A quantum feedback loop that returns information from a forward simulation to an earlier internal time induces a completely positive trace-preserving map on a message register; iterating that map eventually destroys its ability to carry entanglement. The entanglement-breaking index N = n_EB(Phi), the round at which this happens, is studied here along two axes: the interaction and the bath. On the interaction axis we prove that strict contraction onto a full-rank fixed point forces N finite in every dimension, via an explicit separable ball around the limiting Choi state; the qubit closed form overshoots the exact answer by the factor ln(2 sqrt 3) / ln 3 = 1.1309... on isotropic unital channels, and across 3997 sampled channels index and stability gap are one clock, N [1 - rho(A)] in [0.60, 1.50]. On the bath axis, replacing the zero-temperature ancilla by a thermal one at polarisation p = tanh(hbar omega / 2 k_B T) makes every channel quantity an exact quadratic polynomial in p, and the same one-sided certificates (no eigensolver) establish N = 3 at the reference circuit at every temperature, uniformly on a parameter box. The infinite-temperature index is an architecture floor with closed form and exact measure (46.427% of circuits break entanglement in one round however hot the bath). The floor is not a bound: certified circuits dip below it in interior temperature windows, unit-quantised in depth; over 4.1 million circuits their rate follows a Gaussian cutoff in epsilon = 1 - ||A||_2 with exponent 2.02 [1.90, 2.16], and 12 million more certified circuits show valleys surviving to epsilon = 0.125. Certified endpoints make comparisons exact: temperature moves the index by at most 21/14 = 3/2; the interaction moves it by 1213/3.
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33 pages, 1 figure, 9 tables. Companion to arXiv:2608.13764. Ancillary files in two subfolders: zero-temperature/ and thermal/
A quantum feedback loop that returns information from a forward simulation to an earlier internal time induces a completely positive trace-preserving map on a message register; iterating that map eventually destroys its ability to carry entanglement. The entanglement-breaking index N = n_EB(Phi), the round at which this happens, is studied here along two axes: the interaction and the bath. On the interaction axis we prove that strict contraction onto a full-rank fixed point forces N finite in every dimension, via an explicit separable ball around the limiting Choi state; the qubit closed form overshoots the exact answer by the factor ln(2 sqrt 3) / ln 3 = 1.1309... on isotropic unital channels, and across 3997 sampled channels index and stability gap are one clock, N [1 - rho(A)] in [0.60, 1.50]. On the bath axis, replacing the zero-temperature ancilla by a thermal one at polarisation p = tanh(hbar omega / 2 k_B T) makes every channel quantity an exact quadratic polynomial in p, and the same one-sided certificates (no eigensolver) establish N = 3 at the reference circuit at every temperature, uniformly on a parameter box. The infinite-temperature index is an architecture floor with closed form and exact measure (46.427% of circuits break entanglement in one round however hot the bath). The floor is not a bound: certified circuits dip below it in interior temperature windows, unit-quantised in depth; over 4.1 million circuits their rate follows a Gaussian cutoff in epsilon = 1 - ||A||_2 with exponent 2.02 [1.90, 2.16], and 12 million more certified circuits show valleys surviving to epsilon = 0.125. Certified endpoints make comparisons exact: temperature moves the index by at most 21/14 = 3/2; the interaction moves it by 1213/3.
The entanglement-breaking index of a channel is the number of self-compositions after which all entanglement with a reference system is destroyed. In a thermal qubit collision loop whose round is affine in the bath polarisation p, some circuits have thermal valleys: on an interior window of p the index falls below its infinite-temperature value. Rounds at different p compose exactly, so every schedule of polarisations defines a composite channel whose entanglement-breaking status can be certified in ball arithmetic. A gated search of 33 million random circuits finds no valley with contrast above about 3e-3, with an interior maximum near index 90, and yields six certified targets. At the reference target, held inside its window for 88 rounds, a centred excursion to p = 0.05 restores entanglement at exactly eight rounds: every shorter centred pulse certifiably fails, and so do the same eight rounds at the start. A 53-round pulse certifiably beats the static protocol, a 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 at fixed population. Closing the switch costs 21 to 35 rounds. The asymmetry is not one of local rates, which agree to within about two per cent at matched circuit positions (certified). It is one of distance and response: closing must remove an entanglement margin 2.2 times larger, and while hot rounds added to a window hold act almost independently, window rounds added to a hot hold act collectively. Near the crossing, the lever of the log-margin is that of a linear approach to the entanglement-breaking boundary, with no free parameter. At the reference target, depolarising noise of about 4ν/n per round kills a signature of size ν read after n rounds: 1.3e-4 for the static signal, 6.5e-6 for the eight-round switch.
A feedback loop that returns a message into the process it describes defines a quantum channel on the message. We ask when such a loop preserves entanglement with a reference, that is, when its channel is not entanglement breaking. For any qubit loop, the negativity of the Choi state is at most half the fidelity between the states that the two values of the message leave in the discarded registers, in any basis. This channel analogue of the duality between fringe visibility and which-way information is attained by pure dephasing, and a perfect record of the message in the discarded registers makes the loop entanglement breaking. We then solve a three-qubit loop in which the message steers a future event, a probe reads the event, and a partial swap feeds the probe back. Every partial feedback is a strict contraction, and the loop is entanglement breaking exactly when an explicit trigonometric polynomial in its coupling angles is not positive. A partial record of the message in the future event can restore entanglement that the feedback alone destroys. Stability, a coherent fixed point, information about the future event and entanglement preservation coexist on about three quarters of the parameter space.
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.
The paper studies quantum processes in which information extracted from a forward simulation is returned as an input to an earlier internal time of the simulated dynamics: externally the protocol is an ordinary causally ordered circuit, but internally it is future-referential. Contracting a process tensor with a leakage instrument and a controller induces a completely positive trace-preserving map on a message register, and we classify its fixed points by five operational properties: stability, informativeness, feedability, coherence, and preservation of quantum correlations.
At a fixed thermal exposure, the same stabilizer processor can be classically simulable or quantum universal, depending on which environmental records its controller retains. We give an exact computational classification of energy-counting thermal-idle instruments in a quantum processor with ideal stabilizer control and independent local Markov baths. The relaxation time T1, homogeneous coherence time T2, and equilibrium excited-state population pe determine an exact computational boundary at (1−pe)T2/T1=1. If every location lies at or below it, branchwise nonnegative stabilizer decompositions give an explicit efficient classical sampler for the adaptive circuit and its full time-resolved exchange record. Above it at a single repeatedly accessible location, a suitable idle duration and no-exchange conditioning supply distillable ancillas and enable universal quantum computation with polynomial overhead. At finite temperature on the resource side, erasing the record at a sufficiently long, unsplit exposure makes the averaged channel stabilizer measure-and-prepare, and even a terminal parity check then yields only simulable branches. Yet at that same exposure, retaining only the bit recording whether any exchange occurred still heralds distillable ancillas, because a thermal round trip restores the parity after its first exchange has already removed the coherence.
How much memory does it take to apply the same quantum channel many times, when each output must be released before the next input arrives and every retained qubit counts? One-use invariants do not decide it. Two explicit channels on dimension 1664 have identical normalized Choi spectra (32 eigenvalues 1/32), the same maximal complementary entropy 5, and zero asymptotic purity cost. At the same optimal exchange rate, one is implemented exactly with five memory qubits for any number of uses, while the other needs Omega(n^(1/(2 log_2(107)+2))) memory qubits for n uses, at every fixed error at most 1/16 and whatever purity is available; memory O(n^(1/3) log^(4/3) n) suffices for it. The hard channel is built from Houghton's group H_3. Behind the example is a general theory: the achievable exchange and purity rates, the purity cost kappa, which vanishes exactly on the closure of finite tracial factorizations, and memory bounds from how well a channel's environment is approximated by finite ones, and, for channels built from finite presentations, from the approximation profile of the group. Version 1.1 adds where the expensive channels lie: on the relative boundary of the closure; any fixed depolarizing admixture brings them down to polylogarithmic memory at the rank rate; on a face containing the Houghton channel a single moment of a trace on the group decides expense; and one free fermion per site already forces polynomial memory. Version 1.1.1 adds a map of what the main separation depends on (published inputs, companion results and new arguments), sets finite-budget and asymptotic-rate statements side by side, cites the companion papers at their published versions, pins the tested software versions, and makes every check refuse to run under optimized Python. The mathematical results are unchanged. Version 1.1.2 corrects attributions and makes two headline statements precise. The separation of Theorem A is stated at the exchange budget 5n/2, and the channels that need polynomial memory at every exchange rate are those with purity limited to O(log n). Credits added or corrected: Bisio, D'Ariano, Perinotti and Chiribella, and Bisio, D'Ariano, Perinotti and Sedlak; Boes et al. for the square-root dephasing construction; Lie, Son, Boes, Ng and Wilming for the exact case of the tracialization lemma; Faist, Berta and Brandao for the relation to the thermodynamic capacity; Haagerup, Musat and Rordam for the moment criteria; Thoma, Johnson, Watrous, Horodecki-Oppenheim-Winter and Lie-Jeong; Cornulier for the Baumslag-Solitar bound. Known results are moved out of numbered statements. The mathematical results and the numbering are unchanged. All operational bounds use complete-experiment error against adaptive observers with quantum references. The general rate-region achievability result uses a cited closed-device theorem; the explicit separation does not. This preprint archive includes the PDF, TeX source, finite verification scripts, and scoped Lean arithmetic proofs. The artifacts do not formally verify the entire manuscript; their scope and external dependencies are documented in verification/README.md.