The Observer Notebook · 2026-10-07

Undo-precision budget

How precisely must Wigner know his friend's mind to rewind her measurement?

Wigner's friend sees a result, and her memory scrambles it through many small two-qubit steps (gates). To undo the observation, Wigner must run those steps backwards. Here is the finding. Small random errors on every gate are cheap: the rewind survives while the error per gate stays below about one over the square root of the number of gates the record actually touched (its light cone). Not knowing even one of those gates is fatal.

Partly known

01

Rewind a noisy brickwork circuit

Below, the friend writes one bit into qubit 0 and thinks with a random brickwork circuit. Wigner rewinds with his model of that circuit, where every gate carries a small random error of size ε. Each tick draws a fresh circuit and error, runs both branches exactly, and scores restored coherence C: 1 means a perfect undo, and above 0.707 the rewound friend can still violate a Bell-type test.

Live · exact statevector · n = 6, depth 12, 30 gates DEMO SCALE · THE REAL RUN USED n = 4–12 QUBITS, DEPTHS 4–24
the friend's circuit · ember = in the record's light cone · cursor = Wigner's rewind
restored coherence C vs per-gate error ε

starting…

What runs: each true gate G is a random 4×4 unitary. Wigner's copy is G·exp(iεH), with H a random Hermitian of size 1. Coherence is C = |⟨0|V†X0VX0|0⟩| with V = U′†U, the same rule as the notebook's exact sim. The dashed curve is the real run's mean at this exact size (n = 6, depth 12). The dotted curve is the entry's fit, exp(−0.3 ε² Glc). The toggle swaps one light-cone gate in Wigner's model for a random guess.

0.95–1.12ε* × √Glc, n = 4–12, depths 4–24
EXACT SIM
~1/16coherence left by one unknown gate, scrambled
EXACT SIM
~2×precision to undo vs to read (1.05 vs 2.0–2.35)
EXACT SIM
× 2−kk glimpses multiply in, within 0.005 (n = 10)
EXACT SIM
02

In plain words

The budget has two parts. Fuzz is cheap: coherence falls like exp(−0.3 ε² Glc), so Wigner only needs ε around 1/√Glc. Errors on gates outside the record's light cone cost exactly nothing. Ignorance is expensive: one gate he cannot name, deep in the scrambled region, leaves about 1/16 of the coherence.

Two surprises. Undoing needs about twice the precision that merely reading the record needs. And same-direction (systematic) errors did not pile up linearly as we predicted. The random circuit twirls them into effectively random errors. Glimpses and model error multiply: C(ε, k) = C(ε, 0) × 2−k.

Prior artThe echo is an OTOC (Yan, Cincio, Zurek, 1903.02651). Only light-cone noise matters for OTOCs (Kechedzhi et al, 2306.15970). Reading is about as hard as swapping (Aaronson, Atia, Susskind, 2009.07450). No Wigner's-friend reversal error budget was found (Wiseman, Cavalcanti, Rieffel, 2209.08491; Zeng, Labib, Russo, 2409.15302). All fetched.

Next clickAn unlearning analog. Does a model's "light cone" (the weights a datum touched) set the same split between cheap fuzz and fatal ignorance?

These are exact simulations at small sizes, n = 4 to 12 qubits. The 0.3 constant and the systematic-error result still need a 20 to 24 qubit check. A Floquet circuit (the same two layers repeated) gave no clean scaling at these sizes. The combined formula is a synthesis of known pieces, and it is not a new law. The error model is one choice: an independent random Hermitian kick per gate. The live panel is one demo size (n = 6) with its own sampling noise, and the 1/16 line comes from the larger runs (n = 8 and 12), so the demo's floor sits a little higher.

Nearby on the bench