Coherence Dispersion and Temperature Scales in a Quantum-Biology Toy Model

In this work, we investigate how quantum coherence can scatter among the several off-diagonal elements of an arbitrary quantum state, defining coherence dispersion ($Δ_{\rm c}$). It turns out that this easily computable quantity is maximized for intermediate values of an appropriate entropy, a prevalent signature of complexity quantifiers across different fields, from linguistics and information science to evolutionary biology. By focusing on out-of-equilibrium systems, we use the developed framework to address a simplified model of cellular energetics, involving remanent coherence. Within the context of this model, the precise energy of 30.5 kJ/mol (the yield of ATP-ADP conversion) causes the temperature range where $Δ_{\rm c}$ is maximized to be compatible with temperatures for which unicellular life is reported to exist. Low levels of coherence suffice to support this conclusion.

💡 Research Summary

This paper presents an interdisciplinary study bridging quantum information theory and biology, introducing a novel metric called “coherence dispersion (Δc)” and applying it to a toy model of cellular energetics to find a intriguing connection with the temperature range of life.

The first part of the work is theoretical. The authors propose Δc as a measure of the variability (variance) among the absolute values of the off-diagonal elements (coherences) of a quantum state in a chosen basis. It is defined using the ℓ1 and ℓ2 norms of coherence. Δc exhibits key characteristics often associated with complexity measures across various fields: it vanishes for both maximal disorder (maximally mixed state) and minimal disorder (certain pure states), reaching a maximum for intermediate values of an appropriate entropy, specifically the relative entropy of coherence. The states that maximize Δc are pure states that are superpositions of only a subset of the basis states, not all of them. The authors also demonstrate that Δc is a convex function and possesses the property of “scalability,” meaning the dispersion for a system composed of n non-interacting copies can be calculated solely from the purity, predictability, and ℓ1 norm of coherence of a single copy.

The second part connects this concept to thermodynamics and biology. For a system in thermal equilibrium (a Gibbs state), Δc is zero. To model an open, non-equilibrium system that may retain some quantum coherence, the authors construct “partially coherent Gibbs states,” which are mixtures of the equilibrium Gibbs state and a coherent Gibbs pure state. For such states, and considering a system of n independent d-level subsystems with equally spaced energy levels, Δc as a function of dimensionless temperature τ exhibits a sharp global maximum at a specific temperature τ*.

The biological application comes in the form of a toy model for cellular energy consumption. Each energy consumption site (ECS) within a cell is modeled as a two-level system (d=2). The energy gap ε is set to the precise energy released by ATP hydrolysis (approximately 0.316 eV or 30.5 kJ/mol). The state of n such sites is described by the n-copy partially coherent Gibbs state. The number of sites n and the residual coherence level λ are varied over wide ranges informed by biological estimates (e.g., the number of ATP molecules in an E. coli cell). Remarkably, the temperature τ* at which Δc is maximized, when converted to real temperature via T = τϵ/4k_B, consistently falls within an interval of approximately -37°C to 125°C across the parameter ranges. This interval tightly contains the reported temperature range for viable unicellular life, approximately -20°C to 122°C. The result is robust to large variations in λ and the site dimension d but critically sensitive to the value of the energy gap ε; doubling ε shifts the predicted temperature window completely out of the biological range.

The study suggests that the combination of a specific biological energy scale (ATP-ADP conversion) and the presence of even very low levels of quantum coherence in out-of-equilibrium cellular processes could be linked to the emergence of a temperature window conducive to life, as quantified by the maximization of coherence dispersion. It serves as a proof-of-concept, highlighting a potential quantum-thermodynamic signature in biology.