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.
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13 pages, 4 figures. v2: rewritten around a general bound on the entanglement preserved by a qubit channel and an exact solution of the feedback loop; new title. The closed forms supersede the interval certificate of v1. Ancillary files: exact symbolic and numerical checks, a ball-arithmetic certificate and the figure scripts
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.
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.
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.