On the emergence of quantum Darwinism and pointer states for non-commuting evolutions

TL;DR

This paper addresses how objectivity and pointer states emerge when the system and environment evolve under non-commuting Hamiltonians, i.e., $[H_ ext{S},H_ ext{I}] eq 0$. It employs a qubit model with a dephasing-type interaction and analyzes objectivity via accessible information $I_{ m acc}$ and redundancy $ ext{Red}$, along with an SBS-based definition of pointer states. The main finding is that objectivity can still arise in the non-commuting regime, and pointer states can be identified through the SBS form, though their exact basis depends on the initial state and the relative speeds of system and interaction dynamics, with fast-environment interactions ($ ext{γ} o ext{ω}$ or higher) helping preserve objectivity. This work thus extends quantum Darwinism to more realistic, non-ideal dynamics and clarifies how pointer states are defined and manifested when commutativity is absent, offering insights for the quantum-to-classical transition in complex environments.

Abstract

Quantum systems achieve objectivity by redundantly encoding information about themselves into the surrounding environment, through a mechanism known as quantum Darwinism. When this happens, observes measure the environment and infer the system to be in one of its pointer states. We study the emergence of objectivity whenever the Hamiltonian of the system and the interaction Hamiltonian between system and environment do not commute, a condition which is thought to be incompatible with the presence of pointer states. We show that, not only non-commuting evolutions allow for the emergence of objective states, but it is possible to give a more relaxed definition of pointer states, that is always well defined whenever there is redundancy of information, and coincides with the usual one for commuting evolutions.

Figures (5)

• Figure 1: Accessible information as a function of time. We report the accessible mutual information $I_{\rm acc}(\mathcal{S}:\mathcal{E}_1)$ between system and an individual environmental qubit as a function of time, for different values of $p$. The model parameters are fixed at $\omega=\gamma=0.1, n=8$. The reported accessible information is rescaled over $S(\rho_\mathcal{S})$.
• Figure 2:
• Figure 3: Accessible information as a function of time. We report the accessible mutual information $I_{\rm acc}(\mathcal{S}:\mathcal{E}_1)$ between system and an individual environmental qubit as a function of time, for different values of the $\omega/\gamma$ ratio, fixing $p=1$ and $n=8$. The reported accessible information is rescaled over $S(\rho_\mathcal{S})$.
• Figure 4: Maximal redundancy changing model parameters We report $\mathrm{Red}(\rho_{\mathcal{S}\mathcal{E}})$ and $I_{\rm acc}(\mathcal{S}:\mathcal{E}_1)$ computed at the time of maximal redundancy, as a function of $\omega/\gamma$, fixing $p=1$, $n=8$ and $\delta=0.9$.
• Figure 5: Fidelity, with respect to the computational basis, for the pointer states. Fidelity between one of the pointer states emerging from the dynamics $|\varphi^0\rangle$ and $|0\rangle$ as a function of the $p$ parameter. Panel a): the initial state is of the form $\sqrt{x_0}\ket{0}+i\sqrt{1-x_0}\ket{1}$ with $\gamma=\omega=0.1$. Panel b): the initial state is of the form $\frac{1}{\sqrt{2}}\ket{0}+\phi\frac{1}{\sqrt{2}}\ket{1}$ with $\gamma=\omega=0.1$. Panel c): the initial state is $|\circlearrowleft\rangle$ with different degrees of the $\omega/\gamma$ ratios. For all panels $n=6$ and $\delta=0.9$. The plot shows that