2 log(2 pi V) written once; the gap between n = 2 and n = 3 is half log 2 minus a quarter log 3, nothing fitted (no free parameter).
Put n copies of the table on a loop with a Gaussian bump of the same width at every junction: like n beads on a loop of string it vibrates in n ways; the product of the n - 1 moving modes' frequencies is n, the mode that does not move has length root n, and their ratio is the 1/root n in front of (2 pi V)^{-(n-1)/2}: the gold offset. Two cards on faith: equal widths (the replica ring), and n -> 1 as a limit (card again in 4.3).
Three things to take home. Sort the Grid by charge: inside a block nothing turns, outside every block spins, and erasing the outside is the friend's secret measurement. The asymmetry is the Meter's rise, and it needs no entanglement. For a pure window it is the entropy of the charge histogram, and for coins half the log of four pi V.
The Caveats, read by title only, page one: Delta S >= 0 asserted, the proof not shown; the circle-average identity checked on the napkin toy and a random two-spin table, the rest is App. A; the pure-state shortcut of 2.7-2.10, the mixed case is 4.2; circle extended to the line, bump replaced by its Gaussian (the gap is 1/(8 ell) and smaller terms); the replica loop and n -> 1 on faith.
The Caveats, page two: the rotation sign (arrows -alpha, Grid needles +alpha); odd windows; the small magnet is a toy (a short ring, exactly solved, pinned to one stripe pattern) and the paper's infinite-chain mark sits beside it, never on its curve; the historical name said once, the Mpemba effect named, not explained; chi_zz flashed, not defined.
The hole, V: for coins it is ell over four, a binomial's variance (the gold token slides into the hole and out again while Octo comes in with every needle along x, the coins; chi_zz flickers beside the hole, not defined yet). Every second needle swings round into stripes alternating along x: a cartoon of the ground state of a magnet that prefers x over y; its real ground state is these stripes, blurred. What is its variance? Who sets it? And when the window is mixed, is V a variance at all? Part 3 (third dot pulsing).