Entanglement in the quantum Game of Life

We investigate the quantum dynamics of a spin chain that implements a quantum analog of Conway's Game of Life. We solve the time-dependent Schrödinger equation starting with initial separable states and analyze the evolution of quantum correlations across the lattice. We report examples of evolutions leading to all-entangled chains and/or to time oscillating entangling structures and characterize them by means of entanglement and network measures. The quantum patterns result in structures quite different from the classical ones, even in the dynamics of local observables. A peculiar instance is a structure behaving as the quantum analog of a blinker, but that has no classical counterpart.

Figure 1

Time evolution of the initial state $|\psi \left(0\right)\rangle ={|0\rangle }^{\otimes 13}\otimes |101\rangle \otimes {|0\rangle }^{\otimes 16}$ (the quantum blinker) according to (a) the classical $F_{12}$ rule and (b)–(d) the Schrödinger equation, Eq. (1). (a) The alive cells as a function of time according to the classical GOL. (b) and (c) Displayed, respectively, are the expectation value $n_i$ over the quantum state, Eq. (3), and the corresponding discretized population ${\mathcal{D}}_i\left(t\right)$, Eq. (4). (d) The surface plot of the clustering function ${\mathcal{C}}_{\ell }\left(t\right)$, Eq. (6), as a function of time $t$ and cluster size $\ell$. The time evolution is performed over a lattice of $L=32$ sites using TDVP with bond dimension $m=32$ and time step $\mathrm{\Delta }t=0.01$. Note that in the classical dynamics (a) the evolution occurs over a discrete time grid of step $\pi /2$.

Figure 2

Time evolution of the initial state $|\psi \left(0\right)\rangle ={|0\rangle }^{\otimes 4}\otimes |101\rangle \otimes {|0\rangle }^{\otimes 4}$ (the quantum blinker) according to the Schrödinger equation, Eq. (1). (a) The expectation value $n_i$ over the quantum state, Eq. (3), and the corresponding (b) single-site entropy $S_i$. (c)The average concurrence $C\left(d\right)$, Eq. (12). The time evolution is performed over a lattice of $L=11$.

Figure 3

Quantum dynamics of correlations across the lattice with size $L=20$ and for the initial state $|\psi \left(0\right)\rangle ={|0\rangle }^{\otimes 9}\otimes |101\rangle \otimes {|0\rangle }^{\otimes 8}$ (quantum blinker). (a) Central bond entropy $S_{\mathrm{Bond}}(t,\frac{L}{2})$, Eq. (10), as a function of time and (b) bond entropy $S_{\mathrm{Bond}}(t,j)$ as a function of the cell $j$ and evaluated at time 100. The curves are determined using TDVP and bond dimensions $m=32,64,128,256$ (darker colors correspond to higher bond dimension see legends). The data linked by the dashed line correspond to a simulation using ordinary differential equations (ODE). The time step of the evolution is $\mathrm{\Delta }t=0.01$.

Figure 4

Characterization of the dynamics of a quantum blinker using network measures. (a) Density $D\left(t\right)$, Eq. (13), (b) clustering $C\left(t\right)$, Eq. (15), and (c) disparity $Y\left(t\right)$, Eq. (14). The parameters and legends are the same as in Fig. 3.

Figure 5

Dynamics of the interaction between two quantum blinkers. Subplot (a) is the surface plot of the density of each site as a function of time for the dynamics of the double-quantum-blinker (“two quantum blinkers”) state $\left|\psi \right(0)\rangle =|0000\rangle \otimes |101\rangle \otimes |00000\rangle \otimes |101\rangle \otimes |0000\rangle$ and (b) the corresponding discretized population. Subplot (c) gives the evolution of the clustering function for the two quantum blinkers. The size of the lattice is $L=19$. The time step is $\mathrm{\Delta }t=0.01$.

Figure 6

Dynamics of the central bond entropy (a) and equilibrium bond entropy as a function of the lattice sites (b) for one (yellow, light curve, $L=11$) and two (brown, dark curve, $L=19$) quantum blinkers. The bond entropy for a single quantum blinker is shown for comparison.

Figure 7

Characterization of the time evolution of the initial state $|\psi \left(0\right)\rangle ={|0\rangle }^{\otimes 4}\otimes {|1\rangle }^{\otimes 24}\otimes {|0\rangle }^{\otimes 4}$. The subplots and parameters are the same as Fig. 1, except that here for the TVDP simulations the bond dimension is $m=256$.

Figure 8

Quantum dynamics of correlations across the lattice with size $L=16$ (top row) and $L=32$ (bottom row) and for the initial state $|\psi \left(0\right)\rangle ={|0\rangle }^{\otimes 4}\otimes {|1\rangle }^{\otimes 12}\otimes {|0\rangle }^{\otimes 4}$ (top row) and $|\psi \left(0\right)\rangle ={|0\rangle }^{\otimes 4}\otimes {|1\rangle }^{\otimes 24}\otimes {|0\rangle }^{\otimes 4}$ (bottom row). (a)–(c) Central bond entropy $S_{\mathrm{Bond}}(t,\frac{L}{2})$, Eq. (10), as a function of time and (b)–(d) bond entropy $S_{\mathrm{Bond}}(t,j)$ as a function of the cell $j$. The curves are determined using TDVP and bond dimensions $m=32,64,128,256$ (darker colors correspond to higher bond dimension; see legends). The data linked by the dashed line correspond to a simulation using ordinary differential equations. The time step of the evolution is $\mathrm{\Delta }t=0.01$.

Figure 9

Time evolution of the network measures for the initial state $|\psi \left(0\right)\rangle ={|0\rangle }^{\otimes 4}\otimes {|1\rangle }^{\otimes 24}\otimes {|0\rangle }^{\otimes 4}$. The subplots show (a) the density $D\left(t\right)$, Eq. (13), (b) the clustering $C\left(t\right)$, Eq. (15), and (c) the disparity $Y\left(t\right)$, Eq. (14). The parameters and legends are the same as in Fig. 8 for $L=32$.

Figure 10

Characterization of the time evolution of an initial random Fock state with density $\rho \left(0\right)=0.25$. The subplots and parameters are the same as Fig. 1, except that here for the TVDP simulations the bond dimension is $m=64$ and the time step is $\mathrm{\Delta }t=0.1$.

Figure 11

Characterization of the asymptotic dynamics of a random Fock state as a function of the initial density $\rho \left(0\right)$ and for a chain with $L=16$ (top) and $L=32$ sites (bottom). Subplots (a)–(d) display the equilibrium density ${\rho }_{\text{equi.}}$ [Eq. (5)], (b) and (e) display the equilibrium diversity ${\mathrm{\Delta }}_{\text{equi.}}$ [Eq. (8)], (c) and (f) the equilibrium improved diversity ${\mathrm{\Delta }}_{\text{equi.}}^{\text{impr.}}$ [Eq. (9)]. The black symbols indicate the results of the classical dynamics, the colored symbols the results of TDVP simulations with bond dimensions $m=8,16,32,64$ [darker colors correspond to higher bond dimension [see legends in (e)], and time step $\mathrm{\Delta }t=0.1$ [see legends in (b)]. The equilibrium value is extracted by taking the time average over the time interval where the numerical results seem to have reached a stationary value. In the quantum case the interval is [25,30]; for the classical evolution it is [83,100]. (Top row) The orange symbols are the results of the quantum dynamics integrated with a standard ODE algorithm with time step $\mathrm{\Delta }t=0.01$. In the classical simulation, we analyzed all $65536$ different initial configurations. (Bottom row) Each point of the classical simulation is obtained by averaging over about 5000 initial states. In the quantum case, each point is determined by evolving 32 different initial Fock states (we took the same set of states when varying the bond dimension $m$).