The fact horizon
How much of a friend's memory must leak before her observation can never be undone?
Known, reproduced
A friend measures a qubit in a sealed lab and stores the answer in her memory, then her thinking scrambles it across n memory qubits. Wigner, outside, can still run everything backwards and undo the measurement, as long as he holds the whole lab. Now let the environment take memory qubits one at a time. F is the most coherence Wigner can still restore. A Local Friendliness test (a Bell-type check on the friend story) needs F above 0.707. The exact run found the line is crossed at a sharp point: just over half the memory.
Leaking the memory, one qubit at a time
starting
left · the friend's memory (brass lines are her scrambling thoughts). Grey tethers mark the m qubits that were already tangled with the outside before she looked (an "old friend"). Ember qubits have been taken by the environment · right · F after k qubits are taken, from the exact run. Dashed line: 0.707. Dotted line: π/4, the best a blind look can do. Moss marker: k* = ⌊(n−m)/2⌋ + 1. Which qubits go first is cosmetic here: the run averaged over random scramblers, so only the count matters.
EXACT SIM
EXACT SIM
EXACT SIM
EXACT SIM
In plain words
With half the memory gone, Wigner can still restore F = 3/4, which beats 0.707. One more qubit and F drops to about 0.41. At that point the outcome is a fact: no operation by anyone can undo it well enough to pass the test.
An old friend, whose memory was already partly tangled with the outside, crosses sooner. In the run, a friend with m = n−1 old qubits became a fact on the first qubit taken (F = 0.60).
Looking is not enough. Measuring every memory qubit in a basis the friend's dynamics does not fix leaves F = π/4, still above 0.707. A look acts like taking the qubits only when the looker knows the scrambler and can decode it. Copying the outcome into many qubits makes it a fact after one qubit; scrambling erases that redundancy and pushes every copy count back to k* = n/2 + 1.