Reply to Rourk, C. Comment on "Albantakis et al. Computing the Integrated Information of a Quantum Mechanism. Entropy 2023, 25, 449".

In response to a comment by Chris Rourk on our article Computing the Integrated Information of a Quantum Mechanism, we briefly (1) consider the role of potential hybrid/classical mechanisms from the perspective of integrated information theory (IIT), (2) discuss whether the (Q)IIT formalism needs to be extended to capture the hypothesized hybrid mechanism, and (3) clarify our motivation for developing a QIIT formalism and its scope of applicability.

Computing the Integrated Information of a Quantum Mechanism.

Originally conceived as a theory of consciousness, integrated information theory (IIT) provides a theoretical framework intended to characterize the compositional causal information that a system, in its current state, specifies about itself. However, it remains to be determined whether IIT as a theory of consciousness is compatible with quantum mechanics as a theory of microphysics. Here, we present an extension of IIT's latest formalism to evaluate the mechanism integrated information (φ) of a system subset to discrete, finite-dimensional quantum systems (e.g., quantum logic gates). To that end, we translate a recently developed, unique measure of intrinsic information into a density matrix formulation and extend the notion of conditional independence to accommodate quantum entanglement. The compositional nature of the IIT analysis might shed some light on the internal structure of composite quantum states and operators that cannot be obtained using standard information-theoretical analysis. Finally, our results should inform theoretical arguments about the link between consciousness, causation, and physics from the classical to the quantum.

Fusions of Consciousness.

What are conscious experiences? Can they combine to form new experiences? What are conscious subjects? Can they combine to form new subjects? Most attempts to answer these questions assume that spacetime, and some of its particles, are fundamental. However, physicists tell us that spacetime cannot be fundamental. Spacetime, they say, is doomed. We heed the physicists, and drop the assumption that spacetime is fundamental. We assume instead that subjects and experiences are entities beyond spacetime, not within spacetime. We make this precise in a mathematical theory of conscious agents, whose dynamics are described by Markov chains. We show how (1) agents combine into more complex agents, (2) agents fuse into simpler agents, and (3) qualia fuse to create new qualia. The possible dynamics of n agents form an n(n-1)-dimensional polytope with nn vertices-the Markov polytopeMn. The total fusions of n agents and qualia form an (n-1)-dimensional simplex-the fusion simplexFn. To project the Markovian dynamics of conscious agents onto scattering processes in spacetime, we define a new map from Markov chains to decorated permutations. Such permutations-along with helicities, or masses and spins-invariantly encode all physical information used to compute scattering amplitudes. We propose that spacetime and scattering processes are a data structure that codes for interactions of conscious agents: a particle in spacetime is a projection of the Markovian dynamics of a communicating class of conscious agents.

Explanatory profiles of models of consciousness - towards a systematic classification.

Models of consciousness aim to inspire new experimental protocols and aid interpretation of empirical evidence to reveal the structure of conscious experience. Nevertheless, no current model is univocally accepted on either theoretical or empirical grounds. Moreover, a straightforward comparison is difficult for conceptual reasons. In particular, we argue that different models explicitly or implicitly subscribe to different notions of what constitutes a satisfactory explanation, use different tools in their explanatory endeavours and even aim to explain very different phenomena. We thus present a framework to compare existing models in the field with respect to what we call their 'explanatory profiles'. We focus on the following minimal dimensions: mode of explanation, mechanisms of explanation and target of explanation. We also discuss the empirical consequences of the discussed discrepancies among models. This approach may eventually lead to identifying driving assumptions, theoretical commitments, experimental predictions and a better design of future testing experiments. Finally, our conclusion points to more integrative theoretical research, where axiomatic models may play a critical role in solving current theoretical and experimental contradictions.

Fact, Fiction, and Fitness.

A theory of consciousness, whatever else it may do, must address the structure of experience. Our perceptual experiences are richly structured. Simply seeing a red apple, swaying between green leaves on a stout tree, involves symmetries, geometries, orders, topologies, and algebras of events. Are these structures also present in the world, fully independent of their observation? Perceptual theorists of many persuasions-from computational to radical embodied-say yes: perception veridically presents to observers structures that exist in an observer-independent world; and it does so because natural selection shapes perceptual systems to be increasingly veridical. Here we study four structures: total orders, permutation groups, cyclic groups, and measurable spaces. We ask whether the payoff functions that drive evolution by natural selection are homomorphisms of these structures. We prove, in each case, that generically the answer is no: as the number of world states and payoff values go to infinity, the probability that a payoff function is a homomorphism goes to zero. We conclude that natural selection almost surely shapes perceptions of these structures to be non-veridical. This is consistent with the interface theory of perception, which claims that natural selection shapes perceptual systems not to provide veridical perceptions, but to serve as species-specific interfaces that guide adaptive behavior. Our results present a constraint for any theory of consciousness which assumes that structure in perceptual experience is shaped by natural selection.

Consciousness and topologically structured phenomenal spaces.

There are strong reasons to believe that our conscious inner life is structured, suggested both by introspection as well as scientific psychology. One of the most salient structural characteristics of conscious experiences is known as unity of consciousness. In this contribution, we wish to demonstrate how features of experience that pertain to the unity of consciousness could be made precise in terms of mathematical relations that hold between phenomenal objects. Based on phenomenological considerations, we first outline three such features. These are (i) environmental embedding, (ii) the mutual constraint between local and global representations, and (iii) a top-down process of object formation in consciousness. We then introduce a formal model based on the notion of phenomenal space, defined in terms of a set of quasi-elementary and extended entities. We describe the structure of phenomenal space by appealing to mereological and topological concepts, and we outline a projector-based calculus to account for the idea that the structure of phenomenal space is ultimately dynamical. Using the above concepts, one could approach the mind-matter problem by relating environmentally embedded agents to topologically well-defined objects that result from decompositions of phenomenal space. We conclude our discussion by putting it into the context of some recent conceptual questions that appear in cognitive science and consciousness studies. We opt for the possibility to regard the phenomenon of consciousness not in terms of a singular transition that happens between "brain" and "mind" but rather in terms of a series of transitions between structured layers of experience.

A molecular quantum switch based on tunneling in meta-d-phenol C6H4DOH.

We introduce the concept of a molecular quantum switch and demonstrate it with the example of meta-d-phenol, based on recent theoretical and high-resolution spectroscopic results for this molecule. We show that in the regime of tunneling switching with localized low-energy states and delocalized high-energy states the molecular quantum switch can be operated in two different ways: (i) a quasiclassical switching by coherent infrared radiation between the two isomeric structures syn- and anti-m-d-phenol; and (ii) a highly nonclassical switching making use of bistructural quantum superposition states of the syn and anti structures, which can be observed by their time-dependent spectra after preparation.

Wavepacket Dynamics of the Axially Chiral Molecule Cl-O-O-Cl under Coherent Radiative Excitation and Including Electroweak Parity Violation.

We report detailed calculations of the quantum wavepacket dynamics of Cl-O-O-Cl, which serves as a prototype molecule for the stereomutation dynamics of an axially chiral molecule. We include the effects both from electroweak parity violation and from the interaction with a coherent monochromatic laser field. We use the quasiadiabatic channel reaction path Hamiltonian approach to approximately solve the six-dimensional Schrödinger equation describing the vibrational motion, including rotation by an effective Hamiltonian. We calculate time-dependent wave functions based on the time-dependent Schrödinger equation. We study stereomutation dynamics due to tunneling motion and laser-induced population transfer and show results on efficient methods for selectively populating single molecular states in chiral molecules by frequency-modulated laser pulses. We also discuss laser-induced stereomutation (LISM) and a process that may be called resonance Raman induced stereomutation (RRISM). The results are discussed in relation to current experimental attempts to measure the parity violating energy difference ΔpvE between the enantiomers of chiral molecules. Furthermore, we show detailed quantitative simulations of a selection of well-defined parity levels in chiral molecules ("parity isomers") that form the basis of a possible measurement of ΔpvE by the time evolution of parity.

Tunneling and tunneling switching dynamics in phenol and its isotopomers from high-resolution FTIR spectroscopy with synchrotron radiation.

Tunneling and chemical reactions by tunneling switching are reported for phenol and ortho-deuterophenol on the basis of high-resolution FTIR spectroscopy. Tunneling splittings are measured for the torsional motion in the ground and several vibrationally excited states of phenol. Tunneling times range from 10 ns to 1 ps, depending on excitation. For more-highly excited torsional levels in ortho-deuterophenol, delocalization and chemical reaction by tunneling switching is found.