Measuring Rényi entropy using a projected Loschmidt echo

TL;DR

We establish a direct link between the second Rényi entropy $S^{(2)}=-\log \mathrm{Tr}(\hat{\rho}_A^2)$ and a projected Loschmidt-echo (LE) protocol, showing that the purity $\mathrm{Tr}_A[\hat{\rho}_A^2(t)]$ can be written as a sum over projected LE terms $M(t,m_1,m_2)$. This yields $S^{(2)}=-\log \sum_{m_1,m_2} M(t,m_1,m_2)$ and provides practical, LE-based measurement schemes that require fewer resources than previous two-copy approaches. The authors present efficient LE measurement protocols, including a simplified version using a single ancilla $B$, and a random-unitary variant that obviates full basis preparation by Haar randomness; they also derive an OTOC–projected LE relation without random-noise averaging via a diagrammatic approach. Applications to superconducting qubits and cavity QED holographic Hamiltonians demonstrate experimental feasibility, and the work discusses time-reversal challenges, imperfections, and connections to randomized measurements as alternatives. Overall, the paper closes the gap between entanglement quantification, scrambling diagnostics, and experimentally accessible protocols, with potential impact on quantum simulators and holographic models.

Abstract

We present efficient and practical protocols to measure the second Rényi entropy (RE), whose exponential is known as the purity. We achieve this by establishing a direct connection to a Loschmidt echo (LE) type measurement sequence, applicable to quantum many-body systems. Notably, our approach does not rely on random-noise averaging, a feature that can be extended to protocols to measure out-of-time-order correlation functions (OTOCs), as we demonstrate. By way of example, we show that our protocols can be practically implemented in superconducting qubit-based platforms, as well as in cavity-QED trapped ultra-cold gases.

Figures (16)

• Figure 1: The protocol for measuring the Loschmidt echo defined in Eq. \ref{['Echo_closed_def']}. Note that this protocol involves a time-inversion step, which we will come back to further below.
• Figure 2: The protocol for each round of the measurement of projected LE $M(t,m_1,m_2)$ defined in Eq. \ref{['LE_projeted_def']}. We start with the initial state $|m_1,\psi_0,B_0\rangle$ and let subsystems $A$ and $B_2$ evolve forward in time for $t$. Then, we evolve subsystems $A$ and $B_1$ backward in time for the same duration $t$. Finally, we perform a measurement on subsystems $B_1$,$A$, and $B_2$. The protocol is discussed in detail in subsection \ref{['measure_projected_LE_subsection']}.
• Figure 3: Single-round protocol for measuring the second Rényi entropy via the projected LE begins with the given initial state $|m_1,\psi_0,B_0\rangle$. We start with the initial state $|m_1,\psi_0,B_0\rangle$ and let subsystems $A$ and $B_2$ evolve forward in time for $t$. Then, we evolve subsystems $A$ and $B_1$ backward in time for the same duration $t$. Finally, we perform a measurement on subsystems $B_1$ and $A$. This protocol is discussed in detail in subsection \ref{['measure_Renyi_entropy_subsection']}.
• Figure 4: Simplified single-round protocol for measuring the second Rényi entropy via the projected LE with a given initial state. Starting from $\ket{\psi_0}_A\otimes\ket{B_0}_B$, we evolve $A$ and $B$ forward for time $t$, discard the final state of $B$, reset $B$ to $\ket{m_1}$, and then evolve $A$ and $B$ backward for the same duration $t$. Finally, we measure subsystems $A$ and $B$. See Sec. \ref{['measure_Renyi_entropy_subsection']} for details.
• Figure 5: Single-round protocol for measuring the second Rényi entropy via the projected LE with a fixed initial state each round. Starting from $\ket{\psi_0}_A\otimes\ket{B_0}_B$, we evolve $A$ and $B$ forward for time $t$, apply a random unitary on subsystem $B$, and then evolve $A$ and $B$ backward for the same duration $t$. Finally, we measure subsystems $A$ and $B$. See Sec. \ref{['measure_Renyi_entropy_subsection']} for details.