Dorje Brody

The probability of a given candidate winning a future election is worked out
in closed form as a function of (i) the current support rates for each
candidate, (ii) the relative positioning of the candidates within the political
spectrum, (iii) the time left to the election, and (iv) the rate at which noisy
information is revealed to the electorate from now to the election day, when
there are three or more candidates. It is shown, in particular, that the
optimal strategy for controlling information can be intricate and nontrivial,
in contrast to a two-candidate race. A surprising finding is that for a
candidate taking the centre ground in an electoral competition among a
polarised electorate, certain strategies are fatal in that the resulting
winning probability for that candidate vanishes identically.

The cognitive state of mind concerning a range of choices to be made can be modelled efficiently by use of an element of a high-dimensional Hilbert space. The dynamics of the state of mind resulting from information acquisition can be characterised by the von Neumann–Lüders projection postulate of quantum theory. This is shown to give rise to an uncertainty-minimising dynamical behaviour equivalent to Bayesian updating, hence providing an alternative approach to representing the dynamics of a cognitive state, consistent with the free energy principle in brain science. The quantum formalism, however, goes beyond the range of applicability of classical reasoning in explaining cognitive behaviour, thus opening up new and intriguing possibilities.

The detection and processing of environmental information is a fundamental attribute of all living systems. This article approaches the question of how efficiently plants process that information, considering it likely that differential efficiency among different species may help explain differential survival. The primary routes of information transfer, that is, signal transduction, are relatively well understood. It is pointed out, based on current understanding, that erasure of such information may be of equal importance to its acquisition. This in turn could provide a simple means of quantifying the acquisition of information by a plant and the efficiency in doing so. However, wild plants live in an environment that is noisy. A useful analogy to deal with such situations can be found in quantum theory of open systems, wherein processes are both well-characterized and well understood, unlike those in plants. This paper develops this theme from quantum information processing and provides a mean of transferring such quantum characterization to biological dynamical situations. The article concludes with discussion on communication theory, indicating that there is a dearth of experimental results that can currently be used to investigate information processing but suggesting how progress can be made in this important programme.

Stochastic Schrödinger equations that govern the dynamics of open quantum systems are given by the equations for signal processing. In particular, the Brownian motion that drives the wave function of the system does not represent noise, but provides purely the arrival of new information. Thus the wave function is guided by the optimal signal detection about the conditions of the environments under noisy observations. This behaviour is similar to biological systems that detect environmental cues, process this information, and adapt to them optimally by minimising uncertainties about the conditions of their environments. It is postulated that information-processing capability is a fundamental law of nature, and hence that models describing open quantum systems can equally be applied to biological systems to model their dynamics. For illustration, simple stochastic models are considered to capture heliotropic and gravitropic motions of plants. The advantage of such dynamical models is that it allows for the quantification of information processed by the plants. By considering the consequence of information erasure, it is argued that biological systems can process environmental signals relatively close to the Landauer limit of computation, and that loss of information must lie at the heart of ageing in biological systems.

The phase space of a relativistic system can be identified with the future tube of complexified Minkowski space. As well as a complex structure and a symplectic structure, the future tube, seen as an eight-dimensional real manifold, is endowed with a natural positive-definite Riemannian metric that accommodates the underlying geometry of the indefinite Minkowski space metric, together with its symmetry group. A unitary representation of the 15-parameter group of conformal transformations can then be constructed that acts upon the Hilbert space of square-integrable holomorphic functions on the future tube. These structures are enough to allow one to put forward a quantum theory of phase-space events. In particular, a theory of quantum measurement can be formulated in a relativistic setting, based on the use of positive operator valued measures, for the detection of phase-space events, hence allowing one to assign probabilities to the outcomes of joint space-time and four-momentum measurements in a manifestly covariant framework. This leads to a localization theorem for phase-space events in relativistic quantum theory, determined by the associated Compton wavelength.