Non-Abelian symmetry can increase entanglement entropy

The pillars of quantum theory include entanglement and operators' failure to commute. The Page curve quantifies the bipartite entanglement of a many-body system in a random pure state. This entanglement is known to decrease if one constrains extensive observables that commute with each other (Abelian “charges”). Non-Abelian charges, which fail to commute with each other, are of current interest in quantum thermodynamics. For example, noncommuting charges were shown to reduce entropy-production rates and may enhance finite-size deviations from eigenstate thermalization. Bridging quantum thermodynamics to many-body physics, we quantify the effects of charges' noncommutation—of a symmetry's non-Abelian nature—on Page curves. First, we construct two models that are closely analogous but differ in whether their charges commute. We show analytically and numerically that the noncommuting-charge case has more entanglement. Hence charges' noncommutation can promote entanglement.

Figure 1

Analogous noncommuting-charge and commuting-charge models. Each model consists of $N$ sites. A site consists of two qubits, $a$ and $b$. The local noncommuting observables of interest include $Q_1$; and the local commuting observables, $C_1$.

Figure 2

Page curves constructed from microcanonical subspaces. ${\langle S_{\mathrm{E}}\rangle }_{\mathcal{S}}$ denotes any Page curve restricted by charges; and ${\langle S_E\rangle }_{\mathcal{H}}$, the unrestricted Page curve. The red x's form the noncommuting-charge model's Page curve, and the circular blue markers form the commuting-charge model's Page curve. The connecting lines guide the eye. We calculated the top panel's ($N=4$) Page curves from ${10}^4$ samples each and the bottom panel's ($N=8$) Page curves from ${10}^3$ samples each. The $x$ axis ends at $N_A=N/2$ for conciseness; the Page curves are symmetric according to numerics.

Figure 3

Testing the additivity ansatz. ${\langle S_{\mathrm{E}}\rangle }_{\mathcal{S}}$ denote any Page curve restricted by charges; and ${\langle S_E\rangle }_{\mathcal{H}}$, the unrestricted Page curve. The red x's form the noncommuting-charge model's Page curve, and the circular blue markers form the commuting-charge model's Page curve. Both curves were calculated using microcanonical subspaces. The gray triangles illustrate the additivity ansatz.