Measurement Induced Phase Transitions

For low measurement rate $p$ the system is in volume law. There is a critical measurement rate $p_c$ after which the system is in an area law phase. In the volume law entanglement, entropy scales as $S\propto L^d$ while for area law it scales as $S\propto L^{d-1}$. At the phase transition $S\propto \log L$.

Diagnostic	Indicator of Transition
$S(p)$ vs $L$	Curve crossing or slope sharpening
$S(L/2)$ vs $L$ at fixed $p$	Switch from volume to area law
$\text{std}[S(p)]$	Sharp peak near $p_c$
$I(i:j)$	Decay to zero in area law
$S(t)$	Linear growth vs early saturation

Quantum Trajectories

A quantum trajectory is one possible sequence of measurement outcomes, and the resulting pure state of the system after those measurements and unitaries. Think of it like a single history of how the system evolved with randomness in measurement.

To simulate MIPTs:

• You simulate many such trajectories (say, 1000)
• Each one corresponds to a random pattern of which qubits were measured (and what outcomes you got)
• For each trajectory, the system evolves as a pure state, and you compute entanglement entropy on that state.

Then, you average the entropy over all trajectories.

Post Selection Problem

After performing a measurement, you only keep the outcomes you want, and throw away all the others. For example, if you measure a qubit, you may get 0 half of the time and 1 the other half. In post-selection, you choose only the trajectories where the outcome was 1. This leads to a conditional state — the state of a system given a particular outcome. Averaging over all the trajectories results in a mixed state.

To measure MIPT of a particular trajectory, you have to post-select a particular measurement record, so that the output is a pure state. That way, the entanglement entropy is well-defined.

Circuit Implementation

Qiskit Limitations

• Mid-circuit measurements in Qiskit collapse the wavefunction, and you can’t directly access the post-measurement pure state for entanglement.
• You can simulate measurements or statevectors — not both meaningfully together.
• You don’t get post-selected pure trajectories automatically.

QuTip QuTiP lets you

• Evolve a pure state
• Explicitly collapse it by projectors like $|0⟩⟨0|,|1⟩⟨1|$
• Manually apply post-selection (i.e., you keep the collapsed state)
• Track the trajectory as a pure quantum state, allowing:
  • Reduced density matrices
  • Entanglement entropy
  • Mutual information
  • Time-dependent entanglement $S(t)$