Symmetry-protected entanglement in random mixed states

Symmetry is an important property of quantum mechanical systems which may dramatically influence their behavior in and out of equilibrium. In this paper, we study the effect of symmetry on tripartite entanglement properties of typical states in symmetric sectors of Hilbert space. In particular, we consider Abelian symmetries and derive an explicit expression for the logarithmic entanglement negativity of systems with ${\mathbb{Z}}_N$ and $\text{U}\left(1\right)$ symmetry groups. To this end, we develop a diagrammatic method to incorporate partial transpose within the random matrix theory of symmetric states and formulate a perturbation theory in the inverse of the Hilbert space dimension. We further present entanglement phase diagrams as the subsystem sizes are varied and show that there are qualitative differences between systems with and without symmetries. We also design a quantum circuit to simulate our setup.

Figure 1

Tripartite geometry of random pure states. In this paper, we are interested in the entanglement negativity between two parties (e.g., $A_1$ and $A_2$) out of the three. There is no notion of locality in this setup.

Figure 2

Entanglement phase diagram of symmetry-projected random mixed states for $\text{U}\left(1\right)$ symmetric systems. The mixed state is obtained from a random pure state via partial tracing. Different panels correspond to different subsystem filling factors as indicated. Dashed lines [given by $N_{A}f\left({\nu }_A\right)=N_{B}f\left({\nu }_B\right)$] separate the upper part of the phase diagram, where the volume-law term in the negativity is suppressed (but not fully PPT), from the lower part where the negativity obeys a volume-law form. The lower region, $N_{A}f\left({\nu }_A\right)>N_{B}f\left({\nu }_B\right)$, consists of two phases: First, the two corners are called maximal entanglement regimes where $\langle {\mathcal{E}}_{A_1:A_2}\rangle =min(N_{A_1},N_{A_2})f\left({\nu }_A\right)$; second, the middle region, where the dominant diagrams break the replica symmetry leading to $\langle {\mathcal{E}}_{A_1:A_2}\rangle =N_{A}f\left({\nu }_A\right)-N_{B}f\left({\nu }_B\right)$ (to the leading order). The shaded regions represent the critical phase between the replica symmetry breaking and maximal entanglement phases where the entanglement negativity spectrum diverges at zero. Red curves are shown as a reference for critical line of nonsymmetric states.

Figure 3

Entanglement negativity spectrum of projected density matrices ${\widehat{\rho }}_A^{\left(q_A\right)}$ for (a) ${\mathbb{Z}}_2$ symmetric qubit systems $R=2$, (b) ${\mathbb{Z}}_3$ symmetric qutrit systems $R=3$, and (c) ${\mathbb{Z}}_4$ symmetric ququart systems $R=4$, in the semicircle regime. Solid lines are the random matrix theory result given in Eq. (4.19). Numerical data are represented by colored circles. The agreement between theory and numerics is evident. Here, $N_{A_1}=N_{A_2}=3$ and ensemble average is performed over ${10}^4$ samples.

Figure 4

Entanglement negativity spectrum of ${\widehat{\rho }}_A^{\left(q_A\right)}$ for a $\text{U}\left(1\right)$ symmetric system in the replica symmetry-breaking regime. Here, $N_{A_1}=N_{A_2}=5$ and $N_B=12$. Total particle number is $q_A+q_B=11$ and projected sector is labeled by $q_A$. Solid lines are random matrix theory results given by Eq. (4.23) which are in good agreement with the exact numerical simulations (colored circles).

Figure 5

Critical exponents for the negativity spectrum at the transition point in ${\mathbb{Z}}_3$ symmetric states. For a better numerical accuracy, we calculate the cumulative distribution function instead of the spectral density. $N_{A_1}=2,N_{A_2}=6$, and $N_B=5$. Inset shows the spectral density in linear scale.

Figure 6

Evolution of the negativity spectrum as the size of subsystem $B$ is increased for $\text{U}\left(1\right)$ symmetric systems. This trend corresponds to sweeping a vertical path from bottom to top in the phase diagram as shown in the first panel. The colored circles in each panel are numerical simulations (averaged over ${10}^4$ samples) and solid lines correspond to the numerical solutions to Eqs. (4.53) and (4.46) from random matrix theory. Here, we set $N_{A_2}=3$ and $N_{A_1}=6$ and the filling fractions are ${\nu }_A=\frac{1}{3}$ and ${\nu }_B=\frac{1}{2}$. Note that the points within the critical region (shaded area) of the phase diagram are characterized by a diverging spectral density at the origin [(a)–(c)].

Figure 7

Quantum circuit to measure the ${\mathbb{Z}}_2$ charge of subsystem $B$.

Figure 8

The circuit that is used to measure the charge of the $B$ subsystem modulo $2^n$.