Probing Entanglement and Symmetries in Random States Using a Superconducting Quantum Processor

TL;DR

This work experimentally investigates universal features of random many-body quantum states by using a shallow Floquet circuit to generate Haar-like ensembles on a superconducting processor. They measure the Rényi-2 Page curve and entanglement asymmetry, and map the entanglement phase diagram via partially transposed moments, demonstrating alignment with Haar-random predictions for both pure and mixed states. The authors leverage classical-shadow techniques and batch-shadow processing to perform efficient, scalable measurements on modest hardware. The results provide experimental access to the typical entanglement and symmetry properties of chaotic quantum dynamics and set the stage for exploring universal features under conserved quantities and larger system sizes.

Abstract

Quantum many-body systems display an extraordinary degree of complexity, yet many of their features are universal: they depend not on microscopic details, but on a few fundamental physical aspects such as symmetries. A central challenge is to distill these universal characteristics from model-specific ones. Random quantum states sampled from a uniform distribution, the Haar measure, provide a powerful framework for capturing this typicality. Here, we experimentally study the entanglement and symmetries of random many-body quantum states generated by evolving simple product states under ergodic Floquet models. We find excellent agreement with the predictions from the Haar-random state ensemble. First, we measure the Rényi-2 entanglement entropy as a function of the subsystem size, observing the Page curve. Second, we probe the subsystem symmetries using entanglement asymmetry. Finally, we measure the moments of partially transposed reduced density matrices obtained by tracing out part of the system in the generated ensembles, thereby revealing distinct entanglement phases. Our results offer an experimental perspective on the typical entanglement and symmetries of many-body quantum systems.

Figures (4)

• Figure 1: Experimental protocol for generating and characterizing quantum random states.a, Experimental sequence. Floquet driving, Eq. \ref{['eq:floquet_operator']}, is applied for $\tau$ cycles to generate random states. Before performing projective measurements, random single-qubit gates $\{U_{l}\}$, drawn from CUE(2), are applied to each qubit. The measured bitstrings, together with the applied random gates, provide classical shadows of the density matrix. Properties of subsystem $A$ can then be estimated from these classical shadows obtained for various sets of random single qubit gates (see Methods). b, Visualization of the Floquet dynamics. Quantum states (blue arrows) gradually scramble with the number of Floquet cycles $\tau$, yielding an approximately uniform distribution over the Hilbert space (gray sphere). c, Average fidelity between the generated random states as a function of $\tau$. As the number of Floquet cycles increases, the average fidelity converges toward the prediction for the Haar-random ensemble $2^{-L}$ (dash-dotted lines). Dashed lines: noiseless numerical simulations. Dots and error bars: measured values.
• Figure 2: Experimental Page curves of entanglement entropy and entanglement asymmetry. Measurements of the average Rényi-2 entanglement entropy $\mathbb{E}[S_A^{(2)}]$ ( a) and entanglement asymmetry $\mathbb{E}[\Delta S_A^{(2)}]$ ( b) as functions of the subsystem size $L_A$ are shown for different total system sizes $L$. The dashed curves are the theoretical prediction for Haar-random states for the corresponding $L$, while the gray dashed curves indicate the thermodynamic limit $L\to\infty$. In the insets, the symbols are the measured values for system size $L=11$. We observe deviations from the theoretical expectations (dashed curves) due to decoherence during the Floquet evolution that generates the random states, impacting the unitarity of the evolution. These errors can be mitigated as explained in the main text and SM. In the main panels, the symbols are the error-corrected results, which perfectly follow prediction for the Haar-random ensemble. Error bars denote the standard error of the mean (SEM) over different states (see SM Section S3E for the error bars due to finite number of measurement bases). We average over $15$ random states for $L = 5,7$ and $10$ random states for $L= 9,11$.
• Figure 3: Entanglement phase diagram from partial transpose moments.a, Schematic illustration of partitioning the whole qubit chain into three subsystems: $A$ (blue), $B$ (green), and $C$ (yellow). Subsystem $C$ is taken as the environment, and the partial transpose with respect to $B$ is computed for the reduced density matrix of $AB$. b, Theoretical phase diagram expected for the Haar-random state ensemble using the ratio $\tilde{r}_2$, defined in Eq. \ref{['eq:r2']}, in the thermodynamic limit $L\to\infty$. The asymptotic values are $\tilde{r}_2 = 1$ for the ME and PPT phases, and $\tilde{r}_2 = 3/2$ for the ES phase. c, Experimental phase diagram of the ratio $\tilde{r}_2$ for a finite system size $L = 9$. The top and bottom panels show the error-mitigated and raw results, respectively. White crosses indicate measured data points, used to draw the colored contour plot. d, Horizontal (top) and vertical (bottom) slices of the phase diagram. The horizontal slice corresponds to $L_C / L \approx 0.22$, and the vertical slice is taken at $L_A / L_{AB} = 0.5$. Circles and squares represent error-mitigated and raw data, respectively. Dashed lines indicate theoretical predictions for the Haar-random ensemble of $L = 9$. Error bars are obtained via jackknife resampling (see SM Section S3E).
• Figure 4: Dynamics of Rényi-2 entanglement entropy (a) and entanglement asymmetry (b) in the Floquet circuit. Each experimental data point is the error-corrected average over 10 random realizations of the circuit, for various sizes $L_A$ of subsystem $A$ and total system size $L=7$. The error bars represent the SEM. Dashed curves were obtained from the noiseless numerical simulation of the Floquet circuit.