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

TL;DR

This work defines the coherence-dispersion measure $Δ_{ m c}$ as the variance of the absolute off-diagonal density-matrix elements in a fixed basis and shows it peaks at intermediate coherence entropy $S_{ m c}$, a signature of complexity-like behavior. The authors derive exact one- and multi-copy expressions for $Δ_{ m c}$ using scalable quantities $P^2$, $C_1$, and the purity $ ext{Π}$, and connect $Δ_{ m c}$ to quantum athermality via coherent Gibbs states, giving a compact formula $Δ_{ m c}(G(T))$ in terms of the equilibrium purity $ ext{Π}_{ m eq}(T)$. They apply this framework to a two-level toy model of cellular energetics with ATP-ADP energy scale, predicting a finite-temperature maximum $τ^*$ for $Δ_{ m c}$ that falls within the temperature range supporting unicellular life, robust to the number of coherent energy consumption sites and insensitive to some structural details. The results imply that even modest remnant coherence can be functionally relevant in warm, open systems and motivate further exploration of coherence-dispersion concepts in quantum biology and related resource theories.

Abstract

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.

Figures (11)

• Figure 1: Summary of statistical properties of the entries of $\varrho$. Eq. (\ref{['V']}) fills the gap related to the variance of the absolute values of the coherences.
• Figure 2: $S_{\rm c}$ against the dimension $D$, for the states $f$ and $\Phi_{\rm M}$. Inset: ${\Delta_{\rm c}}$ versus $S_{\rm c}$ for the state $|\psi_x\rangle \langle \psi_x|$ with $D=10$. All plotted quantities are dimensionless.
• Figure 3: $\Delta_{\rm c}$ as a function of $\tau$ and $n$ (starting from $5000$) for $\lambda=0.1$. The contours are essentially the same for any value of $\lambda \in [0.001,0.3]$ and for $d=2, \dots, 20$.
• Figure 4: Rank $r(D)$ which maximizes $\Delta_{\rm c}$ as a function of $D$.
• Figure 5: Plot of $\Delta_{\rm c}$ as a function of $n$ and the dimensionless temperature $\tau$, for $n$ ranging from $5.000$ to $50.000$ and $\lambda=0.1$. $\Delta_{\rm c}$ vanishes in the light gray regions and the green stripe marks the locus of the maxima. The section $n=5.000$ is also shown (the orange line corresponds to the horizontal interval in the contour plot).