Certified targets for measuring and controlling an entanglement-breaking index on programmable hardware

Abstract

The entanglement-breaking index of a quantum channel is the number of self-compositions after which the channel destroys all entanglement with any reference system. For qubit channels it is decided by a partial transpose, and to our knowledge it has never been measured. We give a certified design for measuring and controlling it on programmable hardware, using one round of a collision model: a message qubit meets two fresh thermal ancillas. The ancilla states are diagonal, so the bath polarisation p is programmed exactly by classical sampling; the round is affine on the Bloch vector, so composite channels are computed exactly, and every quantity called certified is a two-sided enclosure or a proven integer in 256-bit ball arithmetic. We certify a target list for the integer staircase n_EB(p), with decision margins between 0.025 and 0.109 at twelve points, show that it survives per-round depolarising noise of strength 5e-3 as certified integers, and give exact and tight gate counts (seven CNOTs all-to-all, five native arbitrary-angle gates, ten CNOTs on heavy-hex). A gated search of 33 million circuits shows that the contrast of thermal valleys, circuits whose index dips below its infinite-temperature value on an interior window of p, has an interior maximum near index 90 at a certified 2.85e-3, below what standard tomography resolves. Exact composition over rounds of different polarisation then turns a valley into a switch: a centred pulse of exactly eight rounds restores entanglement, a 53-round pulse beats the static protocol by a certified margin, a certified depth-three valley is walked down its staircase by pulse duration, and a 1.2 per cent coherent tilt of the ancilla opens the switch and triples the static signal at fixed population. Valley signatures die at per-round noise near 1e-4. Certificates, code, search output and an independent re-derivation script are deposited.