Precision's arrow of time

Abstract

The arrow of time is usually attributed to two mechanisms: decoherence through environmental entanglement, and chaos through nonlinear dynamics. Here we demonstrate a third route, Precision-Induced Irreversibility (PIR), requiring neither. No entanglement. No nonlinearity. Just three ingredients: amplification, non-normality, and finite dynamic range, whose interplay yields an operational arrow of time; remove any one and reversibility can be restored. Non-Hermitian evolution remains mathematically invertible, yet beyond a sharp temporal predictability horizon scaling linearly with available precision, distinct states collapse onto identical representations. Echo-fidelity tests confirm this transition across arbitrary-precision calculations and hardware, revealing where formal invertibility and physical reversibility diverge.

Figures (6)

• Figure 1: The dynamic-range crisis and predictability horizon. (a) Under non-Hermitian evolution, the amplified mode $|c_p(t)|$ (red) grows while suppressed mode $|c_q(t)|$ (blue) decays exponentially, driving dynamic range ratio $r(t) = |c_p|/|c_q|$ across orders of magnitude. The shaded area (precision shadow) marks the relative precision threshold $\varepsilon \cdot |c_p(t)|$ below which the subdominant component becomes numerically unresolvable. At the overflow time $T_{\text{of}}$, $|c_q|$ falls into the precision shadow, forcing a many-to-one mapping and information evaporates. (b) Many-to-one mapping mechanism. Left: Before $T_{\text{of}}$, the available precision bits $m$ suffice to represent both components during quantum operations ($D(t) < m$). Right: After $T_{\text{of}}$, the dynamic range overflows the precision capacity ($D(t) > m$), and quantum operations collapse distinct initial states to identical representations, rendering dynamics operationally irreversible. The addition of a tiny term to a large one corresponds to an off-diagonal coupling $H_{ij}\psi_j$ feeding into component $i$. In a purely diagonal system this mixed-scale addition never occurs, which is why diagonal dynamics shows no PIR in floating-point-like arithmetic (per-component exponent) despite the same precision limit.
• Figure 2: Sharp-knee signatures of precision-induced reversibility breakdown. (a) Loschmidt-echo or fidelity $F(t) = |\langle\psi_0|\psi_{\text{rec}}(t)\rangle|^2/(\langle\psi_0|\psi_0\rangle\langle\psi_{\text{rec}}(t)|\psi_{\text{rec}}(t)\rangle)$ for precision values $m \in \{15, 50, 90\}$ bits using mpmath at $m$-bit precision with time step $\Delta t = 0.4$. Sharp transition at overflow time $T_{\text{of}}$ separates the reversible regime ($F \approx 1$) from irreversible collapse ($F \ll 1$). Vertical dashed lines mark predicted $T_{\text{of}} = m\ln(\beta)/\Delta b$ with $\Delta b = 1.327$ (time in units of $1/g$, where $g$ is the coupling strength). (b) Work-echo ratio $\eta_W(t) = W_{\text{rec}}(t)/W_{\text{out}}(t)$ for the same precision values, revealing three distinct regimes: Regime 1 ($t < T_{\text{of}}$): Curves collapse to an $m$-independent reversible plateau $\eta_W^{\text{rev}} \approx 1.2$, determined by initial state and measurement Hamiltonian—precision is irrelevant while dynamics remain reversible. Regime 2 ($t \approx T_{\text{of}}$): Sharp knee with transition occurring at $m$-dependent $T_{\text{of}}$ values that scale linearly with precision. Regime 3 ($t \gg T_{\text{of}}$): Saturation to $\eta_W^{\infty} \approx 0.3$, independent of both precision $m$ and initial state—the system has forgotten its preparation.
• Figure 3: Universal scaling of overflow time with precision. Main panel: Overflow time $T_{\text{of}}$ versus precision bits $m$, measured via Loschmidt fidelity $F$ (circles) and work-echo ratio $\eta_W$ (crosses) using onset detection (1% deviation from reversible plateau). Data from arbitrary-precision stepped evolution (mpmath, orange) and native hardware arithmetic (float32, pink; float64, green) follow the theoretical prediction $T_{\text{of}} \propto m\ln(\beta)/\Delta b$ (dashed line) with $\Delta b = 2\sqrt{\gamma^2 - g^2} = 1.327$ for $\gamma=1.2$, $g=1.0$ (gain/loss and coupling strengths, respectively). Linear scaling confirms that reversibility breakdown is precision-limited; agreement between fidelity and work-echo demonstrates observable-independence. Inset: Condition number $\ln\kappa(U(t))$ (natural logarithm) grows exponentially at rate $\Delta b$, crossing precision thresholds $\ln\kappa_{\text{th}} = m\ln(\beta)$ at the predicted $T_{\text{of}}$ (vertical dotted lines) for $m \in \{30, 60, 90\}$ bits.
• Figure S1: Non-normality benchmark for PIR. (a) Propagator condition number $\log\kappa(U(\tau))$ versus time. The PT-symmetric and normal diagonal systems show identical exponential growth, while the Hermitian system maintains $\kappa(U) = 1$. (b) Loschmidt echo fidelity $F(\tau)$. Only the PT-symmetric (non-normal) system shows fidelity collapse at $T_{\mathrm{of}}$. The diagonal (normal) system maintains $F \approx 1$ despite identical $\kappa(U)$ growth. Parameters: $\gamma = 1.2$, $g = 1.0$, $\Delta_b = 1.33$, $m = 53$ bits. Computed using stepped evolution at $m$-bit precision ($\Delta t = 0.4$), matching the methodology of Fig. 2 in the main text.
• Figure S2: Linear cavity geometry for PIR detection. A PT-symmetric coupled-waveguide dimer (one waveguide with gain $+\gamma$, one with loss $-\gamma$, evanescent coupling $g$) is placed inside a linear cavity. Light enters through a partial mirror (left), propagates through the dimer of length $L$, and reflects from an end element that implements the $i\sigma_y$ transformation (waveguide crossing plus $\pi$ phase shift). The return path traverses the same physical structure, ensuring that forward evolution under $\mathcal{H}$ is followed by backward evolution under $-\mathcal{H}$. This geometry eliminates fabrication mismatches between forward and backward paths.