Fig 1.
A predator and its prey balancing on the archetypal wavefunction Ψ, a symbol of their entangled relationship.
Fig 2.
Four types of interactions between a cell (Alice) and a virus (Bob).
The top row illustrates successful bindings when the cell receptors correlate with the virus ligands, C = V = 1 or C = V = −1. The bottom row shows unsuccessful binding attempts when the opposite occurs, C ≠ V. This setting is reminiscent of the Alice-Bob coordination game underlying bipartite Bell inequalities.
Fig 3.
Population dynamics throughout a simulated game.
The effect of nonlocal correlations on cell (red) and virus (blue and green) communities is depicted by comparing the evolution of the species’ populations over time in a simulation. On the left, viruses from both communities, the blue and green, are equally correlated with their host cells. In this case the entire system is stable and the communities all survive. The right plot illustrates the case where viruses from one community (green) are coordinated with their host cells to the extent permitted by quantum mechanics (above any classical correlation). Due to monogamy of correlations, the blue virus community exhibit lack of coordination with their host cells which ultimately leads to the extinction of the blue specie.
Fig 4.
An agent-based simulation of monogamy of survival in a 3-specie population dynamics game.
The axes represent the values of the Bell parameters and
. For each value of these parameters that lie on the circle defined by the saturation of Eq (8), the extinction probabilities of species
(greenish) and
(pinkish) are represented by the length of the corresponding pin. The classical correlations are bounded by the square. The more one virus specie violates Bell’s inequality, the smaller become the odds of the other virus specie to survive.
Fig 5.
Top left: The real and imaginary parts (up to sign) of the three significant Lyapunov exponents, plotted as a function of θ, with parameters n = 100, β = 1 and δ = 4. Note there is a range of θ values where two of the Lyapunov exponents form a complex conjugate pair (here we only depict the one with positive imaginary part). The positivity of the Lyapunov exponents depends on the value of ζ. Since 0 < ζ < ζ + γ = δ (= 4 in this case), the answer generally depends on the relation between γ and ζ, i.e. the cell reproduction rate and virus decay rate, respectively. Top right: The real and imaginary parts (up to sign) of the two significant Lyapunov exponents λ as a function of δ, plotted for β = 1 and θ = 0; i.e., the maximally-entangled case. In this case the Lyapunov exponents no longer depend on n. Note the sharp transition occuring at : for δ ≤ δc, ℜ(λ) = (γ − ζ)/2, and the two Lyapunov exponents comprise a complex conjugate pair; thus, we may expect unstable behavior iff γ > ζ. However, for δ > δc this is no longer true, as the upper branch exceeds the continuation of the line ℜ(λ) = δ/2 − ζ. Bottom left: the real parts of the two significant Lyapunov exponents λ as a function of n and δ, plotted for θ = θmax and β = 1 (the equally-correlated case). The plot to the bottom right depicts the (δ = 4)-cut of the left plot, and illustrates both the real and imaginary parts (up to sign) of the Lyapunov exponents. The Lyapunov exponents form a complex conjugate pair for δ− < δ < δ+, where
, and the imaginary part obtains its maximal absolute value for
. Note that for small values of n, the plot resembles the maximally-entangled case. This is not surprising, since for n = 1 the two cases converge. Large n behavior can be observed as well: note how the larger Lyapunov exponent approaches −ζ, while the smaller one approaches
.