Toy Model Challenging Prevailing Definitions of Classicality

Abstract

We analyze a toy model that obeys environmentally induced decoherence and quantum Darwinism and satisfies the decoherent histories criterion and Leggett-Garg inequalities with respect to the pointer basis. Yet, the resulting "classical" dynamics are extremely fragile and recohere after a seemingly innocent control operation. This challenges the idea that classical behaviour or the Multiverse branches can be identified by looking at the universal wave function alone. It also demonstrates that quantum Darwinism is compatible with strong non-Markovianity. Possible solutions related to Markovianity, non-integrability and an operational definition of pointer states are briefly discussed.

Figures (2)

• Figure 1: Numerical test of the predictability sieve by plotting the time evolution of von Neumann entropy for different initial states $|\psi(0)\rangle = e^{-i\phi\sigma_y}|0\rangle$ with $\phi\in\{0,\pi/8,\pi/6,\pi/4\}$. We see a rapid rise of entropy, i.e., loss of predictability, for all states except the pointer state $|0\rangle$. Parameters of the model are $g_j = 1$ and $\gamma_j = 1$ for all $j$.
• Figure 2: Effect of the control operation ${\mathfrak{C}}$ on the coherences ${\langle {\sigma_x}\rangle}(t)$ as a function of time $t$ for different numbers of environmental particles. ${\mathfrak{C}}$ happens at $\Gamma_{{\cal{E}}}t^*=2$ (dashed vertical red line) and causes perfect recoherence at $t = 2t^*$ (blue lines). The green lines display the evolution without ${\mathfrak{C}}$. The initial state was $|\psi(0)\rangle_{{\cal{S}}} = |+\rangle_{{\cal{S}}} \equiv (|0\rangle_{{\cal{S}}}+|1\rangle_{{\cal{S}}})/\sqrt{2}$ and model parameters are as in Fig. \ref{['fig 1']}.

Theorems & Definitions (1)

• Definition : $(\epsilon,\tau)$-pointer states