Information-Theoretic Equilibration: The Appearance of Irreversibility under Complex Quantum Dynamics

The question of how irreversibility can emerge as a generic phenomenon when the underlying mechanical theory is reversible has been a long-standing fundamental problem for both classical and quantum mechanics. We describe a mechanism for the appearance of irreversibility that applies to coherent, isolated systems in a pure quantum state. This equilibration mechanism requires only an assumption of sufficiently complex internal dynamics and natural information-theoretic constraints arising from the infeasibility of collecting an astronomical amount of measurement data. Remarkably, we are able to prove that irreversibility can be understood as typical without assuming decoherence or restricting to coarse-grained observables, and hence occurs under distinct conditions and time scales from those implied by the usual decoherence point of view. We illustrate the effect numerically in several model systems and prove that the effect is typical under the standard random-matrix conjecture for complex quantum systems.

Figure 1

We plot $\mathrm{Pr}\mathbf{(}k|\rho (t)\mathbf{)}$ for Pauli-$Z$ measurements given by quantum theory (blue) and the uniform distribution (red) for a random local Hamiltonian acting on 10 qubits at $t=0$ and for $t>t_{\mathrm{eq}}$. ITE arises from the difficulty in distinguishing quantum fluctuations from sampling errors for $t>t_{\mathrm{eq}}$.

Figure 2

The RLH ensemble average of the probabilities $\mathrm{Pr}(k\ne 0|e^{-iHt}|0⟩⟨0|e^{iHt})$ of the evolved state for 250 random Hamiltonians plotted as a function of time for 5,…,10 qubit systems. The circles show $t_{\mathrm{eq}}$ for each $n$, which scale roughly as $O\mathbf{(}\log (D)\mathbf{)}$.

Figure 3

Numerically computed expectation values and variances over the RLH ensemble of the outcome variance computed at $t=10$ where $t_{\mathrm{eq}}⪅2$ for $\rho (0)=|0⟩⟨0|$. The data were obtained for 250 randomly chosen Hamiltonians, and show that ${\mathbb{V}}_{E_H}[{\mathbb{V}}_k\{\mathrm{Pr}\mathbf{(}k|\rho (t)\mathbf{)}\}]\approx 0.05D^{-4}$ and ${\mathbb{E}}_{E_H}[{\mathbb{V}}_k\{\mathrm{Pr}\mathbf{(}k|\rho (t)\mathbf{)}\}]\approx D^{-2}$.

Figure 4

Evidence of ITE for $t\ge t_{\mathrm{eq}}$ in an extremely simple many-body system with nearest-neighbor interactions for increasing number of spins $n$ ($D=2^n$). The measurement consists of readout of each spin along the $z$ axis. The plot shows ${\mathbb{V}}_k\{\mathrm{Pr}\mathbf{(}k|\rho (t)\mathbf{)}\}\approx 1.6D^{-2}$ for the Hamiltonian, Eq. (5), with $t=20$ where $t_{\mathrm{eq}}\simeq 15$ (see Supplemental Material [26]).