Quantum Information Biology: From Information Interpretation of Quantum Mechanics to Applications in Molecular Biology and Cognitive Psychology

Abstract

We discuss foundational issues of quantum information biology (QIB)—one of the most successful applications of the quantum formalism outside of physics. QIB provides a multi-scale model of information processing in bio-systems: from proteins and cells to cognitive and social systems. This theory has to be sharply distinguished from “traditional quantum biophysics”. The latter is about quantum bio-physical processes, e.g., in cells or brains. QIB models the dynamics of information states of bio-systems. We argue that the information interpretation of quantum mechanics (its various forms were elaborated by Zeilinger and Brukner, Fuchs and Mermin, and D’ Ariano) is the most natural interpretation of QIB. Biologically QIB is based on two principles: (a) adaptivity; (b) openness (bio-systems are fundamentally open). These principles are mathematically represented in the framework of a novel formalism— quantum adaptive dynamics which, in particular, contains the standard theory of open quantum systems.

Appendices

Appendix 1: Quantum Nonlocality in Physics and Biology

Nonlocal interpretation of violation of Bell’s inequality led to revolutionary rethinking of foundations of QM. The impossibility to create theories with local hidden variables makes quantum randomness even more mystical than it was seen by fathers of QM, for example, by von Neumann [54]. In biology we have not yet seen violations of Bell’s inequality in so to say nonlocal setting. Nevertheless, it may be useful to comment possible consequences of such possible violations. Consider, for example, quantum modeling of cognition and the issue of mental nonlocality. The quantum-mental analogy has to be used with some reservations. The brain is a small physical system (comparing with distances covered by propagating light). Therefore “mental-nonlocality” (restricted to information states produced by a single brain) is not as mystical as physical nonlocality of QM. One may expect that in future cognition can be “explained”, e.g., in terms of “nonlocal” hidden variables. This position was presented by J. Acacio de Barros in [21] and it is reasonable.

“Quantum nonlocality” is still the subject of intense debates about interpretational and experimental issues. From the interpretation side, the main counter-argument against the common nonlocal interpretation is that, in fact, the main issue is not nonlocality, but contextuality of Bell’s experiments [66]: s tatistical data collected in a few (typically four) experimental contexts $C_i$ are embedded in one inequality. However, from the viewpoint of CP each context $C_i$ has to be represented by its own probability space. It is not surprising or mystical that such contextual data can violate the inequality which was derived for a single probability space [14, 59]. This viewpoint is confirmed by experiments in neutron interferometry [67] showing a violation of Bell’s inequality for a single neutron source, but in multi-context setting. In cognitive psychology a similar experiment, on recognition of ambiguous figures, was performed by Conte et al. [16], see [12] for the theoretical basis of this experiment.

The experimental situation in quantum physics is not so simple as it is presented by the majority of writers about violations of Bell’s inequality. There are known various loopholes in Bell’s experimental setups. A loophole appears in the process of physical realization of the ideal theoretically described experimental setup. One suddenly finds that what is possible to do in reality differs essentially from the textbook description. To close each loophole needs tremendous efforts and it is costly. The main problem is to close a few (in future all possible) loopholes in one experiment. Unfortunately, this “big problem” of combination of loophole closing is practically ignored. The quantum physical community is factually fine with the situation that different loopholes are closed in different types of experiments. Of course, for any logically thinking person this situation is totally unacceptable. The two main loopholes which disturb the project of the experimental justification of violation of Bell’s inequality are the detection efficiency loophole also known as fair sampling loophole and the locality loophole.

Locality loophole: It is very difficult, if possible at all, to remove safely two massive particles (i.e., by escaping decoherence of entanglement) to a distance which is sufficiently large to exclude exchange by signals having the velocity of light and modifying the initially prepared correlations. Therefore the locality loophole was closed by Weihs et al. [68] for photons, massless particles.

Detection efficiency/fair sampling loophole: Photo-detectors (opposite to detectors for massive particles) have low efficiency, an essential part of the population produced by a source of entangled photons is undetected. Thus a kind of post-selection is in charge. In terms of probability spaces, we can say that each pair of detectors (by cutting a part of population) produces random output described by its own probability space. Thus we again fall to the multi-contextual situation.⁷ Physicists tried to “solve” this problem by proceeding under the assumption of fair sampling, i.e., that the detection selection does not modify the statistical features of the initial population. Recently novel photo-detectors of high efficiency (around 98%) started to be used in Bell’s tests. In 2013 two leading experimental groups closed the detection loophole with the aid of such detectors, in Vienna [69] (Zeilinger’s group) and in Urbana-Champaign [70] (Kwiat’s group).

Experiments in 2013 led to increase of expectations that finally both locality and detection loopholes would be soon closed in a single experiment. However, it seems that the appearance of super-sensitive detectors did not solve the problem completely. Photons disappear not only in the process of detection; they disappear in other parts of the experimental scheme: larger distance - more photons disappear. It seems that the real experimental situation matches with the prediction of A. Khrennikov and I. Volovich that the locality and detection loopholes would be never closed in a single experiment [71] (where by detection efficiency we understand the efficiency of the complete experimental setup): an analog of Heisenberg’s uncertainty relation for these loopholes.

Another problem of the experimental verification of violation of Bell’s inequality is that experiments closing one of previously known loopholes often suffer of new loopholes which were not present in “less advanced experiments”.

In general one has not to overestimate the value of Bell’s test; other test of nonclassicality, for example, violation of LTP, may be less controversial.

Finally, we make a philosophic remark: Bell’s story questions the validity of Popper’s principle of falsification of scientific theories. It seems that it is impossible to falsify the CP-model of statistical physics in any concrete experiment. It seems that one has to accept that the QP-formalism is useful not because it was proven that the CP-formalism is inapplicable, but because the QP-formalism is operationally successful and simple in use. (We remark that QP is based solely on linear algebra which much easier mathematically than the measure theory serving as the basis of CP).

Appendix 2: Exotic Quantum-Like Models: Possible Usefulness for Biological Information Theory

Quantum adaptive dynamics, although essentially deviating from the standard quantum formalism, is still based on complex Hilbert space. Coming back to the question posed in section (“why the quantum formalism?”) we remark that, in principle, there is no reason to expect that the biological information theory should be based on the representation of probabilities by complex amplitudes, normalized vectors of complex Hilbert space. Therefore one may try to explore models which are quite exotic, even comparing with quantum adaptive dynamics.

Some of these models are exotic only from the viewpoint of using of rather special mathematical tools; otherwise they arise very naturally from the probabilistic viewpoint. For example, we point to a novel model, so-called hyperbolic quantum mechanics, which was applied to a series of problems of cognition and decision making [14]. Here, the probability amplitudes are valued not in the field of the complex numbers, but in the algebra of the hyperbolic numbers, numbers of the form $z=x+jy,$ where x, y are real numbers and j is the generator of the algebra satisfying the equation $j^2=+1.$ Hyperbolic amplitudes describe interference which is stronger than one given by the complex amplitudes used in QM: the interference term is given by the hyperbolic cosine, opposite to the ordinary trigonometric cosine in QM. In [14] there can be found the classification of various types of (probabilistic) interference exhibited in the form of violation of LTP; here the interference term is defined as the magnitude of deviation from LTP. It was found that the mathematical classification leads to only two basic possibilities: either the standard trigonometric interference or the hyperbolic one. Probabilistic data exhibiting the trigonometric interference can be represented in the complex Hilbert space and the data with the hyperbolic interference in the hyperbolic Hilbert space. The third possibility is mixture of the two types of interference terms. This leads to representation of probabilities in a kind of Hilbert space over hyper-complex numbers. We remind that in quantum foundations extended studies were performed to check usefulness of various generalizations of the complex Hilbert space formalism; for example, quaternionic QM or non-Archimedean QM [72]. It seems that such models were not so much useful in physics; in any event foundational studies did not lead to concrete experimental results. However, one might expect that such models, e.g., quanternionic QM, can find applications in quantum-like biological information. Possible usefulness of non-Archimedean, especially p-adic QM, will be discussed later in relation with unconventional probability models.

One of the widely known unconventional probabilistic model is based on relaxation of the assumption that a probability measure has to value in the segment [0, 1] of the real line. So-called “negative probabilities” appear with strange ragularity in a variety of physical problems , see, e.g., [59], for very detailed presentation (we can mention, for examples, the contributions of such leading physicists as P. Dirac, R. Feynman, A. Aspect). Recently negative probabilities started to be used in cognitive psychology and decision making [65, 73].

Negative probabilities appears naturally in p-adic quantum models as limits of relative frequencies with respect to p-adic topology on the set of rational numbers (frequencies are always rational). P-adic statistical stabilization characterizes a new type of randomness which is different from both classical and quantum randomness. P-adic probabilities were used in some biological applications [62]: cognition, population dynamics, genetics.