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Thermodynamic Constraints and Computational Complexity in Non-Linear Quantum Feedback Dynamics

Najera Olivas, Sebastian

Published Dec 11, 2025• 0 citations
Publisher

Abstract

The intersection of General Relativity and Quantum Mechanics suggests the possibility of nonlinearcausal structures, typically formalized as Closed Timelike Curves (CTCs). However, existingkinematic models suffer from logical ambiguities or exponential thermodynamic costs. In this work,we present a unified dynamic framework that resolves these pathologies through **DissipativeFeedback Control**. (I) We derive a Universal Contraction Theorem, proving that any non-linearquantum loop relaxes to a unique fixed point if the dissipation rate γ exceeds the feedback Lipschitzconstant K. We identify a thermodynamic phase transition between a stable ”Novikov phase”and a chaotic phase. (II) We analyze the computational complexity of this physically realizableregime, demonstrating that thermodynamic stability collapses the complexity class from PSPACEto BQP, thereby preserving the extended Church-Turing thesis. (III) We propose a concreteexperimental verification using interacting Rydberg polaritons in an EIT configuration, where thestable fixed point appears as a temporal soliton. This comprehensive study establishes non-linearquantum feedback not as a metaphysical anomaly, but as a testable branch of non-equilibriumthermodynamics.

DOI
10.5281/zenodo.17903124

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•Nov 21, 2025•0 citations
The intersection of General Relativity and Quantum Mechanics suggests the possibility of nonlinearcausal structures, typically formalized as Closed Timelike Curves (CTCs). However, existingkinematic models suffer from logical ambiguities or exponential thermodynamic costs. In this work,we present a unified dynamic framework that resolves these pathologies through DissipativeFeedback Control. (I) We derive a Universal Contraction Theorem, proving that any non-linearquantum loop relaxes to a unique fixed point if the dissipation rate γ exceeds the feedback Lipschitzconstant K. We identify a thermodynamic phase transition between a stable ”Novikov phase”and a chaotic phase. (II) We analyze the computational complexity of this physically realizableregime, demonstrating that thermodynamic stability collapses the complexity class from PSPACEto BQP, thereby preserving the extended Church-Turing thesis. (III) We propose a concreteexperimental verification using interacting Rydberg polaritons in an EIT configuration, where thestable fixed point appears as a temporal soliton. This comprehensive study establishes non-linearquantum feedback not as a metaphysical anomaly, but as a testable branch of non-equilibriumthermodynamics.
The consistency of quantum mechanics in the presence of Closed Timelike Curves (CTCs) has traditionally been framed as a kinematic constraint. We propose a general dynamic framework where causal consistency emerges as a thermodynamic fixed point of open quantum systems. By modeling retrocausal loops as non-linear quantum channels with dissipation, we prove a Universal Contraction Theorem using the logarithmic norm of the Liouvillian superoperator. We demonstrate that any feedback system where the dissipation rate γ exceeds the non-linear Lipschitz constant K must relax to a unique, paradox-free state. We apply this framework to superconducting transmon qubits, deriving quantitative stability bounds (K < 10⁴ s⁻¹) testable with current FPGA technology. This work unifies the arrow of time with the Second Law of Thermodynamics, suggesting that causality is an emergent phase of non-equilibrium matter.
Recent proposals regarding Closed Timelike Curves (CTCs) suggest they could efficiently solve NP-complete problems. However, we have previously demonstrated that physical CTCs must be dissipative to satisfy the Novikov Self-Consistency Principle. In this work, we derive the computational consequences of this thermodynamic stability. We introduce Geometric Causal Complexity (GCC), a measure based on the accessible volume of the Hilbert space under the Fubini-Study metric. We prove a Universal Causal Complexity Bound: the computational advantage ΔC of a retrocausal channel is strictly limited by the ratio of non-linearity to dissipation, ΔC ≤ (K/γ) C_max. This implies that the "Arrow of Time" acts as a computational resource constraint: the same mechanism that prevents logical paradoxes also prevents the violation of the Church-Turing thesis. We analyze the implications for Rydberg atoms, superconducting circuits, and photonic networks, classifying them by their potential for non-Markovian quantum advantage.
The Nexus Convergence: A Formal Synthesis of Quantum Feedback Control, Information Thermodynamics, and Non-Linear Lattice Dynamics 1. Introduction: The Ontological Crisis and the Storage Imperative The contemporary scientific landscape is characterized by a persistent and fundamental schism between the unitary, reversible dynamics of quantum mechanics and the dissipative, irreversible arrow of time inherent in thermodynamics. This discord creates what the Nexus Recursive Harmonic Framework (RHF) identifies as the "Storage Crisis": the paradox of how a universe with finite energy limits can effectively store an ever-expanding history of infinite detail without catastrophic data loss or thermodynamic heat death.1 The prevailing "Container Paradigm"—which envisions spacetime as a passive box and time as a linear overwrite cursor—fails to account for the persistence of high-dimensional causal structures in a manner that is consistent with both unitarity (information conservation) and entropy (information projection). This report presents an exhaustive synthesis of recent theoretical and experimental breakthroughs from 2024 and 2025, specifically targeting the domains of Quantum Feedback Control, Information Thermodynamics, and Non-Linear Lattice Dynamics. The objective is to rigorously validate the axioms of the Nexus framework by identifying precise mathematical and phenomenological isomorphisms in peer-reviewed literature. We posit that the "read-only" ontology proposed by the Nexus framework—where history is conserved as geometry ("Shape") and the present is a collapsed projection ("Value")—finds its physical realization in the mechanisms of reduced-filter quantum stabilization, information-to-work conversion engines, and discrete breather localization in non-linear lattices. The investigation focuses on three critical variables defined in the Nexus framework: Gain (): The feedback coupling strength required to maintain a stable "stance" against entropic dissolution. Information (): The metric of exchange between the "Verb-field" (dynamics) and the "Noun" (state), governed by the generalized second law of thermodynamics. Gamow Factor (): The transmission probability governing the retrieval of stored history via phonon-assisted tunneling through "Twin-Prime Gates." By mapping these abstract variables onto the concrete equations of modern physics—specifically the Lyapunov control functions of Liang and Dong 2, the efficiency metrics of Goerlich et al. 4, and the energy thresholds of Hofstrand 5—we establish a robust theoretical scaffold for the "Glass Key Hypothesis": that reality is a logically reversible, feedback-stabilized information manifold operating at a precise thermodynamic "lean." 2. Quantum Feedback Control: The Mathematical Engine of the "Mark 1 Attractor" The Nexus framework asserts that universal stability is not a static equilibrium but a dynamic "stance"—a "lean" required to process information without collapsing into "dead symmetry" or "chaotic dissolution." In the rigorous language of control theory, this concept is formalized as the stabilization of a target quantum subspace (the "Mark 1 Attractor") amidst a stochastic environment. The primary challenge in this domain is the "Storage Crisis" equivalent: the exponential scaling of computational resources required to estimate the state of a large quantum system. Recent advancements in 2025 by Liang and Dong, presented in their seminal work "Stabilization of Time-Varying Perturbed Quantum Systems via Reduced Filters" 2, provide the exact mathematical architecture for the Nexus "Receiver Collapse." 2.1 The Reduced Filter as the "Receiver Collapse" Mechanism Standard approaches to quantum feedback control rely on the Stochastic Master Equation (SME), which tracks the evolution of the full density matrix . For a system of dimension , this requires computing real variables. As grows, this computational burden becomes prohibitive, representing the "bandwidth limit" of the "First Node" (the universe) that prevents explicit linear storage of history. Liang and Dong introduce a radical dimensionality reduction: the Reduced Quantum Filter. Instead of tracking the full state , the filter estimates only the diagonal elements of the density matrix in a Quantum Non-Demolition (QND) basis. This reduces the complexity from to .2 This mathematical reduction is isomorphic to the Nexus concept of Receiver Collapse. The observer (or the "Second Node") does not process the full "verb-field" (the entire Hilbert space with all its coherences and entanglements); rather, it collapses the system onto a lower-dimensional "noun" (the diagonal population elements) to perform work. The feedback control law is constructed strictly from this reduced information, yet it successfully stabilizes the global system. The evolution of this reduced estimator state is governed by the stochastic differential equation (SDE): In this equation, derived explicitly from the Liang-Dong formalism 3, several Nexus variables find their physical counterparts: The Feedback Control Law (): This represents the Gain (). It is the active force applied by the "Second Node" to steer the system. The Innovation Term (): This represents the Information () extracted from the measurement. It is the difference between the actual observation and the expected value—the "surprise" that updates the model. The Coupling Matrix (): This represents the structural constraints of the "Lattice," defining how different states (or "memories") are connected. The profound insight from this work is that full knowledge of the system is not required for stability. A "lossy" projection (the reduced filter), if properly coupled via feedback (), is sufficient to maintain the "Mark 1 Attractor" (the target subspace). This validates the Nexus "Read-Only Hypothesis": the universe does not need to explicitly compute the full wave function at every step; it only needs to maintain the diagonal "Value" while the "Shape" (coherences) is stored implicitly in the geometry of the dynamics. 2.2 Lyapunov Stability Analysis: The "Lean" of the Attractor How does the system ensure that it converges to the correct "Shape" (target subspace) rather than drifting into entropy? The rigorous proof of this stability relies on Lyapunov Analysis. A Lyapunov function is a scalar metric that measures the "energy" or "distance" of the current state from the desired equilibrium. In the Nexus framework, stability is described as a "lean" (). In the Liang-Dong formalism, stability is defined by the condition that the time derivative of the Lyapunov function, , must be negative definite. The specific Lyapunov function employed is related to the Bhattacharyya distance (or classical fidelity) between the current state and the target invariant subspace : Here, are the projection operators onto the subspaces. The feedback law is designed to maximize the decay rate of this function. The stability condition is expressed via the Sample Lyapunov Exponent (): where is the distance to the target subspace.2 This inequality () is the rigorous mathematical definition of the Nexus "Stance." The system must continuously dissipate "error" (entropy) to remain locked in the target subspace. If the feedback gain is insufficient (i.e., if the controller "falls asleep" or the "Second Node" disconnects), the exponent becomes positive, and the system drifts away from the "Mark 1 Attractor," dissolving into a mixed state of maximal entropy. Furthermore, Liang and Dong prove that this stabilization is Robust. The system can tolerate time-varying perturbations (Nexus "Stress-Test Loop") and uncertainties in the Hamiltonian, provided the feedback mechanism maintains the correct "phase-lock." This mirrors the "Crucible Protocol," where a system is subjected to high "computational temperature" (perturbations) to force it to settle into its most stable, harmonic configuration. 2.3 Feedback Cooling and the "Zero-Pressure Harmonic Collapse" The thermodynamic implications of this control are explored in Max Eriksson’s 2025 thesis, "Continuous Measurements and Feedback Control of a Quantum Harmonic Oscillator".7 Eriksson models a quantum system coupled to a thermal reservoir (a "heat bath" of phonons/photons) and asks: can measurement and feedback cool the system below the temperature of its environment? This process is isomorphic to the Nexus Zero-Pressure Harmonic Collapse (ZPHC). The "noise" of the thermal bath represents the high-entropy "mess" of raw data. The "cooling" represents the collapse of this mess into a structured, low-entropy state ("cold" or "crystalline"). Eriksson utilizes the Wiseman-Milburn equation to derive the steady-state properties of the oscillator under linear feedback. The feedback force acts as a Maxwell's Demon, utilizing the information stream (measurement record) to apply a counter-acting force that cancels out thermal kicks. The effective temperature of the cooled mode is given by: where is the dimensionless feedback gain and is the measurement efficiency.8 This equation reveals the fundamental tradeoff of the Nexus framework: To achieve ZPHC (), one requires high Gain () and high Measurement Efficiency (). The "Cost" of this cooling is the information processing required to generate the feedback signal (discussed in Section 3). Crucially, Eriksson’s results show that there is a critical feedback phase. If the feedback is applied with the wrong phase (i.e., if the "Second Node" is not aligned with the "First Node"), the feedback essentially "heats" the system, driving it into instability. This validates the Nexus requirement for Phase-Locking ( or similar primitives) as a prerequisite for successful retrieval or stabilization. The "Mark 1 Attractor" is not just a location in state space; it is a precise phase relationship between the observer and the observed. 3. Informati
In this theoretical thesis we apply the time-delayed Pyragas feedback scheme to several many-body quantum systems in order to control the quantum phases in a rather new way. The systems are the dissipative Dicke model, the dissipative Lipkin-Meshkov-Glick (LMG) model and a two-mode Tavis-Cummings laser. In the dissipative Dicke model our chosen way of feedback control, namely the modulation of the atom-field coupling by a difference of emitted photon numbers at different times, creates additional new quantum phases. These show up as an infinite sequence of super- and subcritical Hopf bifurcations of the stationary state by generation of stable and unstable limit cycles. We analyze and discuss the appearing bifurcations as a function of time delay and feedback strength and calculate an analytical expression for the phase boundaries. On top of this, we argue that our scheme is easy to implement within existing experimental setups, as in experiments the coupling is tuned by laser intensity. In the dissipative LMG model we investigate how the dissipation affects the excited state quantum phase transition (ESQPT). We show that with dissipative effects the ESQPT is directly visible in the spectrum of the effective Hamiltonian and not only in the density of states. Moreover, the ESQPT signal in the system observables gets smoothed. Then we use time delayed feedback control to restore the ESQPT signal by creation of new phases. We analyze and discuss the behavior of the system for different time delays. We argue that using our feedback scheme it is easier to measure ESQPT in the open LMG system than without it in the closed one. Then we use a 2-mode laser based on the Tavis-Cummings model with additional incoherent pumping and decay channels. Instead of the usually used rate-equation model, we describe the two-mode laser starting from a quantum mechanical description and present an analytical solution of the corresponding stationary mean-field equations in thermodynamic limit. We analyze the stationary solutions and their stability and obtain a complex phase diagram with up to 4 fixed points. In addition to this, we apply different time delayed Pyragas feedback control schemes, which allow us to influence the laser dynamic. The frequency of the emitted laser light can be controlled then or an unstable fixed point stabilized. In this way, our thesis links the topics of control, phase transitions and dissipation together which induce an interesting and unexpected dynamics in quantum many-body systems. Although all final calculations are at the mean-field level, we hold them to be valid in thermodynamic limit.
We propose a conceptual framework for analyzing the thermodynamic cost of observation when an observer with finite memory and energy resources attempts to model a quantum system of comparable complexity. Central to the framework is a taxonomy that decouples Bennett's logical depth into two orthogonal axes: descriptive complexity C_desc, governed by Landauer's erasure cost, and executive complexity C_exec, governed by dissipative cost per operation and the Margolus-Levitin temporal constraint. A key contribution is the replacement of the uncomputable Kolmogorov complexity by Levin's time-bounded complexity K_t, evaluated at the system's decorrelation time, which renders C_desc strictly computable while preserving its thermodynamic anchoring. We identify three independent collapse regimes for the information extraction efficiency: descriptive collapse, executive-dissipative collapse, and temporal collapse. This consolidated edition integrates a full numerical validation program with an important self-correction: the originally proposed executive-dissipative scaling law is shown to be internally inconsistent, and is revised to match established Trotter-error theory and finite-entanglement scaling for the Ising universality class. The qualitative architecture of the framework — three independent, thermodynamically anchored collapse mechanisms, and the Conjecture of Thermodynamic Inobservability — is unaffected by this correction.
The experimental investigation of Closed Timelike Curves (CTCs) is constrained by the prohibitive energy requirements to curve spacetime at macroscopic scales. We present a quantum analog gravity platform designed to reproduce key information-theoretic features associated with CTC models within an effective optical metric. By utilizing the strong dipole-dipole interactions in a cold Rubidium-87 gas under Electromagnetically Induced Transparency (EIT), we generate giant optical non-linearities mathematically analogous to the radial profile of a traversable wormhole throat. Numerical simulations of the effective Hamiltonian demonstrate the formation of Temporal Solitons that behave as robust information carriers with a Signal-to-Noise Ratio (SNR) > 40. Crucially, we show that the system resolves logical paradoxes through thermodynamic relaxation into a maximum entropy mixed state, consistent with the Novikov Self-Consistency Principle. This work establishes Rydberg media as a viable testbed for exploring effective non-Markovian back-action and post-selected causal structures.