Disentangling strategies and entanglement transitions in unitary circuit games with matchgates

Abstract

In unitary circuit games, two competing parties, an "entangler" and a "disentangler", can induce an entanglement phase transition in a quantum many-body system. The transition occurs at a certain rate at which the disentangler acts. We analyze such games within the context of matchgate dynamics, which equivalently corresponds to evolutions of non-interacting fermions. We first investigate general entanglement properties of fermionic Gaussian states (FGS). We introduce a representation of FGS using a minimal matchgate circuit capable of preparing the state and derive an algorithm based on a generalized Yang-Baxter relation for updating this representation as unitary operations are applied. This representation enables us to define a natural disentangling procedure that reduces the number of gates in the circuit, thereby decreasing the entanglement contained in the system. We then explore different strategies to disentangle the systems and study the unitary circuit game in two different scenarios: with braiding gates, i.e., the intersection of Clifford gates and matchgates, and with generic matchgates. For each model, we observe qualitatively different entanglement transitions, which we characterize both numerically and analytically.

Figures (13)

• Figure 1: (a) Illustration of the unitary circuit game: Blue boxes represent random matchgates and red boxes are unitary gates chosen to disentangle the bond. (b) Right standard form (RSF) of fermionic Gaussian states: any pure FGS can be expressed as a matchgate circuit with this form. (c) Example of entangling and disentangling operations within the RSF formalism of FGS.
• Figure 2: Example of the application of the (a) absorption algorithm, where the dark blue gate is absorbed into the RSF, and (b) disentangling algorithm, where the light gray gate is removed from the RSF, by applying a gate (depicted in red) in the second bond. The numbers above the arrows indicate which step of the algorithm is applied (see main text).
• Figure 3: Numerical results of the braiding gate model. (a) Averaged half-chain entanglement entropy normalized by the system size in the steady state of the unitary game as a function of the disentangling probability $p$. The results indicate an area law phase for every nonzero probability. The inset shows the data collapse for critical exponent $\nu=1$. (b) Evolution of the averaged half-chain entanglement entropy $\overline{S_{L/2}}$ as a function of time $t$ with a random braiding evolution (left panel) followed by a disentangling evolution (right panel), for different system sizes $L$. The time axis is normalized by $L^2$ and by $L$ in the left and right panels respectively to indicate the convergence time. The inset of the left panel shows the diffusive spread of the entanglement entropy in the $p=0$ case, with $\overline{S_{L/2}}\propto \sqrt{t}$.
• Figure 4: Numerical results for the unitary game with von Neumann disentangler. (a) Averaged half-chain von Neumann entanglement entropy normalized by the system size in the steady state of the unitary game as a function of the disentangling probability $p$. We find a decay with system size for any nonzero disentangling probability, but with the system sizes achieved we do not observe convergence. (b) Evolution of the averaged half-chain von Neumann entropy $\overline{S_{L/2}^{(1)}}$ as a function of time $t$ with a random matchgate evolution (left panel) followed by a disentangling evolution with von Neumann disentangler (right panel), for different system sizes $L$. The time axis is normalized by $L^2$ and by $L$ in the left and right panels respectively. The inset of the left panel shows the diffusive spread of the entanglement entropy in the $p=0$ case, with $\overline{S_{L/2}^{(1)}}\propto \sqrt{t}$.
• Figure 5: Entangling (left to right) and disentangling (right to left) rules for the Bell pair model. In rules 5 and 6, the entangler increases the distance of the longest Bell pair at the expense of shortening the other pair, while the disentangler does the opposite. If none of the rules apply, the state is not modified.

Theorems & Definitions (11)

• Definition 1: Right standard form (RSF)
• Lemma 1
• proof
• Corollary 1
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• Corollary 2