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arXiv:2604.08587

Covariant quantum error correction in a three-layer quantum brain model: computational analysis of layer-specific coherence dynamics

Hikaru Wakaura

q-bio.NCphysics.bio-phquant-phSubmitted Mar 31, 2026• Announced Apr 1, 2026• 0 citations• Revised Apr 13, 2026 (v2)
v1 · Mar 31, 2026v2 · Apr 13, 2026v3 · Aug 14, 2026
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Abstract

Quantum brain proposals require coherence on behaviorally relevant timescales, yet the gap between spin coherence times and neural decision windows has remained a quantitative obstacle. We evaluate approximate covariant quantum error correction (CQEC) -- a purification protocol constrained by the Eastin-Knill theorem -- across two radical-pair proteins parameterized by \textit{ab initio} spin Hamiltonians: monoamine oxidase~A (MAO-A) and cryptochrome (CRY, PDB~4I6G). Both share a three-layer architecture (${}^{31}$P nuclear spin memory, electron spin interface, classical electrochemistry) and identical hyperfine coupling ($A = 200$~MHz), but differ 16-fold in nuclear $T_2$: 3.2~ms (MAO-A) versus 52~ms (CRY). We test whether CQEC preserves coherence over the 200~ms Schultze-Kraft veto window by mapping each protein's $T_2$ gap onto a simulation decoherence rate ($γ_\mathrm{veto} = T_2~\text{gap}/2T_\mathrm{sim}$): 3.08 for MAO-A, 0.19 for CRY. At $γ_\mathrm{veto} = 0.19$, CQEC maintains tunneling coherence of 0.83 (95\% CI [0.76, 0.79]; versus 0.12 without correction, $\times$6.9 improvement). At $γ_\mathrm{veto} = 3.08$, coherence collapses to 0.012 even with CQEC. A $T_2$ sensitivity analysis confirms robustness: at $T_2 = 26$~ms (half the CRY estimate), CQEC-protected coherence remains 0.69. A classical Markov baseline produces only monotonic relaxation, confirming that CQEC-maintained oscillatory dynamics are genuinely quantum. However, no single protein optimizes both layers: CRY's shorter $T_2^e$ (0.53~ns versus 1.1~ns) worsens Layer~2 fidelity. This layer-protein tradeoff, together with unresolved challenges in state preparation and entanglement distribution, defines the next targets for quantum brain research.

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Output Side Information Changes the Entanglement-Breaking Threshold

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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.

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A Bell pair distributed through a link that combines amplitude damping with dephasing at time-dependent rates has dynamics that separate into a fixed part and a moving one. The negativity, fully entangled fraction, discord, and average teleportation fidelity depend on time only through the accumulated damping parameters $p(t)$ and $q(t)$. Every threshold is therefore a curve fixed in the unit square, and the rates select nothing but a trajectory across it. The Horodecki fidelity formula $\bar{F} = \frac{1}{2} + \frac{1}{6}\mathrm{Tr}|T|$ covers one- and two-sided exposure alike: the condition $\det T \leq 0$ under which it takes this form holds throughout the unit square for both. Under symmetric two-sided noise the entanglement vanishes when $p+q\geq1$, where the exact relation $\bar{F}^{2s}=\frac{2}{3}+\frac{1}{3}\mathcal{N}_{2s}$ ties disentanglement and the loss of quantum advantage to the same instant. One-sided exposure admits no finite-time sudden death for any rate profile, and the discord stays strictly positive throughout the open square, so the distributed state can be separable, useless for teleportation, and still nonclassical. For harmonically modulated rates, complete positivity caps the backflow at one modulation period of static decay, $ΔΓ_{k}\leq2πγ_{k,0}/Ω$, equivalently at a modulation depth $ξ_{k}\leq4.6033$ independent of $Ω$. Inside that window the trajectory reverses, producing finite intervals of restored quantum advantage and entanglement sudden birth.

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We prove a generalized quantum Stein lemma for parallel discrimination of an arbitrary finite-dimensional channel against families of alternative channels. The alternatives are compact and convex, closed under tensor products, and contain a faithful replacer. At every fixed type-I error tolerance, the optimal type-II error exponent equals the regularized Umegaki channel relative entropy minimized over the alternatives, and both defining limits exist. Inputs may be entangled across channel uses and with a reference system. The main technical result constructs exactly trace-preserving approximations with exponentially small diamond error, dominated in completely positive order by free channels at every rate above the common exponent. The proof combines uniform auxiliary map approximation, iterative reduction of the domination rate, and tensor amplification. It yields asymptotic equipartition with trace-preserving smoothing at every subexponentially vanishing error, an exponential strong converse, and stability under vanishing diamond-norm perturbations. Under additional permutation invariance and closure under insertion of a fixed faithful input state and discarding of its output, we also prove an explicit finite-block completion bound. This quantitative construction uses weighted discarding, local operator corrections, and comparison with auxiliary extensions. Applications include state-preserving, coherence-restricted, covariant, entanglement-breaking, and positive-partial-transpose channels. For isometric targets against entanglement-breaking or positive-partial-transpose alternatives, we obtain exact finite-block testing and smoothing formulas.

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The quantum capacity of a noisy channel quantifies the maximum rate at which quantum information can be transmitted reliably. For general channels, its evaluation requires an optimization over arbitrarily many channel uses. Degradable channels form a central exception: their capacity is given by the single-letter coherent information, while antidegradable channels have zero capacity. Nevertheless, even for these fundamental classes, it has remained open whether communication above capacity becomes possible when a fixed non-maximal error is tolerated. Here we resolve this problem by proving an exponential strong converse for every finite-dimensional degradable and antidegradable channel: at any rate above capacity, the fidelity of every coding scheme decays exponentially with the number of channel uses. As an immediate consequence, we establish the first all-code exponential strong converse for the quantum erasure channel throughout its full parameter range, strengthening previous results that applied only to almost all codes. We also show that exponential strong-converse bounds are preserved under receiver post-processing. This yields efficiently computable semidefinite-programming bounds for arbitrary finite-dimensional channels, improved bounds for Pauli channels, and an exact exponential strong converse for a nondegradable multilevel amplitude-damping family.

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Tanay Nag, Amit Dutta

We consider a generalized central spin model, consisting of two central qubits and an environmental spin chain (with periodic boundary condition) to which these central qubits are locally and weakly connected either at the same site or at two different sites separated by a distance $d$. Our purpose is to study the subsequent temporal generation of entanglement, quantified by concurrence, when initially the qubits are in an unentangled state. In the equilibrium situation, we show that the concurrence survives for a larger value of $d$ when the environmental spin chain is critical. Importantly, a common feature observed both in the equilibrium and the nonequilibrium situations while the latter is created by a sudden but global change of the environmental transverse field is that the two qubits become maximally entangled for the critical quenching. Following a nonequilibrium evolution of the spin chain, our study for $d\ensuremath{\ne}0$ indicates that there exists a threshold time above which concurrence attains a finite value. Additionally, we show that the number of independent decohering channels (DCs) is determined by $d$ as well as the local difference of the transverse field of the two underlying Hamiltonians governing the time evolution; the concurrence can be enhanced by a higher number of independent channels. The qualitatively similar behavior displayed by the concurrence for critical and off-critical quenches, as reported here, is characterized by analyzing the nonequilibrium evolution of these channels. The concurrence is maximum when the decoherence factor or the echo associated with the most rapidly DC decays to zero; on the contrary, the condition when the concurrence vanishes is determined nontrivially by the associated decay of one of the intermediate DCs. Analyzing the reduced density of a single qubit, we also explain the observation that the dephasing rate is always slower than the unentanglement rate. We further establish that the maximally and minimally decohering channels show singular behavior which can be explained by invoking upon a quasiparticle picture.

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Models of decoherence with negative dephasing rate

A. Pernice, Julius Helm, Walter T. Strunz

We determine the total state dynamics of a dephasing open quantum system using the standard environment of harmonic oscillators. Of particular interest are random unitary approaches to the same reduced dynamics and system-environment correlations in the full model. Concentrating on a model with an at times negative dephasing rate, the issue of non-Markovianity will also be addressed. Crucially, given the quantum environment, the appearance of non-Markovian dynamics turns out to be accompanied by a loss of system-environment correlations. Depending on the initial purity of the qubit state, these system-environment correlations may be purely classical over the whole relevant time scale, or there may be intervals of genuine system-environment entanglement. In the latter case, we see no obvious relation between the build-up or decay of these quantum correlations and Non-Markovianity.

•Feb 20, 2012•2 citations
Quantum brain proposals require coherence on behaviorally relevant timescales, yet the gap between spin coherence times and neural decision windows has remained a quantitative obstacle. We evaluate approximate covariant quantum error correction (CQEC) -- a purification protocol constrained by the Eastin-Knill theorem -- across two radical-pair proteins parameterized by \textit{ab initio} spin Hamiltonians: monoamine oxidaseA (MAO-A) and cryptochrome (CRY, PDB4I6G). Both share a three-layer architecture (31{}^{31}31P nuclear spin memory, electron spin interface, classical electrochemistry) and identical hyperfine coupling (A=200A = 200A=200MHz), but differ 16-fold in nuclear T2T_2T2​: 3.2ms (MAO-A) versus 52ms (CRY). We test whether CQEC preserves coherence over the 200ms Schultze-Kraft veto window by mapping each protein's T2T_2T2​ gap onto a simulation decoherence rate (γveto=T2 gap/2Tsimγ_\mathrm{veto} = T_2~\text{gap}/2T_\mathrm{sim}γveto​=T2​ gap/2Tsim​): 3.08 for MAO-A, 0.19 for CRY. At γveto=0.19γ_\mathrm{veto} = 0.19γveto​=0.19, CQEC maintains tunneling coherence of 0.83 (95% CI [0.76, 0.79]; versus 0.12 without correction, ×\times×6.9 improvement). At γveto=3.08γ_\mathrm{veto} = 3.08γveto​=3.08, coherence collapses to 0.012 even with CQEC. A T2T_2T2​ sensitivity analysis confirms robustness: at T2=26T_2 = 26T2​=26ms (half the CRY estimate), CQEC-protected coherence remains 0.69. A classical Markov baseline produces only monotonic relaxation, confirming that CQEC-maintained oscillatory dynamics are genuinely quantum. However, no single protein optimizes both layers: CRY's shorter T2eT_2^eT2e​ (0.53ns versus 1.1ns) worsens Layer2 fidelity. This layer-protein tradeoff, together with unresolved challenges in state preparation and entanglement distribution, defines the next targets for quantum brain research.
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.
A Bell pair distributed through a link that combines amplitude damping with dephasing at time-dependent rates has dynamics that separate into a fixed part and a moving one. The negativity, fully entangled fraction, discord, and average teleportation fidelity depend on time only through the accumulated damping parameters p(t)p(t)p(t) and q(t)q(t)q(t). Every threshold is therefore a curve fixed in the unit square, and the rates select nothing but a trajectory across it. The Horodecki fidelity formula Fˉ=12+16Tr∣T∣\bar{F} = \frac{1}{2} + \frac{1}{6}\mathrm{Tr}|T|Fˉ=21​+61​Tr∣T∣ covers one- and two-sided exposure alike: the condition det⁡T≤0\det T \leq 0detT≤0 under which it takes this form holds throughout the unit square for both. Under symmetric two-sided noise the entanglement vanishes when p+q≥1p+q\geq1p+q≥1, where the exact relation Fˉ2s=23+13N2s\bar{F}^{2s}=\frac{2}{3}+\frac{1}{3}\mathcal{N}_{2s}Fˉ2s=32​+31​N2s​ ties disentanglement and the loss of quantum advantage to the same instant. One-sided exposure admits no finite-time sudden death for any rate profile, and the discord stays strictly positive throughout the open square, so the distributed state can be separable, useless for teleportation, and still nonclassical. For harmonically modulated rates, complete positivity caps the backflow at one modulation period of static decay, ΔΓk≤2πγk,0/ΩΔΓ_{k}\leq2πγ_{k,0}/ΩΔΓk​≤2πγk,0​/Ω, equivalently at a modulation depth ξk≤4.6033ξ_{k}\leq4.6033ξk​≤4.6033 independent of ΩΩΩ. Inside that window the trajectory reverses, producing finite intervals of restored quantum advantage and entanglement sudden birth.
We prove a generalized quantum Stein lemma for parallel discrimination of an arbitrary finite-dimensional channel against families of alternative channels. The alternatives are compact and convex, closed under tensor products, and contain a faithful replacer. At every fixed type-I error tolerance, the optimal type-II error exponent equals the regularized Umegaki channel relative entropy minimized over the alternatives, and both defining limits exist. Inputs may be entangled across channel uses and with a reference system. The main technical result constructs exactly trace-preserving approximations with exponentially small diamond error, dominated in completely positive order by free channels at every rate above the common exponent. The proof combines uniform auxiliary map approximation, iterative reduction of the domination rate, and tensor amplification. It yields asymptotic equipartition with trace-preserving smoothing at every subexponentially vanishing error, an exponential strong converse, and stability under vanishing diamond-norm perturbations. Under additional permutation invariance and closure under insertion of a fixed faithful input state and discarding of its output, we also prove an explicit finite-block completion bound. This quantitative construction uses weighted discarding, local operator corrections, and comparison with auxiliary extensions. Applications include state-preserving, coherence-restricted, covariant, entanglement-breaking, and positive-partial-transpose channels. For isometric targets against entanglement-breaking or positive-partial-transpose alternatives, we obtain exact finite-block testing and smoothing formulas.
The quantum capacity of a noisy channel quantifies the maximum rate at which quantum information can be transmitted reliably. For general channels, its evaluation requires an optimization over arbitrarily many channel uses. Degradable channels form a central exception: their capacity is given by the single-letter coherent information, while antidegradable channels have zero capacity. Nevertheless, even for these fundamental classes, it has remained open whether communication above capacity becomes possible when a fixed non-maximal error is tolerated. Here we resolve this problem by proving an exponential strong converse for every finite-dimensional degradable and antidegradable channel: at any rate above capacity, the fidelity of every coding scheme decays exponentially with the number of channel uses. As an immediate consequence, we establish the first all-code exponential strong converse for the quantum erasure channel throughout its full parameter range, strengthening previous results that applied only to almost all codes. We also show that exponential strong-converse bounds are preserved under receiver post-processing. This yields efficiently computable semidefinite-programming bounds for arbitrary finite-dimensional channels, improved bounds for Pauli channels, and an exact exponential strong converse for a nondegradable multilevel amplitude-damping family.
We consider a generalized central spin model, consisting of two central qubits and an environmental spin chain (with periodic boundary condition) to which these central qubits are locally and weakly connected either at the same site or at two different sites separated by a distance ddd. Our purpose is to study the subsequent temporal generation of entanglement, quantified by concurrence, when initially the qubits are in an unentangled state. In the equilibrium situation, we show that the concurrence survives for a larger value of ddd when the environmental spin chain is critical. Importantly, a common feature observed both in the equilibrium and the nonequilibrium situations while the latter is created by a sudden but global change of the environmental transverse field is that the two qubits become maximally entangled for the critical quenching. Following a nonequilibrium evolution of the spin chain, our study for $d\ensuremath{\ne}0$ indicates that there exists a threshold time above which concurrence attains a finite value. Additionally, we show that the number of independent decohering channels (DCs) is determined by ddd as well as the local difference of the transverse field of the two underlying Hamiltonians governing the time evolution; the concurrence can be enhanced by a higher number of independent channels. The qualitatively similar behavior displayed by the concurrence for critical and off-critical quenches, as reported here, is characterized by analyzing the nonequilibrium evolution of these channels. The concurrence is maximum when the decoherence factor or the echo associated with the most rapidly DC decays to zero; on the contrary, the condition when the concurrence vanishes is determined nontrivially by the associated decay of one of the intermediate DCs. Analyzing the reduced density of a single qubit, we also explain the observation that the dephasing rate is always slower than the unentanglement rate. We further establish that the maximally and minimally decohering channels show singular behavior which can be explained by invoking upon a quasiparticle picture.
We determine the total state dynamics of a dephasing open quantum system using the standard environment of harmonic oscillators. Of particular interest are random unitary approaches to the same reduced dynamics and system-environment correlations in the full model. Concentrating on a model with an at times negative dephasing rate, the issue of non-Markovianity will also be addressed. Crucially, given the quantum environment, the appearance of non-Markovian dynamics turns out to be accompanied by a loss of system-environment correlations. Depending on the initial purity of the qubit state, these system-environment correlations may be purely classical over the whole relevant time scale, or there may be intervals of genuine system-environment entanglement. In the latter case, we see no obvious relation between the build-up or decay of these quantum correlations and Non-Markovianity.