Whose superposition?
If one particle is "in two places at once", is that a fact about the particle, or about where you are standing?
We were wrong
Put three particles, A, B and C, on a ring. Physics only fixes where they are relative to each other. A quantum reference frame (QRF) is the view from one particle's seat: "where are the other two, measured from me?"
Switching seats is an exact reshuffle of the same quantum state. Yet it changes what looks entangled (linked so that neither particle has a state of its own). From C's seat, A is fuzzy and B is sharp, with no entanglement. From A's seat, that same fuzziness shows up as entanglement between C and B. One ratio sets how much.
Our first write-up said only two kinds of state stay unentangled from every seat: both particles sharply placed, or both moving as plane waves. That rule was wrong.
Grid states also qualify. If A and B each sit on an evenly spaced comb of spots with the same spacing and equal weights (like the GKP grid states used in quantum error correction), entanglement is exactly zero in all three seats. Pick grid (comb) in the panel below to watch it. Make one tooth heavier and it breaks. Source: sims/qrf/grid_states_check.py.
Hop between seats
starting…
Left · every dot is one joint position of the other two particles, seen from the current seat. Size shows probability. At each hop the dots slide to their new coordinates under the exact relabelling: from A's seat, C sits at −a and B at b − a. A rectangle of dots means the two are unentangled; a slanted pattern means they are linked. Right · entanglement between the other two, in bits, from each seat. It is computed live from the state's Schmidt spectrum. In two bumps, A sits at two sharp spots d apart and B is a bump of width s = 2; the brass tick is the closed form h((1+o)/2).
In plain words
"Is this particle in a superposition?" has no seat-free answer. The fuzziness of one particle and the entanglement of the other two are the same thing, written in different coordinates. The amount depends only on d/s: how far apart A's two spots are, compared with how wide B is. When B is wide enough to cover both shifts, nothing gets linked.
The corrected rule says when a state looks unentangled from every seat. Each particle has to be unchanged when shifted by any gap found inside the other one. A sharp spot passes trivially, and so does a plane wave. An evenly weighted comb passes too, as long as both combs share one spacing. Change one tooth's weight, or give the combs different spacings, and some seat sees entanglement.
Prior artPartly known. De la Hamette, Ludescher and Müller (PRL 2022) work out entanglement across reference frames. The grid-state case was our own correction.
Next clickQuantum Darwinism seen from a superposed seat.
- This is a toy: three particles on a discrete ring, one dimension, no forces, and pure states only.
- The page's ring has 32 spots; the notebook's run used 64. The formula and the grid-state result hold at both sizes.
- The original "only two families" claim was our error. The grid states were found by a direct check, and the old rule should not be quoted.
- "GKP-like" is an analogy in shape only. These are discrete combs on a ring, and they are not real oscillator codes.