Rényi Entropies from Random Quenches in Atomic Hubbard and Spin Models

We present a scheme for measuring Rényi entropies in generic atomic Hubbard and spin models using single copies of a quantum state and for partitions in arbitrary spatial dimensions. Our approach is based on the generation of random unitaries from random quenches, implemented using engineered time-dependent disorder potentials, and standard projective measurements, as realized by quantum gas microscopes. By analyzing the properties of the generated unitaries and the role of statistical errors, with respect to the size of the partition, we show that the protocol can be realized in existing quantum simulators and used to measure, for instance, area law scaling of entanglement in two-dimensional spin models or the entanglement growth in many-body localized systems.

Figure 1

Measuring Rényi entropies via random quenches. (a) Experimental sequence: For a given reduced density matrix ${\rho }_A={\mathrm{Tr}}_{\mathcal{S}\backslash A}[\rho ]$, we apply (i) a random unitary $U_A$ realized by a series of $\eta$ random quenches [cf. Eq. (3)], implemented using (spin-dependent) disorder potentials [cf. Eq. (4)]; this is followed by (ii) a projective measurement (readout) with a quantum gas microscope, to obtain $S^{(n)}({\rho }_A)$ from Eq. (6). (b) Within our protocol, we illustrate for the ground state of a 2D Heisenberg model ($8\times 8$ sites) area law scaling of $S^{(2)}\propto \partial A$ (with $\partial A$ the perimeter of area $A$), showing convergence with increasing $\eta$ to the exact value (black line). (c) For the many-body localized phase of the 1D Bose-Hubbard model (ten sites and five particles), we illustrate a measurement of the logarithmic growth of $S^{(2)}({\rho }_A)$ at half partition as a function of time. The exact value of $S^{(2)}({\rho }_A)$ (solid lines) is compared to the estimated values (dots). The dashed lines are linear fits. The simulated experiments in (b) and (c) assume $N_U=100$ random unitaries and $N_M=100$ measurements per random unitary (see the text).

Figure 2

Creation of approximate 2-designs in the Heisenberg model. (a) Average error of the estimated purity $|(p_2{)}_e-(p_2)|$ for a unidimensional partition of size $L=8$ and various test states: an antiferromagnetic state $|{\psi }_{\mathrm{AFM}}⟩$, the phase separated state $|{\psi }_{\mathrm{PS}}⟩=\prod _{\mathbf{i},i_x\le L_x/2}|\downarrow {⟩}_{\mathbf{i}}\prod _{\mathbf{i},i_x>L_x/2}|\uparrow {⟩}_{\mathbf{i}}$, a pure random state $|{\psi }_{\mathrm{rand}}⟩$ with $S_z=0$, and the mixed state ${\rho }_A=\frac{1}{2}(|{\psi }_{\mathrm{AFM}}⟩⟨{\psi }_{\mathrm{AFM}}|+|{\psi }_{\mathrm{PS}}⟩⟨{\psi }_{\mathrm{PS}}|)$. (b),(c) Error for ${\rho }_A=|{\psi }_{\mathrm{AFM}}⟩⟨{\psi }_{\mathrm{AFM}}|$ for (b) unidimensional partitions ($L=L_x$) and (c) two-dimensional partitions ($L=L_x{L}_y$). (d) Optimization of the quench time $JT$ for fixed total time $T_{\mathrm{tot}}$ and disorder strength $\delta =J$. For all panels, we average over $N_U=500$ unitaries and consider $N_M=\infty$.

Figure 3

Scaling of statistical errors. (a) Average statistical error of the estimated purity as a function of $N_U$ for various $N_M$, ${\mathcal{N}}_A=256$. (b) Error as a function of $N_M$, for different ${\mathcal{N}}_A$, showing birthday paradox scaling $N_M/\sqrt{{\mathcal{N}}_A}$. Circles represent $N_U=100$ and triangles $N_U=1000$. The unitaries are sampled from the CUE numerically [42]. The black lines represent the expressions given in the text and Ref. [31].

Figure 4

Protocol with local unitaries. Purity of all (sub)systems $A^{\prime }\subseteq A$ with $N_U=2N_M=100$. The numbers refer to the indices $i=1,\dots ,L$ contained in $A^{\prime }$, the green bar to $A^{\prime }=A$. The black lines indicate the exact values.