Entanglement transition in the projective transverse field Ising model

Discrete quantum trajectories of systems under random unitary gates and projective measurements have been shown to feature transitions in the entanglement scaling that are not encoded in the density matrix. In this paper, we study the projective transverse field Ising model, a stochastic model with two noncommuting projective measurements and no unitary dynamics. We numerically demonstrate that their competition drives an entanglement transition between two distinct steady states that both exhibit area law entanglement, and introduce a classical but nonlocal model that captures the entanglement dynamics completely. Exploiting a map to bond percolation, we argue that the critical system in one dimension is described by a conformal field theory, and derive the universal scaling of the entanglement entropy and the critical exponent for the scaling of the mutual information of two spins exactly. We conclude with an interpretation of the entanglement transition in the context of quantum error correction.

Figure 1

Projective time evolution. A few typical time steps for $p\approx 0.5$ on a chain of $L=10$ spins with periodic boundaries. Blue boxes on edges denote measurements ${\mathcal{M}}_e^{zz}$ on adjacent spins $e=(i,i+1)$ and red circles measurements ${\mathcal{M}}_i^x$ on a single spin $i$. Each time step comprises one row of ${\mathcal{M}}_e^{zz}$ measurements followed by a row of ${\mathcal{M}}_i^x$ measurements. Note that the order of ${\mathcal{M}}_e^{zz}$ measurements does not affect the dynamics as their projectors commute. The system is initialized in the product state $\left|+\cdots +\right.〉$.

Figure 2

Colored cluster model. Single time step of the colored cluster model, split into two substeps: (1) the application of ${\mathrm{\Pi }}_{\mathit{z}}^{zz}$ on edges (crosses on faces in space-time) that merges/grows/nucleates clusters and (2) ${\mathrm{\Pi }}_{\mathit{x}}^x$ on sites (crosses on vertical edges in space-time) that erodes clusters. Sites with dashed boundary represent the intermediate state between $t$ and $t+1$ and gray sites denote single-site clusters.

Figure 3

Entanglement transition. Numerical results for the entanglement entropy $S_L(L/2)$ (red) and the mutual information $I(1,1+L/2)$ (blue) as functions of the rate $p$ for systems of length $L=100,300,500$ (circles, squares, diamonds) with periodic boundary conditions (PBC). At the critical point $p=0.5=p_c$, the entanglement entropy grows logarithmically with the system size (black markers). Each point is based on ${10}^5$ sampled trajectories.

Figure 4

Entanglement structure. Various quantities characterizing the entanglement structure of a chain with length $L=500$ above, at, and below the critical point; all results are based on ${10}^6$ sampled trajectories: (a) entanglement entropy $S_L\left(l\right)$ as a function of the length $l/L$ of the subsystem for a chain with periodic boundary conditions. The system obeys an area law in both phases, $p=0.45<p_c$ and $p=0.525>p_c$, with a logarithmic contribution at the critical point $p=0.5=p_c$. (b) Mutual information $I(i,j)=I\left(\right|i-j\left|\right)$ as a function of the distance $|i-j|/L$ for a chain with periodic boundary conditions. $I\left(\right|i-j\left|\right)$ vanishes exponentially for $p>p_c$ and saturates at a finite value for $p<p_c$. At the critical point $p=p_c$, it is well described by the algebraic decay $I\left(\right|i-j\left|\right)={\alpha |i-j|}^{-\kappa }$ with fitted exponent $\kappa \approx 0.66$ and nonuniversal fit parameter $\alpha$ (dashed black line). (c) Probability density $W\left(n\right)\times L$ of the relative weight $n/L$ of Bell clusters for a chain with open boundary conditions. At the entanglement transition, clusters of extensive weight emerge. (d) Probability density $D\left(n\right)\times L$ of the relative diameter $n/L$ of Bell clusters for a chain with open boundary conditions. At the entanglement transition, clusters with diameters on all length scales exist.

Figure 5

Critical scaling. Entanglement entropy $\mathrm{\Delta }S(l/L)$ as a function of the subsystem size $l/L$ for chains of different length $L$ with periodic boundary conditions at the critical point $p=0.5=p_c$. The data collapse is almost perfect and the prediction for critical systems of the form $\mathrm{\Delta }S(l/L)=\tilde{c}/3{log}_2\xi (l/L)$ with $\xi \left(x\right)=sin\left(\pi x\right)$ describes the dependency remarkably well for the fit parameter $\tilde{c}\approx 0.57$ (dashed black line). Each curve is based on ${10}^6$ sampled trajectories.

Figure 6

Percolation. (a) The same measurement patter as in Fig. 1. Here we mark horizontal edges with ${\mathcal{M}}_e^{zz}$ measurements and vertical edges without ${\mathcal{M}}_i^x$ measurements as active (bold black lines). Then, the probability for both horizontal and vertical edges to be active is $1-p$ and the projective dynamics gives rise to isotropic bond percolation on a square lattice in space-time. (b) A possible history for the CCM state in Fig. 2. Sites at time $t$ have the same color if and only if they are connected via active edges in space-time.

Figure 7

Conformal field theory. (a),(b) Field configurations on the half-infinite cylinder mapped to the complex plane. Vertex operator insertions are indicated by $\otimes$. (a) Scaling of the entanglement entropy $S\left(A\right)$. Field configurations with contours that connect $A$ (red segment) with the environment (black boundary) contribute to the entanglement between them (light red domains). (b) Scaling of the mutual information $I(x_1,x_2)$. Field configurations that connect the points $x_1$ and $x_2$ (light blue domain) contribute to the mutual information between them. (c) Comparison of numerical results (squares and bullets) and theoretical predictions (solid lines) for the PTIM at criticality. The simulations are based on ${10}^7$ trajectories for a system of length $L=500$ with periodic boundary conditions. The CFT predictions are of the form $\mathrm{\Delta }S(l/L)=\tilde{c}/3{log}_2\xi (l/L)+\alpha$ and $I(i,j)=I\left(\right|i-j\left|\right)=\beta {\xi }^{-\kappa }\left(\right|i-j\left|\right)+\gamma$ with $\tilde{c}$ and $\kappa$ as given in the text and $\alpha$, $\beta$, $\gamma$ nonuniversal fit parameters; $\xi \left(x\right)=sin\left(\pi x\right)$ accounts for the finite size and the periodic boundaries. Note that the plot for $\mathrm{\Delta }S$ is logarithmic on the $l/L$ axis; the plot for $I$ is logarithmic on both axes.

Figure 8

Quantum error correction. (a),(b) Time evolution with random stabilizer measurements $S_e$ between adjacent qubits (horizontal bars) and random errors $E_i$ on qubits (circles); time runs upwards. The system is initialized in the code space with a global Bell cluster (bold black vertical lines). (a) The initial cluster survives and the amplitudes are preserved. (b) Adding a few additional errors (black disks) makes the black cluster die out, losing all information about the encoded qubit. The new, independent cluster (red) is trivial and does not carry quantum information. (c) Average cluster lifetime $\tau$ of a global Bell cluster as function of system size $L$ for $p\in \{0.45,0.50,0.60\}$. For $p=0.45<p_c$, $\tau$ grows exponentially so that the amplitudes of the initial cluster are retained almost indefinitely. At criticality $p=0.5=p_c$, $\tau$ grows algebraically with $\tau \sim L^{\beta }$ and $\beta \approx 1$. For $p=0.6>p_c$, the growth is only logarithmic. The solid lines are fits of the form $\alpha e^{\beta L}$ (blue), $\alpha L^{\beta }+\gamma$ (red), and $\alpha log\left(L\right)+\beta$ (yellow). Each point is based on ${10}^5$ sampled trajectories on a chain with open boundary conditions and a cutoff simulation time $t_{\max }=5\times {10}^4$.