Measurement-Induced Entanglement Transition
Take a chain of L qubits. Each time step, scramble neighbouring pairs with random two-qubit gates, then measure every qubit with probability p. Gates spread entanglement; measurements cut it. Below a critical p the gates win and the entanglement between the two halves of the chain grows with L — a volume law. Above it the measurements win and it stops growing — an area law. In between is a genuine phase transition in the structure of a quantum state, found in 2018 and still an active research area.
This page runs the experiment rather than describing it: random Clifford brickwork, projective Z measurements, half-chain entropy averaged over independent circuits. It is the stabilizer-circuit numerics of Li, Chen & Fisher, at sizes a browser tab reaches in seconds.
11 measurement rates from 0 to 0.4, each circuit 2L layers deep from the all-zero state, open boundaries. L = 128 at 100 circuits per point takes a few minutes.
How to read it, and what it does not show
- The dashed line is not computed here. p_c ≈ 0.16 is the value reported for this model from finite-size scaling on chains of hundreds of qubits (Li, Chen & Fisher, Phys. Rev. B 100, 134306, 2019). At the sizes on this page the curves separate over a range of p, not at a point; extracting a critical rate from them would be reading more into the data than is there.
- Why Clifford gates. A generic random circuit on 64 qubits cannot be simulated. A random Clifford circuit can — the state stays a stabilizer state, and its entropy is a matrix rank. The transition exists for both; its critical exponents differ between the two.
- What the entropy is of. Each circuit, with its particular measurement outcomes, ends in a pure state. S(L/2) is that state’s entanglement, then averaged over circuits. The average state — outcomes thrown away — is featureless at every p, which is why this transition hid for so long and why observing it on hardware needs post-selection or a decoder.
- Simplifications. Open boundaries rather than periodic, depth 2L, and von Neumann entropy only (for stabilizer states every Rényi entropy is the same number).
Sources: Li, Chen & Fisher, Phys. Rev. B 98, 205136 (2018); Skinner, Ruhman & Nahum, Phys. Rev. X 9, 031009 (2019). The engine is the Stabilizer Simulator; its random two-qubit Cliffords are drawn uniformly from all 11,520.