Two coins always disagree (HT/TH only); a singlet pair, two dots until the first click, measured along z gives the same tallies; the tables overlap exactly; the question: is there any difference at all, and where is it hiding; the singlet formula with its minus sign.
Window rule: Octo (eight spins up); the blue frame A around the left spins with brace and the symbol ell; the rest dims to grey, B the rest; ell = |A|: today a few spins, by Part 4 a long window in an endless chain (no size is written as a number); progress dots; the frame stays.
One spin as an arrow (side view). Tilted by theta it is cos(theta/2) up + sin(theta/2) down; the odds bars carry cos^2 and sin^2 of half theta, and at a right angle they are one half each. A coin behind a curtain prepares a second spin up or down ten times (entries 1 and 0); the average is a dot with the same odds, one half each: same odds, different objects. The 5 s sphere turn takes the side view to the top view, home of every rotation about z.
The 2x2 table of the tilted spin: odds on the diagonal (from the odds bars), cos sin in the corners, the matrix written beside it; the coin loaded to the same odds has black corners (its matrix has zeros there); the theta sweep (the corners brighten, peak at a right angle and fade, the coin's stay black); corners painted red and hatched; the table is named: the density matrix rho. Question: what colour are the corners of one spin of the singlet?
Two coins, one table (rows left coin, columns right coin; 0, 1/2 / 1/2, 0); a curtain over the right coin; adding across each row gives the left coin alone, 1/2 and 1/2; p_A(i) = sum_j p(i,j), the word 'add' underlined for one beat.
The singlet's 4x4 table: two diagonal cells at one half, two corners at minus one half (red, hatched). Curtain over B. The rule: keep a cell only if its two B letters agree, add the kept cells that share the same A letters, discard the rest. The corners (up-down against down-up) disagree and are discarded; what survives is the 2x2 table for A: one half, one half, corners black, a fair coin, the dot of 1.3; rho_A = Tr_B |Psi>
The singlet as a sum of two paired terms with weights one over root two; the squares of the weights, a half and a half, are the two entries on the diagonal of the 2x2 table: the eigenvalues of rho_A, drawn as eigenvalue bars labelled one half each. Part 3 keeps these bars when it compresses a chain.
Vote. The spin pair with the curtain over B and the singlet written out. Pause card: is the left spin of the singlet in a definite state? Options: yes, one we have not found; no, and no measurement on A alone can tell it from a fair coin; it depends on what B does. Hands up.
The answer: rho_A is one half, one half, corners black, the fair coin's table; nothing done to A alone can tell them apart (option two). The knob: cos(theta/2) up-down + sin(theta/2) down-up; at 0 the left spin is definitely up, at a right angle it is the fair coin, the pair one pure state the whole way. Scoreboard: coins log 2 / log 2, singlet 0 / log 2. Classical ignorance flows downhill; only quantum lets the part know less than the whole. Next: the ruler.
The ruler: eigenvalue bars become -p log p pieces (natural log) that stack in the Meter. Fair coin: two pieces of half log 2, the tube reads log 2 (the tick); S(rho_A) = -sum_a p_a log p_a, the letter S on the Meter. Loaded three to one: pieces 3/4 log 4/3 and 1/4 log 4, the level below log 2. Certain coin: both pieces vanish, 0 (0 log 0 = 0). The trap: the tilted spin's table beside the loaded coin's, same diagonal; a ghost Meter fed the diagonals reads the same for both, crossed out. Eigenvalues, not diagonal entries: the spin's table along its own arrow is 1, 0 / 0, 0, its Meter reads 0 (the red corners said so), the coin's stays below log 2. Pointed at one spin of the singlet: log 2, the most one spin can read; at the whole singlet: 0. The scoreboard of 1.6 read once more: coins log 2 / log 2, singlet 0 / log 2.
Badge: n = 2 is the entropy you read by overlapping two copies; Part 2 needs that trick. Four eigenvalue bars of unequal height, p_1 to p_4, tallest first; the Meter at their entropy S. The gold Dial n and the formula S^(n) = log Tr rho^n / (1 - n), Tr rho^n = sum p_a^n. n = 2: bars squared, their sum less than one, the reading drops; n = 3: further; far end (tallest bar held at its height): only the tallest bar survives, S^(inf) = -log p_max. Back towards 1: zero over zero (log Tr rho^n -> log 1 = 0 over 1 - n -> 0), the dial reads the limit, the Meter of 1.7. Two translucent copies of the chart slide into the unit square cut by p_1 .. p_4 on both edges; the diagonal cells have area sum p_a^2 = Tr rho^2; the dial back at 2, the bars squared again: n = 2 is minus the log of that overlap.
S(ell) against the window size ell for three chains (a sketch: no numbers on the axes): a product state (every arrow up) reads zero for every ell; a random state climbs like ell log 2 (dashed line over the first windows); the pinned ground state of a small ring (two rings, one marker each, on top of each other) rises for the first couple of spins and has stopped by the next even window, just under the dashed ruler at twice the half-chain entropy of the infinite chain (iDMRG, Part 3; no value quoted). Why: a gap means short-range correlations, only the two end spins of the window know the outside, and two ends is a fixed number however long the window gets: the area law (Black-box card, Hastings 2007).
Three things to take home. The density matrix: diagonal is odds, red corners are what a coin never has. The curtain: keep the cells whose B letters agree, add, discard the rest. The Meter: eigenvalue bars, minus p log p, and in a gapped chain it stops growing.
The Caveats, read by title only: same rho_A in every basis (asserted); 0 log 0 = 0, eigenvalues nonnegative and summing to one, S >= 0 with S = 0 only for a pure state (stated); the n -> 1 limit of S^(n) named, not derived; area law and gap -> exponential decay on faith; the random-state curve and the two small rings; the two Bloch views for the sphere; e^{i alpha Q_A} shown, not defined.
Top view, every spin rotated about z by the same angle alpha: the tilted spin's needle swings round; the window's red corners grow purple needles spinning at q_i - q_j while the colours (the sizes of the entries) do not change, so the Meter does not move: it is blind, it cannot even tell the corners exist; the singlet does not change at all. S(e^{i alpha Q_A} rho_A e^{-i alpha Q_A}) = S(rho_A), Q_A defined in Part 2. Which broke a rule? Part 2 builds a meter for the corners and finds it needs no entanglement.