Special Issue on Quantum Information Applied in Neuroscience

1. Introduction

The rapid progress achieved by quantum information science in recent decades was made possible by the realization that genuine quantum phenomena, for which their occurrences are forbidden by classical physics, are not a defect of quantum theory but are useful physical resources [1,2,3]. This was accompanied by a significant increase in the available introductory literature on the subject [4,5,6,7,8]. The application of the full mathematical formalism of quantum information theory to neuroscience, however, has only recently been employed to address different aspects of the mind–brain problem [9,10,11].

Qubits are the smallest physical carriers of quantum information. Quantum information contained in the quantum state $|\Psi \rangle$ of a qubit has some truly remarkable properties. Qubits cannot be observed, read or deduced from experimental data [12], as in the case of classical bits stored on a DVD [9]. If we have a quantum version of a DVD storing a string of qubits, in general, we also cannot copy [13], erase [14], or process any of the stored qubits with irreversible computational gates [7,8]. Qubits can be transported from place to place similarly to classical bits, but each qubit cannot be cloned and delivered to multiple recipients. Because qubits cannot be wholly converted into classical bits, they cannot be broadcasted. Multiple qubits, however, can be used to carry classical bits. Although n qubits can carry more than n classical bits of information, according to Holevo’s theorem, the greatest amount of classical information that can be retrieved by external observers is only n bits [15]. Furthermore, the Bell and Kochen–Specker no-go theorems imply that quantum information is nonlocal, and quantum correlations are enforced with superluminal speed [16,17,18]. These fascinating properties of quantum information may not be reserved for manifestation only in modern quantum technologies but may already have been employed for the enhancement of the survival of evolving biological systems and boosting the power of their neural systems [9,19,20].

Continuous symmetries in the formulation of quantum mechanics are important for the identification of energy as a partial time derivative and momentum as a spatial gradient [21]. Noether’s theorem then establishes that every differentiable symmetry of the action of a physical system is associated with a corresponding conservation law [22], meaning that energy and momentum are conserved quantities. The Schrödinger equation, which is the core of quantum theory, can be derived from the fact that the time-evolution operator must be unitary and must, therefore, be generated by the exponential of a self-adjoint operator, which is the quantum Hamiltonian [23]. The unitarity of quantum evolution leads to the conservation of quantum probabilities and constitutes an essential ingredient in the proofs of many of the quantum information-theoretic no-go theorems, including the celebrated quantum no-cloning theorem. However, the appearance of symmetries in quantum foundations is not the only way that the concept of symmetry can enter neuroscience. The presence of biological order is essential for the operation of living systems, and the spatial organization of repeated structural motifs has been shown to produce favorable boundary conditions for solving the Schrödinger equation that support the extended lifetime of quantum quasiparticles such as solitons in protein $\alpha$-helices [24,25,26] or enable quantum tunneling of protons that trigger the opening of voltage-gated ion channels [27,28,29,30].

2. Quantum Information Applied in  Neuroscience

In light of the above information, this Special Issue presents both theoretical contributions [31,32,33,34,35], which apply quantum information theory as a tool for investigation of open questions in neuroscience, and experimental contributions [36], which probe the submillisecond dynamic physical processes supporting neuronal electric excitation.

In a comprehensive general introduction to the composition of physical systems in quantum mechanics, Gudder describes the quantum measurement process with the use of three types of mathematical entities: quantum observables, quantum instruments and quantum measurement models [31]. After explicitly defining what it means for one mathematical entity to be a part of another entity, the author thoroughly investigates the joint measurability and coexistence of different quantum observables. He also shows that composite quantum systems may contain more information than the individual systems provided that those individual systems are allowed to interact with one another [31].

Georgiev further illustrates how the Schrödinger equation determines the quantum dynamics of a simple two-qubit toy model [32]. After briefly reviewing the theory behind Schmidt decomposition, the author establishes the importance of the non-zero interaction Hamiltonian for the generation of quantum entanglement and the manifestation of observable correlations between different quantum measurement outcomes. He also elaborates on the applicability of the quantum toy model to the actual gating of molecular ion channels in the plasma membranes of electrically active neurons [32].

Kariev and Green discuss in detail the molecular structure and quantum gating of a particular type of voltage-gated potassium ion channel known as the Kv1.2 channel [33]. The authors dedicate their efforts to explain why quantum calculations on ion channels are more useful than classical calculations and focus their attention on the quantum tunneling of protons that trigger the switching of the ion channel conformation from closed to open state. They also acknowledge the need of using supercomputers for performing quantum calculations and point out that current technology is limited to physical systems with a size that is less than a few thousand chemical atoms [33].

Sun and Zhang investigated the effects of the physical environment upon the quantum dynamics of neuronal network that is modeled as an open quantum system [34]. After considering the Born–Markov approximation, the authors study the resulting master equation for the neural dynamics at zero and at room temperature. They observed that, in the presence of thermal noise, the neural state needs a longer time period in order to return back to the initial rest state through depolarization and repolarization processes [34].

Melkikh considers the peculiarities of human mathematical thinking and explores the possibility of its physical implementation in the form of interacting qubits of proteins and other molecules in the neurons of the brain [35]. The author argues that ensuing indeterministic quantum dynamics should be responsible for the uncertainty of human thinking and free will. He also proposes that the consideration of quantum physical principles could help us generalize the already existing branches of mathematics [35].

Singh et al. developed a novel experimental methodology and employed coaxial atom probes in order to perform dielectric resonance imaging of a neuron membrane and its internal structures [36]. The authors are able to record the electric activity of a neuron with microsecond resolution from which they conclude that the neuronal spike is indeed controlled by much faster molecular transitions [36]. This opens the door for a better understanding of cognitive processing of information by neural systems.

Taken together, the contributions collected in this Special Issue provide a comprehensive overview of quantum information theory and illustrate its utility for addressing open questions in neuroscience.