Probing entanglement entropy via randomized measurements (1806.05747v2)

Abstract: Entanglement is the key feature of many-body quantum systems, and the development of new tools to probe it in the laboratory is an outstanding challenge. Measuring the entropy of different partitions of a quantum system provides a way to probe its entanglement structure. Here, we present and experimentally demonstrate a new protocol for measuring entropy, based on statistical correlations between randomized measurements. Our experiments, carried out with a trapped-ion quantum simulator, prove the overall coherent character of the system dynamics and reveal the growth of entanglement between its parts - both in the absence and presence of disorder. Our protocol represents a universal tool for probing and characterizing engineered quantum systems in the laboratory, applicable to arbitrary quantum states of up to several tens of qubits.

• The paper demonstrates a novel protocol that measures second-order Rényi entanglement entropy via statistical correlations from randomized measurements.
• It employs single-qubit unitaries on trapped-ion systems with up to 20 qubits, significantly reducing experimental complexity.
• Simulations confirm the method’s scalability and its ability to capture entanglement dynamics, including transitions between thermalization and localization.

Probing Entanglement Entropy via Randomized Measurements

The paper presented in the paper, "Probing entanglement entropy via randomized measurements," introduces a novel protocol for measuring the entanglement entropy within quantum systems. Specifically, it leverages statistical correlations derived from randomized measurements to experimentally assess the second-order Rényi entropy $S^{(2)}$. This protocol eschews the necessity for two identical quantum system copies, a requirement in previous methodologies that relied on collective measurements, thus streamlining the experimental setup significantly.

Overview of the Protocol

Randomized measurements involve applying a product of independent single-qubit unitaries sampled from the circular unitary ensemble (CUE) and measuring in a fixed logical basis. These measurements are repeated for different unitary instances to build the necessary statistics. The second-order Rényi entropy $S^{(2)}(\rho_A)$ is then derived from ensemble-averaged cross-correlations of excitation probabilities across different random unitary ensembles.

Experimental Implementation

The experiments were conducted utilizing a trapped-ion quantum simulator, enabling manipulation of up to 20 qubits. Here, the focus was on quantifying the entanglement entropies of various subsystem partitions, observing how entropies evolved over time under different conditions—both with and without disorder. The paper reports that the method extends to engineered arbitrary quantum states involving tens of qubits, allowing for entanglement probing beyond current capabilities using alternative methodologies.

Key Findings and Implications

The findings from this research are multifaceted:

1. Entanglement Dynamics: Using the proposed randomized measurement protocol, the growth of entanglement was successfully monitored. It was observed that the proposed methodology could characterize the dynamical behavior of quantum systems faithfully, which is critical for applications in quantum simulation and computation.
2. Scalability: The protocol's advantage is further emphasized by its scalability and applicability to mixed or highly entangled quantum states, without imposing structural assumptions on the quantum states. This strength highlights its potential utility in ongoing research into quantum many-body systems.
3. Numerical Simulations: Simulations corroborated experimental results, indicating the protocol's reliability, particularly its potential to capture the entropy growth associated with many-body localization (MBL). This includes capturing the transition between thermalization and localization in quantum systems, an area of significant interest in condensed matter physics.
4. Practical Utility: By reducing the experimental complexity without a need for identical quantum copies, this protocol enhances practical utility in quantum computing and related fields.

Theoretical and Practical Implications

Theoretically, this approach advances understanding of entropy dynamics—essential for distinguishing thermalization from localization phenomena in interacting quantum systems. Practically, the protocol can be a pivotal tool for experimentalists in configuring and validating large-scale quantum systems.

Future Directions

Future research directions may focus on extending the capability of this protocol for even larger qubit systems, potentially integrating with superconducting qubit setups, which offer high state generation rates. Exploring its applicability to decoherence-prone environments and enhancing algorithmic complexity could also pave the way for developing robust quantum technologies.

In summary, this paper contributes an efficacious approach to measuring entanglement entropy, which will bolster both theoretical investigations and practical applications in quantum mechanics, further solidifying the capability of quantum systems as valid contenders in performing tasks considered intractable for classical systems.