Measuring the entanglement-breaking index of a qubit collision channel on a quantum processor: protocol and worked example

The entanglement-breaking index of a quantum channel is the smallest number of self-compositions after which the channel destroys all entanglement with any reference system. A photonic experiment has fixed an index of 2; to our knowledge no larger index has been located by measuring the n-round channel across rounds. We give a protocol for doing so on a programmable processor, for any qubit channel realised as a collision model whose ancillas are prepared in states diagonal in a known basis, so that the bath polarisation is set exactly by weighting basis-state configurations in the analysis. Single-qubit process tomography of the n-round channel gives the smallest eigenvalue of its partially transposed Choi matrix, read against zero at a pre-registered five standard deviations; entanglement breaking survives further rounds, so the readings are monotone in n and the index follows. Comparing each direct measurement with the composition of the measured single-round map tests the repeated-interaction assumption. We show that a readable step needs margins on both sides, show how depolarising and reset errors move the readings, and give a power-based pilot rule. As a worked example we take one round of a three-qubit feedback loop, compiled to eight CZ gates on a heavy-hex path with mid-circuit reset, with certified ideal indices 3, 2 and 2 for three baths. The run is estimated to cost a few hundred seconds of processor time, and simulations under a device noise model, with the pilot's shot raise at low noise, recover all three indices.