Mixed-State Entanglement in a Minimal Model of Quantum Chaos

Abstract

Understanding the dynamics of quantum correlations in many-body systems is a central problem in non-equilibrium quantum physics. We study the spread of mixed-state entanglement in a minimal model of quantum chaos, the kicked field Ising model. By combining the replica trick with the space-time duality of the model, we determine the exact spectrum of the partially transposed reduced density matrix. The resulting flat spectrum leads to exact relations between entanglement negativity, odd entropy and Rényi mutual information at early times. Numerical results further demonstrate that for equal tri-partitions and at late times, all entanglement measures saturate to the Haar-random values. In contrast, for unequal tri-partitions Rényi mutual information and negativity vanish at late times, implying that the corresponding reduced density matrix is factorizable. Extensive numerical simulations also show that the relation remains quantitatively valid for generic initial states, leading us to conjecture it for all initial states and all times.

Figures (3)

• Figure 1: Rényi-1/2 mutual information ($I_{A:B}^{(1/2)}(t)$), negativity ($2\mathcal{E}(t)$) and odd entropy($\frac{2}{3}\mathcal{E}^{o}(t)$) for initial solvable state. All the systems have total size $L=21$ and subsystem size $L_{A}=L_{B}=L_{C}=7$. We choose the parameters corresponding to: (a) $\mathcal{T}$ class: $\theta=\pi/2$, $\phi=0$,$h_{i}=1$. (b) $\mathcal{L}$ class: $\theta=0$,$\phi=0$,$h_{i}=1$. (c) Integrable: $\theta=\pi/2,\phi=0,h_{i}=0$ and (d) Weak integrability breaking: $\theta=\pi/2,\phi=0,h_{i}=0.1.$ The two horizontal black dashed lines denote the Haar random values for negativity (lower) and mutual information (upper).
• Figure 2: Rényi-1/2 mutual information ($I_{A:B}^{(1/2)}(t)$), negativity ($2\mathcal{E}(t)$) and odd entropy($\frac{2}{3}\mathcal{E}^{o}(t)$) for generic intial state. All the systems have total size $L=21$, subsystem size $L_{A}=L_{B}=L_{C}=7$ and $h_{i}=1$. We choose the parameters corresponding to: (a) $\theta=1,\phi=1$. (b) $\theta=2,\phi=2$ (c) $\theta=2.5,\phi=1$(d) $\theta=2.5,\phi=3$. The two horizontal black dashed lines denote the Haar random value for negativity (lower) and mutual information (upper).
• Figure 3: Rényi-$1/2$ mutual information ($I_{A:B}^{(1/2)}(t)$), negativity ($2\mathcal{E}(t)$) and odd entropy($\frac{2}{3}\mathcal{E}^{o}(t)$) for initial state corresponding to $\mathcal{T}$class in (a) and (b) and generic state with $\theta=\phi=1$ in (c) and (d). All the systems have total size $L=30$. The subsystem sizes are (a) $L_{A}=L_{B}=6$, (b) $L_{A}=L_{B}=7$, (c) $L_{A}=L_{B}=6$, (d) $L_{A}=L_{B}=7$, and $L_{C}=L-(L_{A}+L_{B})$. The horizontal black dashed lines denotes the value $\frac{2}{3}S_{AB}$.

Theorems & Definitions (3)

• Lemma 1
• Theorem 1
• Conjecture 1