Classifying two-body Hamiltonians for Quantum Darwinism

TL;DR

This work provides a broad classification of generic system–environment Hamiltonians with at most two-body interactions regarding their ability to support quantum Darwinism. It identifies two key structural requirements: a time-independent pointer observable that commutes with both the system free dynamics and all system–environment interaction terms, and the preservation of singly-branching states to ensure redundant, accessible information across environment fragments. The authors show that when these conditions hold, objectivity emerges through redundant encoding in the environment, even for nontrivial time dependence and diverse collision models; when they fail, Darwinian objectivity is lost due to intra-environment scrambling or absence of a stable pointer basis. The results provide a practical framework for diagnosing and designing models in quantum thermodynamics and quantum information where quantum-to-classical transitions are central, including explicit demonstrations across qutrit–qubit, alternating interaction, and collision-model scenarios, and they outline paths for extending the analysis to more general (infinite-dimensional or continuous-variable) settings.

Abstract

Quantum Darwinism is a paradigm to understand how classically objective reality emerges from within a fundamentally quantum universe. Despite the growing attention that this field of research as been enjoying, it is currently not known what specific properties a given Hamiltonian describing a generic quantum system must have to allow the emergence of classicality. Therefore, in the present work, we consider a broadly applicable generic model of an arbitrary finite-dimensional system interacting with an environment formed from an arbitrary collection of finite-dimensional degrees of freedom via an unspecified, potentially time-dependent Hamiltonian containing at most two-body interaction terms. We show that such models support quantum Darwinism if the set of operators acting on the system which enter the Hamiltonian satisfy a set of commutation relations with a pointer observable and with one other. We demonstrate our results by analyzing a wide range of example systems: a qutrit interacting with a qubit environment, a qubit-qubit model with interactions alternating in time, and a series of collision models including a minimal model of a quantum Maxwell demon.

Figures (12)

• Figure 1: Schematic representation of the paradigm of quantum Darwinism. A system $\mathfrak{S}$ is immersed in an environment $\mathfrak{E}$, comprised of many subsystems $\{\mathfrak{E}_1, \mathfrak{E_2}, \dots\}$. Observers obtain information about the system indirectly through measurements of fragments of the environment $\mathfrak{F}$. Quantum Darwinism recognizes that the system-environment interaction leads to information about the system being redundantly encoded in the environment such that many observers can recover the classically accessible information about the system through measurements of distinct fragments of the environment (e.g., $\mathfrak{F}_{a,b,c,d}$). Since this process allows many observers to reconstruct the same information with no back-action on the system, that information has become classical and objective.
• Figure 2: Plot of the mutual information between a subset $\mathfrak{F}$ of an environment $\mathfrak{E}$ (here 11 qubits) and a system $\mathfrak{S}$ (a qubit), normalized by the entropy of the reduced system state and averaged over all fragments of that size. The "classical plateau" refers to the flattening of the blue curve about $|\mathfrak{F}|/|\mathfrak{E}| = 1/2$, indicating that the system-environment state allows classical objectivity, whereas the orange curve shows a more typical state which does not. Note that the small size of the simulated environment smooths out the curves; with a sufficiently large environment, the blue curve would be almost perfectly flat at one over almost the entire range of fragment sizes and the orange curve would jump from near zero to near two as the fragment size crosses half the environment. The data for this plot are for the examples E and G discussed in Sec. \ref{['ssec:AlternatingQubit']}.
• Figure 3: (Top) Schematic of the separable interaction structure necessary for a model of a qubit system interacting with an collection of qubits as an environment with two-body interactions to support Quantum Darwinism. (Bottom) Schematic of a generic $\mathfrak{S}\mathfrak{E}$ model demonstrating how the system may interact non-separably with the environment.
• Figure 4: Evolution of the normalized mutual information between the system qutrit and environment fragments of a given size $\mathfrak{F}$ as a function of time, for (a) model A, (b) model B, (c) model C, (d) model D. Each plot shows the mutual information averaged over $100$ simulations with randomized choices of coefficients for each Hamiltonian, all starting from the initial state of Eq. \ref{['eqn:QtInitialState']}.
• Figure 5: Alternate view of Fig. \ref{['fig:QutritQDExamples']}(d) showing the average mutual information fragments of size $1\le|\mathfrak{F}|\le9$ evolving under the Hamiltonian of model D. Beyond $t\approx0.5$ the mixing terms induced by the system-environment interaction cause the information encoded in the environment to spread, precluding the emergence of objectivity in this case.