Do mixed states exhibit deep thermalisation?

Abstract

Deep thermalisation -- where ensembles of pure states on a local subsystem, conditioned on measurement outcomes on its complement, approach universal maximum-entropy ensembles -- represents a stronger form of ergodicity than conventional thermalisation. We show that this framework fails dramatically for mixed initial states, evolved unitarily, even with infinitesimal initial mixedness. To address this, we introduce a new paradigm of deep thermalisation for mixed states, fundamentally distinct from that for pure-state ensembles. In our formulation, the deep thermal ensemble arises by tracing out auxiliary degrees of freedom from a maximum-entropy ensemble defined on an augmented system, with the ensemble structure depending explicitly on the entropy of the initial state. We demonstrate that such ensembles emerge dynamically in generic, locally interacting chaotic systems. For the self-dual kicked Ising chain, which we show to be exactly solvable for a class of mixed initial states, we find exact emergence of the so-defined mixed-state deep thermal ensemble at finite times. Our results therefore lead to fundamental insights into how maximum entropy principles and deep thermalisation manifest themselves in unitary dynamics of states with finite entropy.

Figures (10)

• Figure 1: (a) Schematic showing the construction of ${\cal E}_{\rm ref}[\rho_0]$, defined via Eq. \ref{['eq:Eref-def']}. Note that $\rho_0$ itself is an operator and each black line carries two sets of indices, one from the kets and and other from the bras of $\rho_0$. The operators $U_{\rm Haar}$ and $U_{\rm Haar}^\dagger$ acting on them, respectively, have been folded into the one white box. (b) Trace distance between the $k^{th}$ moments of the $\mathcal{E}_\text{mix}$ and the Haar ensemble plotted against the second Rényi entropy of the initial state for $|A|=2$.
• Figure 2: (a) The kicked Ising chain in Eq. \ref{['eq:UF-kim']} as a circuit with the shaded box denoting $U_F$. The two copies represent $U_t$ and $U_t^\dagger$. The red triangles on subsystem $B$ denote projective measurements in the computational basis whereas the blue triangles on $E$ along with the mixed state on $S$ denotes the initial state in Eq. \ref{['eq:rho0-kim']}. (b) Results for $\Delta_k(t)$ (defined in Eq. \ref{['eq:Delta']}) for $(J,g,h)=(0.8,0.6472,0.7236)$. Different panels correspond to different values of $k$. The initial state is of the form in Eq. \ref{['eq:rho0-kim']} with $|S|=3$. $|A|=2$ is held fixed and different intensities correspond to different $|B|=9,10...17$ with darker colours denoting higher values. The data is averaged over $100$ initial states. (c) Results for $\Delta_k(t)$ at a self-dual point $(J,g,h)=(\pi/4,\pi/9,\pi/4)$, with $|S|=1$ and $\ket{v_j}=\ket{+}$ (see Eq. \ref{['eq:rho0-kim']}) in which case the moments can be computed analytically in the limit of $|B\cap E|\to\infty$.
• Figure 3: Estimated behaviour of the deep-thermalisation timescale $t_k$ with $k$ for the kicked Ising chain. The data show no dependence of $t_k$ on $|S|$ and a slow increase with $k$, consistent with both logarithmic and sub-linear power-law growth. The timescale $t_k$ was estimated by setting a threshold $\epsilon = 3\times10^{-2}$ and identifying the time at which $\Delta_k$ drops below $\epsilon$.
• Figure 4: The PoPs over the projected ensemble for the kicked Ising chain at a late time of $t=1000$ for $L=19$ and $|A|=3$. Different colours correspond to different $|S|=0,1,2,3$ (which also tunes the entropy of the initial state). The histograms correspond to the numerical data whereas the solid lines denote the Erlang distribution (Eq. \ref{['eq:erlang']}) for the corresponding $D_S$. The two panels denote two different bitstrings in $A$.
• Figure 5: The proxy for the second Rényi entropy from the bitstring probabilities (solid line) plotted against time for the kicked Ising model for $L=12$ and $|A|=3$. The dashed line denotes the theoretical value of the second Rényi entropy of the initial state $\rho_0$ which is a random mixed state with its eigenvalues sampled from a uniform Dirichlet distribution and its eigenvectors sampled from the Haar measure on $U(2^L)$.