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arXiv:2608.13764

Certified coherent, informative, and non-entanglement-breaking fixed points of future-referential quantum feedback

Eran Kopel

quant-phcs.ITSubmitted Aug 13, 2026• Announced Aug 17, 2026• 0 citations
v1 · Aug 13, 2026v2 · Oct 4, 2026
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Abstract

We study quantum processes in which information extracted from a forward simulation is returned as 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. Four results separate notions that informal discussions of "information from the future" often conflate. A two-parameter unitary-dilation family yields a closed-form, globally attractive, coherent fixed point (Proposition 1), yet is entanglement breaking whenever future records are perfectly distinguishable (Lemma 1). Releasing that orthogonality, a four-parameter partial-swap family admits a nonempty open non-entanglement-breaking region (Proposition 2), with an explicit Choi partial-transpose neighborhood of half-width $0.0163π$ (Proposition 3). Combining outward-rounded interval enclosures with perturbation bounds tracking the channel and its stationary-state drift, we certify an explicit parameter square of half-width $0.0013π$ on which the feedback channel is simultaneously strictly contractive (margin $\ge 0.237$), coherent ($\ge 0.416$), informative about the designated future variable ($\ge 0.172$ bits), and non-entanglement-breaking (NPT margin $\ge 0.188$) (Proposition 4). Direct evaluation shows all four properties persisting over a region an order of magnitude larger, so the certified square is a proof of principle rather than a phase boundary. All enclosures and margins are confirmed by a machine-verified ball-arithmetic certificate, and the complete code and certificate accompany the paper.

Comments
11 pages, 4 figures. Ancillary files include a machine-verified interval certificate (Arb/python-flint, 256-bit ball arithmetic), an independent second-stack audit (mpmath), and all research code. Submitted to Foundations of Physics

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Discarding an output subsystem is a local post-processing operation, so an entanglement-breaking channel remains entanglement breaking after output reduction. The converse need not hold: a reduced channel can be entanglement breaking even when a richer output still carries reference entanglement. This preprint gives an exactly solvable two-parameter benchmark that quantifies this asymmetry. On two qubits, let (\Pi_\pm=(I\pm Z\otimes I)/2), let (\mathcal L(X)=\sum_s\Pi_sX\Pi_s) be sector dephasing, let (\mathcal M(X)=\sum_s\operatorname{Tr}(\Pi_sX)\Pi_s/2) be measure-and-prepare within the two sectors, and let (\mathcal D_4(X)=\operatorname{Tr}(X)I_4/4). The channel family is [\Phi_{t,\eta}=(1-\eta)\big[(1-t)\mathcal L+t\mathcal M\big]+\eta\mathcal D_4,\qquad 0\le t,\eta\le1.] For the reduced task, a variable second-qubit input is embedded with an arbitrary fixed first-qubit state and the first output qubit is discarded. The induced qubit channel is depolarizing with parameter (q=(1-\eta)(1-t)), and is entanglement breaking exactly when (q\le1/3). For the full four-dimensional output, the partially transposed normalized Choi state has one potentially negative eigenvalue, [\lambda_- = \frac{-4(1-\eta)+6(1-\eta)t+\eta}{16},] with multiplicity two. The block structure makes PPT sufficient for separability in this family. Hence, for (\eta<1), [t_{\rm red}(\eta)=\max!\left{0,1-\frac{1}{3(1-\eta)}\right},\qquadt_{\rm full}(\eta)=\max!\left{0,\frac{4-5\eta}{6(1-\eta)}\right}.] The exact region in which the reduced channel is already entanglement breaking while the full channel is not is (t_{\rm red}(\eta)\le t<t_{\rm full}(\eta)). Its width is [w(\eta)=\begin{cases}\eta/[6(1-\eta)],&0\le\eta\le2/3,\(4-5\eta)/[6(1-\eta)],&2/3\le\eta<4/5,\0,&4/5\le\eta\le1,\end{cases}] and reaches the exact maximum (w_{\max}=1/3) at (\eta=2/3). The Choi negativities are [\mathcal N_{\rm red}=\frac{[3(1-\eta)(1-t)-1]+}{4},\qquad\mathcal N{\rm full}=\frac{[4-5\eta-6(1-\eta)t]_+}{8}.] Therefore the maximal full-output Choi negativity compatible with an already entanglement-breaking reduced output is (1/12), attained at ((\eta,t)=(2/3,0)). The result is a controlled benchmark for output-access dependence: the same physical channel family can cross the entanglement-breaking boundary at different noise values solely because an output subsystem is retained or discarded.
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.
In the conventional picture of quantum feedback control, sensors perform measurements on the system, a classical controller processes the results of the measurements, and actuators supply semiclassical potentials to alter the behavior of the quantum system. In this picture, the sensors tend to destroy coherence in the process of making measurements, and although the controller can use the actuators to act coherently on the quantum system, it is processing and feeding back classical information. This paper proposes an alternative method for quantum feedback control, in which the sensors, controller, and actuators are quantum systems that interact coherently with the system to be controlled. In this picture, the controller gets, processes, and feeds back quantum information. Controllers that operate using such quantum feedback loops can perform tasks such as entanglement transfer that are not possible using classical feedback. Necessary and sufficient conditions are presented for Hamiltonian quantum systems to be controllable and observable using both classical and quantum feedback.
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 along two axes, interaction and bath. On the interaction axis, strict contraction onto a full-rank fixed point forces N finite in every dimension, via an explicit separable ball around the limiting Choi state, 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, a thermal ancilla at polarisation p = tanh(hbar omega / 2 k_B T) makes every channel quantity an exact quadratic in p, and one-sided certificates (no eigensolver) establish N = 3 at the reference circuit at every temperature. The infinite-temperature index is an architecture floor with closed form and exact measure (46.427% of circuits break entanglement in one round at infinite temperature). For exchange collisions (generalised amplitude damping) the index has a closed form and is monotone in p; for the loop the floor is not a bound: certified circuits dip below it in interior temperature windows, at a rate with Gaussian cutoff in epsilon = 1 - ||A_0||_2 (marginal exponent 2.02 [1.90, 2.16], 4.1 million circuits), and shallow-floor valleys survive to epsilon = 0.125. On a certified coupling-temperature surface, temperature moves the index by at most 3/2, the interaction by 1213/3. Applied to quantum-cognition proposals, a per-bath polarisation table at 310 K shows nuclear spins maximally hot and intramolecular vibrations frozen, and a four-input checklist replaces decoherence times by round budgets; Fisher's nuclear-spin proposal specifies its bath and storage coherence, not its processing interaction or rate.
When a quantum system interacts with its environment, the so-called decoherence effect will normally destroy the coherence in the quantum state and the entanglement between its subsystems. We propose a feedback control strategy based on quantum weak measurements to protect coherence and entanglement of the quantum state against environmental disturbance. For a one-qubit quantum system under amplitude damping and dephasing decoherence channels, our strategy can preserve the coherence based on the measured information about the population difference between its two levels. For a two-qubit quantum system disentangled by independent amplitude damping and dephasing decoherence channels, the designed feedback control can preserve coherence between the ground state and the highest excited states by tuning the coupling strength between the two qubits, and at the same time minimize the loss of entanglement between the two qubits. As a consequence of dynamic symmetry, the generalization of these results derives the concept of control-induced decoherence-free observable subspace, for which several criteria are provided.
This paper explores the concept of algorithmic quantum entanglement propagation, a novel approach to maintaining entanglement across interconnected quantum systems, mitigating the effects of decoherence. We propose a dynamic feedback loop algorithm designed to autonomously adjust entanglement states, ensuring robustness and enabling long-distance propagation. The core mechanism centers around subtle, self-correcting quantum feedback loops, independent of classical control, to maintain entanglement integrity. The research investigates the potential of this approach to overcome limitations in current entanglement distribution techniques and address a critical challenge in quantum computing – the degradation of entanglement due to decoherence. This work presents a foundational concept for a robust and adaptable entanglement propagation system.