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Topological quantum field theories (TQFTs) provide a general, minimal-assumption language for describing quantum-state preparation and measurement. They therefore provide a general language in which to express multi-agent communication protocols, e.g. local operations, classical communication (LOCC) protocols. In the accompanying Part I, we construct LOCC protocols using TQFT, and show that LOCC protocols induce quantum error-correcting codes (QECCs) on the agent-environment boundary. Such QECCs can be regarded as implementing or inducing the emergence of spacetimes on such boundaries. Here we investigate this connection between inter-agent communication and spacetime, exploiting different realizations of TQFT. We delve into TQFTs that support on their boundaries spin-networks as computational systems: these are known as topological quantum neural networks (TQNNs). TQNNs, which have a natural representation as tensor networks, implement QECC. We recognize into the HaPPY code a paradigmatic example. We then show how generic QECCs, as bulk-boundary codes, induce effective spacetimes. The effective spatial and temporal separations that take place in QECC enables LOCC protocols between spatially separated observers. We then consider the implementation of QECCs in BF and Chern-Simons theories, and show that QECC-induced spacetimes provide the classical redundancy required for LOCC. Finally, we consider topological M-theory as an implementation of QECC in higher spacetime dimensions., Comment: 37 pages

We study the relationship between assumptions of state separability and both preparation and measurement contextuality, and the relationship of both of these to the frame problem, the problem of predicting what does not change in consequence of an action. We state a quantum analog of the latter and prove its undecidability. We show how contextuality is generically induced in state preparation and measurement by basis choice, thermodynamic exchange, and the imposition of a priori causal models, and how fine-tuning assumptions appear ubiquitously in settings characterized as non-contextual., Comment: 21 pp

Topological quantum field theories (TQFTs) provide a general, minimal-assumption language for describing quantum-state preparation and measurement. They therefore provide a general language in which to express multi-agent communication protocols, e.g. local operations, classical communication (LOCC) protocols. Here we construct LOCC protocols using TQFT, and show that LOCC protocols generically induce quantum error-correcting codes (QECCs). Using multi-observer scenarios described by quantum Darwinism and Bell/EPR experiments as examples, we show how these LOCC-induced QECCs effectively convert entanglement into classical redundancy. In the accompanying Part II, we show that such QECCs can be regarded as implementing, or inducing the emergence of, spacetimes on the boundaries between interacting systems. We investigate this connection between inter-agent communication and spacetime using BF and Chern-Simons theories, and then using topological M-theory., Comment: 31 pages

Living systems face both environmental complexity and limited access to free-energy resources. Survival under these conditions requires a control system that can activate, or deploy, available perception and action resources in a context specific way. We show here that when systems are described as executing active inference driven by the free-energy principle (and hence can be considered Bayesian prediction-error minimizers), their control flow systems can always be represented as tensor networks (TNs). We show how TNs as control systems can be implmented within the general framework of quantum topological neural networks, and discuss the implications of these results for modeling biological systems at multiple scales., Comment: 44 pgs

We show in this pedagogical review that far from being "an apparent law of physics that stands by itself" (R. Bousso, Rev. Mod. Phys. 74 (2002), 825-874), the holographic principle (HP) is a straightforward consequence of the quantum information theory of separable systems. It provides a basis for the theories of measurement, time, and scattering. Principles equivalent to the HP appear in both computer science and the life sciences, suggesting that the HP is not just a fundamental principle of physics, but of all of science., Comment: 39 pages, 8 figures

Macromolecular protein complexes catalyze essential physiological processes that sustain life. Various interactions between protein subunits could increase the effective mass of certain peptide groups, thereby compartmentalizing protein $\alpha$-helices. Here, we study the differential effects of applied massive barriers upon the soliton-assisted energy transport within proteins. We demonstrate that excentric barriers, localized onto a single spine in the protein $\alpha$-helix, reflect or trap three-spine solitons as effectively as concentric barriers with comparable total mass. Furthermore, wider protein solitons, whose energy is lower, require heavier massive barriers for soliton reflection or trapping. Regulation of energy transport, delivery and utilization at protein active sites could thus be achieved through control of the soliton width, or of the effective mass of the protein subunits., Comment: 28 pages, 13 figures

We show how any system with morphological degrees of freedom and locally limited free energy will, under the constraints of the free energy principle, evolve toward a neuromorphic morphology that supports hierarchical computations in which each level of the hierarchy enacts a coarse-graining of its inputs, and dually a fine-graining of its outputs. Such hierarchies occur throughout biology, from the architectures of intracellular signal transduction pathways to the large-scale organization of perception and action cycles in the mammalian brain. Formally, the close formal connections between cone-cocone diagrams (CCCD) as models of quantum reference frames on the one hand, and between CCCDs and topological quantum field theories on the other, allow the representation of such computations in the fully-general quantum-computational framework of topological quantum neural networks.

We introduce novel methods for implementing generic quantum information within a scale-free architecture. For a given observable system, we show how observational outcomes are taken to be finite bit strings induced by measurement operators derived from a holographic screen bounding the system. In this framework, measurements of identified systems with respect to defined reference frames are represented by semantically-regulated information flows through distributed systems of finite sets of binary-valued Barwise-Seligman classifiers. Specifically, we construct a functor from the category of cone-cocone diagrams (CCCDs) over finite sets of classifiers, to the category of finite cobordisms of Hilbert spaces. We show that finite CCCDs provide a generic representation of finite quantum reference frames (QRFs). Hence the constructed functor shows how sequential finite measurements can induce TQFTs. The only requirement is that each measurement in a sequence, by itself, satisfies Bayesian coherence, hence that the probabilities it assigns satisfy the Kolmogorov axioms. We extend the analysis so develop topological quantum neural networks (TQNNs), which enable machine learning with functorial evolution of quantum neural 2-complexes (TQN2Cs) governed by TQFTs amplitudes, and resort to the Atiyah-Singer theorems in order to classify topological data processed by TQN2Cs. We then comment about the quiver representation of CCCDs and generalized spin-networks, a basis of the Hilbert spaces of both TQNNs and TQFTs. We finally review potential implementations of this framework in solid state physics and suggest applications to quantum simulation and biological information processing., Comment: 61 pages

Protein$\alpha$-helices provide an ordered biological environment that is conducive to soliton-assisted energy transport. The nonlinear interaction between amide I excitons and phonon deformations induced in the hydrogen-bonded lattice of peptide groups leads to self-trapping of the amide I energy, thereby creating a localized quasiparticle (soliton) that persists at zero temperature. The presence of thermal noise, however, could destabilize the protein soliton and dissipate its energy within a finite lifetime. In this work, we have computationally solved the system of stochastic differential equations that govern the quantum dynamics of protein solitons at physiological temperature, T=310 K, for either a single isolated $\alpha$-helix spine of hydrogen bonded peptide groups or the full protein $\alpha$-helix comprised of three parallel $\alpha$-helix spines. The simulated stochastic dynamics revealed that although the thermal noise is detrimental for the single isolated $\alpha$-helix spine, the cooperative action of three amide I exciton quanta in the full protein $\alpha$-helix ensures soliton lifetime of over 30 ps, during which the amide I energy could be transported along the entire extent of an 18-nm-long $\alpha$-helix. Thus, macromolecular protein complexes, which are built up of protein $\alpha$-helices could harness soliton-assisted energy transport at physiological temperature. Because the hydrolysis of a single adenosine triphosphate molecule is able to initiate three amide I exciton quanta, it is feasible that multiquantal protein solitons subserve a variety of specialized physiological functions in living systems., Comment: 27 pages, 12 figures

Conceptual and mathematical models of neurons have lagged behind empirical understanding for decades. Here we extend previous work in modeling biological systems with fully scale-independent quantum information-theoretic tools to develop a uniform, scalable representation of synapses, dendritic and axonal processes, neurons, and local networks of neurons. In this representation, hierarchies of quantum reference frames act as hierarchical active-inference systems. The resulting model enables specific predictions of correlations between synaptic activity, dendritic remodeling, and trophic reward. We summarize how the model may be generalized to nonneural cells and tissues in developmental and regenerative contexts., Comment: 40 pgs, 7 figures

The Free Energy Principle (FEP) states that under suitable conditions of weak coupling, random dynamical systems with sufficient degrees of freedom will behave so as to minimize an upper bound, formalized as a variational free energy, on surprisal (a.k.a., self-information). This upper bound can be read as a Bayesian prediction error. Equivalently, its negative is a lower bound on Bayesian model evidence (a.k.a., marginal likelihood). In short, certain random dynamical systems evince a kind of self-evidencing. Here, we reformulate the FEP in the formal setting of spacetime-background free, scale-free quantum information theory. We show how generic quantum systems can be regarded as observers, which with the standard freedom of choice assumption become agents capable of assigning semantics to observational outcomes. We show how such agents minimize Bayesian prediction error in environments characterized by uncertainty, insufficient learning, and quantum contextuality. We show that in its quantum-theoretic formulation, the FEP is asymptotically equivalent to the Principle of Unitarity. Based on these results, we suggest that biological systems employ quantum coherence as a computational resource and - implicitly - as a communication resource. We summarize a number of problems for future research, particularly involving the resources required for classical communication and for detecting and responding to quantum context switches., Comment: 57 pgs, 6 figs

The electric activities of cortical pyramidal neurons are supported by structurally stable, morphologically complex axo-dendritic trees. Anatomical differences between axons and dendrites in regard to their length or caliber reflect the underlying functional specializations, for input or output of neural information, respectively. For a proper assessment of the computational capacity of pyramidal neurons, we have analyzed an extensive dataset of three-dimensional digital reconstructions from the NeuroMorpho.Org database, and quantified basic dendritic or axonal morphometric measures in different regions and layers of the mouse, rat or human cerebral cortex. Physical estimates of the total number and type of ions involved in neuronal electric spiking based on the obtained morphometric data, combined with energetics of neurotransmitter release and signaling fueled by glucose consumed by the active brain, support highly efficient cerebral computation performed at the thermodynamically allowed Landauer limit for implementation of irreversible logical operations. Individual proton tunneling events in voltage-sensing S4 protein $\alpha$-helices of Na$^{+}$, K$^{+}$ or Ca$^{2+}$ ion channels are ideally suited to serve as single Landauer elementary logical operations that are then amplified by selective ionic currents traversing the open channel pores. This miniaturization of computational gating allows the execution of over 1.2 zetta logical operations per second in the human cerebral cortex without combusting the brain by the released heat., Comment: 18 pages, 4 figures

Biological order provided by $\alpha$-helical secondary protein structures is an important resource exploitable by living organisms for increasing the efficiency of energy transport. In particular, self-trapping of amide I energy quanta by the induced phonon deformation of the hydrogen-bonded lattice of peptide groups is capable of generating either pinned or moving solitary waves following the Davydov quasiparticle/soliton model. The effect of applied in-phase Gaussian pulses of amide I energy, however, was found to be strongly dependent on the site of application. Moving solitons were only launched when the amide I energy was applied at one of the $\alpha$-helix ends, whereas pinned solitons were produced in the $\alpha$-helix interior. In this paper, we describe a general mechanism that launches moving solitons in the interior of the $\alpha$-helix through phase-modulated Gaussian pulses of amide I energy. We also compare the predicted soliton velocity based on effective soliton mass and the observed soliton velocity in computer simulations for different parameter values of the isotropy of the exciton-phonon interaction. The presented results demonstrate the capacity for explicit control of soliton velocity in protein $\alpha$-helices, and further support the plausibility of gradual optimization of quantum dynamics for achieving specialized protein functions through natural selection., Comment: 19 pages, 11 figures, 4 videos

The essential biological processes that sustain life are catalyzed by protein nano-engines, which maintain living systems in far-from-equilibrium ordered states. To investigate energetic processes in proteins, we have analyzed the system of generalized Davydov equations that govern the quantum dynamics of multiple amide I exciton quanta propagating along the hydrogen-bonded peptide groups in $\alpha$-helices. Computational simulations have confirmed the generation of moving Davydov solitons by applied pulses of amide I energy for protein $\alpha$-helices of varying length. The stability and mobility of these solitons depended on the uniformity of dipole-dipole coupling between amide I oscillators, and the isotropy of the exciton-phonon interaction. Davydov solitons were also able to quantum tunnel through massive barriers, or to quantum interfere at collision sites. The results presented here support a nontrivial role of quantum effects in biological systems that lies beyond the mechanistic support of covalent bonds as binding agents of macromolecular structures. Quantum tunneling and interference of Davydov solitons provide catalytically active macromolecular protein complexes with a physical mechanism allowing highly efficient transport, delivery, and utilization of free energy, besides the evolutionary mandate of biological order that supports the existence of such genuine quantum phenomena, and may indeed demarcate the quantum boundaries of life., Comment: 40 pages, 20 figures

Protein clamps provide the cell with effective mechanisms for sensing of environmental changes and triggering adaptations that maintain homeostasis. The general physical mechanism behind protein clamping action, however, is poorly understood. Here, we explore the Davydov model for quantum transport of amide I energy, which is self-trapped in soliton states that propagate along the protein {\alpha}-helix spines and preserve their shape upon reflection from the {\alpha}-helix ends. We study computationally whether the Davydov solitons are able to reflect from massive barriers that model the presence of external protein clamps acting on a portion of the {\alpha}-helix, and characterize the range of barrier conditions for which the Davydov solitons are capable of tunneling through the barrier. The simulations showed that the variability of amino acid masses in proteins has a negligible effect on Davydov soliton dynamics, but a massive barrier that is a hundred times the mass of a single amino acid presents an obstacle for the soliton propagation. The greater width of Davydov solitons stabilizes the soliton upon reflection from such a massive barrier and increases the probability of tunneling through it, whereas the greater isotropy of the exciton-lattice coupling suppresses the tunneling efficiency by increasing the interaction time between the Davydov soliton and the barrier, thereby prolonging the tunneling time through the barrier and enhancing the probability for reflection from the latter. The results as presented, demonstrate the feasibility of Davydov solitons reflecting from or tunneling through massive barriers, and suggest a general physical mechanism underlying the action of protein clamps through local enhancement of an effective protein {\alpha}-helix mass., Comment: 26 pages, 17 figures

The transport of energy inside protein $\alpha$-helices is studied by deriving a system of quantum equations of motion from the Davydov Hamiltonian with the use of the Schr\"odinger equation and the generalized Ehrenfest theorem. Numerically solving the system of quantum equations of motion for different initial distributions of the amide I energy over the peptide groups confirmed the generation of both moving or stationary Davydov solitons. In this simulation the soliton generation, propagation, and stability were found to be dependent on the symmetry of the exciton-phonon interaction Hamiltonian and the initial site of application of the exciton energy., Comment: 18 pages, 9 figures

Chu Spaces and Channel Theory are well established areas of investigation in the general context of category theory. We review a range of examples and applications of these methods in logic and computer science, including Formal Concept Analysis, distributed systems and ontology development. We then employ these methods to describe human object perception, beginning with the construction of uncategorized object files and proceeding through categorization, individual object identification and the tracking of object identity through time. We investigate the relationship between abstraction and mereological categorization, particularly as these affect object identity tracking. This we accomplish in terms of information flow that is semantically structured in terms of local logics, while at the same time this framework also provides an inferential mechanism towards identification and perception. We show how a mereotopology naturally emerges from the representation of classifications by simplicial complexes, and briefly explore the emergence of geometric relations and interactions between objects.

In the context of orientable circuits and subcomplexes of these as representing certain singular spaces, we consider characteristic class formulas generalizing those classical results as seen for the Riemann-Hurwitz formula for regulating the topology of branched covering maps and that for monoidal transformations which include the standard blowing-up process. Here the results are presented as cap product pairings, which will be elements of a suitable homology theory, rather than characteristic numbers as would be the case when taking Kronecker products once Poincar\'e duality is defined. We further consider possible applications and examples including branched covering maps, singular varieties involving virtual tangent bundles, the Chern-Schwartz-MacPherson class, the homology L-class, generalized signature, and the cohomology signature class.

We consider a generalized Riemann-Hurwitz formula as it may be applied to rational maps between projective varieties having an indeterminacy set and fold-like singularities. The case of a holomorphic branched covering map is recalled. Then we see how the formula can be applied to iterated maps having branch-like singularities. Separately, we consider a further application involving the Chern classes of determinantal varieties when the latter are realized as the degeneracy loci of certain vector bundle morphisms., Comment: Some corections and improvements were made in this new version

We present an operator-coefficient version of Sato's infinite-dimensional Grassmann manifold, and tau-function. In this context, the Burchnall-Chaundy ring of commuting differential operators becomes a C*-algebra, to which we apply the Brown-Douglas-Fillmore theory, and topological invariants of the spectral ring become readily available. We construct KK classes of the spectral curve of the ring and, motivated by the fact that all isospectral Burchnall-Chaundy rings make up the Jacobian of the curve, we compare the (degree-1) K-homology of the curve with that of its Jacobian. We show how the Burchnall-Chaundy C*-algebra extension of the compact operators provides a family of operator-valued tau-functions.

Several features of an analytic (infinite-dimensional) Grassmannian of (commensurable) subspaces of a Hilbert space were developed in the context of integrable PDEs (KP hierarchy). We extended some of those features when polarized separable Hilbert spaces are generalized to a class of polarized Hilbert modules, in particular the Baker and tau-functions, which become operator-valued. Following from Part I we produce a pre-determinant structure for a class of tau-functions defined in the setting of the similarity class of projections of a certain Banach *-algebra. This structure is explicitly derived from the transition map of a corresponding principal bundle. The determinant of this map gives a generalized, operator-valued tau-function that takes values in a commutative C*-algebra. We extend to this setting the operator cross-ratio which had been used to produce the scalar-valued tau-function, as well as the associated notion of a Schwarzian derivative along curves inside the space of similarity classes. We link directly this cross-ratio with Fay's trisecant identity for the tau-function (equivalent to the KP hierarchy). By restriction to the image of the Krichever map, we use the Schwarzian to introduce the notion of operator-valued projective structure on a compact Riemann surface: this allows a deformation inside the Grassmannian (as it varies its complex structure). Lastly, we use our identification of the Jacobian of the Riemann surface in terms of extensions of the Burchnall-Chaundy C*-algebra (Part I) to describe the KP hierarchy.

We outline a model for a cognitive epigenetic system based on elements of the Shannon theory of information and the statistical physics of the generalized Onsager relations. Particular attention is paid to the concept of the rate distortion function and from another direction as motivated by the thermodynamics of computing, the fundamental homology with the free energy density of a physical system. A unifying aspect of the dynamic framework involves the concept of a groupoid and of a groupoid atlas. From a stochastic differential equation we postulate a multidimensional Ito process for an epigenetic system from which a stochastic flow may permeate through components of this atlas., Comment: 22 pages

A novel algebraic topology approach to supersymmetry (SUSY) and symmetry breaking in quantum field and quantum gravity theories is presented with a view to developing a wide range of physical applications. These include: controlled nuclear fusion and other nuclear reaction studies in quantum chromodynamics, nonlinear physics at high energy densities, dynamic Jahn-Teller effects, superfluidity, high temperature superconductors, multiple scattering by molecular systems, molecular or atomic paracrystal structures, nanomaterials, ferromagnetism in glassy materials, spin glasses, quantum phase transitions and supergravity. This approach requires a unified conceptual framework that utilizes extended symmetries and quantum groupoid, algebroid and functorial representations of non-Abelian higher dimensional structures pertinent to quantized spacetime topology and state space geometry of quantum operator algebras. Fourier transforms, generalized Fourier-Stieltjes transforms, and duality relations link, respectively, the quantum groups and quantum groupoids with their dual algebraic structures; quantum double constructions are also discussed in this context in relation to quasi-triangular, quasi-Hopf algebras, bialgebroids, Grassmann-Hopf algebras and higher dimensional algebra. On the one hand, this quantum algebraic approach is known to provide solutions to the quantum Yang-Baxter equation. On the other hand, our novel approach to extended quantum symmetries and their associated representations is shown to be relevant to locally covariant general relativity theories that are consistent with either nonlocal quantum field theories or local bosonic (spin) models with the extended quantum symmetry of entangled, 'string-net condensed' (ground) states.

Given a Kaehlerian holomorphic fiber bundle whose fiber is a compact homogeneous Kaehler manifold, we describe the perturbed Hermitian-Einstein equations relative to certain holomorphic vector bundles. With respect to special metrics on the holomorphic bundles, there is a dimensional reduction procedure which reduces these equations to a system of equations on the base, known as the twisted coupled vortex equations., Comment: 13 pages

We construct a Fourier--Mukai transform for smooth complex vector bundles $E$ over a torus bundle $\pi:M \to B,$ the vector bundles being endowed with various structures of increasing complexity. At a minimum, we consider vector bundles $E$ with a flat partial unitary connection, that is families or deformations of flat vector bundles (or unitary local systems) on the torus $T.$ This leads to a correspondence between such objects on $M$ and relative skyscraper sheaves $\cS$ supported on a spectral covering $\Sigma \hra \what M,$ where $\hat\pi:\what{M} \to B$ is the flat dual fiber bundle. Additional structures on $(E,\nabla)$ (flatness, anti-self-duality) will be reflected by corresponding data on the transform $(\cS, \Sigma).$ Several variations of this construction will be presented, emphasizing the aspects of foliation theory which enter into this picture, Comment: 33 pages. Presented at the Workshop on Geometric Integral Transforms and Applications, in Salamanca, Spain. v2: published version. To appear in J Geom Phys

Our main aim is to associate a holonomy Lie groupoid to the connective structure of an abelian gerbe. The construction has analogies with a procedure for the holonomy Lie groupoid of a foliation, in using a locally Lie groupoid and a globalisation procedure. We show that path connections and 2-holonomy on line bundles may be formulated using the notion of a connection pair on a double category, due to Brown-Spencer, but now formulated in terms of double groupoids using the thin fundamental groupoids introduced by Caetano-Mackaay-Picken. To obtain a locally Lie groupoid to which globalisation applies, we use methods of local subgroupoids as developed by Brown-$\dot{\rm I}$\c{c}en-Mucuk., Comment: 10 pages, Latex2E