Entanglement Negativity and Mutual Information after a Quantum Quench: Exact Link from Space-Time Duality

We study the growth of entanglement between two adjacent regions in a tripartite, one-dimensional many-body system after a quantum quench. Combining a replica trick with a space-time duality transformation, we derive an exact, universal relation between the entanglement negativity and Rényi-$1/2$ mutual information that holds at times shorter than the sizes of all subsystems. Our proof is directly applicable to any local quantum circuit, i.e., any lattice system in discrete time characterized by local interactions, irrespective of the nature of its dynamics. Our derivation indicates that such a relation can be directly extended to any system where information spreads with a finite maximal velocity.

Figure 1

Illustration of the time evolution of a product state in a brick-work quantum circuit. Each time step consists of the application of two-site gates, first to odd and then to even pairs of consecutive sites, so that at time $t$ the state is given by $|{\mathrm{\Psi }}_t⟩={\mathbb{U}}^t|{\mathrm{\Psi }}_0⟩$ [cf. Eq. (1)]. The boundary conditions are assumed to be “periodic.”

Figure 2

Folded representation of $\rho (t{)}_{AB}^{t_A}$. To obtain this diagram, we start with a copy of the state $|{\mathrm{\Psi }}_t⟩$ (green, cf. Fig. 1), and its Hermitian conjugate $⟨{\mathrm{\Psi }}_t|$ (red), which is flipped and positioned behind $|{\mathrm{\Psi }}_t⟩$, i.e., the diagram is “folded.” The degrees of freedom in the subsystem $C$ are then traced out, which is represented by connecting the legs of the two copies. Finally, we perform the partial transpose $(·{)}_A^t$ by exchanging the output legs of the two copies in the subsystem $A$.

Figure 3

Schematic illustration of $\exp [{\mathcal{E}}_{2n}(t)]=\mathrm{tr}[(\rho (t{)}_{AB}^{t_A}{)}^{2n}]$. In the left panel the green and red rectangles represent the time-evolved state and its conjugate respectively (i.e., the objects in Fig. 2), and the connections among different copies are depicted in white. For clarity we introduce a simplified representation in the middle panel, where each copy is now represented by a row, and each column corresponds to one of the subsystems (rightmost and leftmost columns coincide due to periodic boundary conditions). Connections are represented by thick black lines, while the thinner horizontal lines connect different subsystems within the same copy. In the right panel we reshuffled the positions of different copies so that the connections always occur between nearest neighbors. This is achieved by performing a permutation of copies at each interface between different subsystems [cf. Eq. (7)].