Measuring Rényi entropy using a projected Loschmidt echo

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.

💡 Research Summary

The paper introduces a practical and scalable protocol for measuring the second Rényi entropy (the logarithm of the purity) in quantum many‑body systems by exploiting a direct connection to a Loschmidt echo (LE) measurement. The authors first define the second Rényi entropy as (S^{(2)} = -\log \operatorname{Tr}\rho_A^2) and the standard LE as (M(t)=|\langle\psi_0|e^{iH_2t}e^{-iH_1t}|\psi_0\rangle|^2). They then consider a bipartition of the total system into subsystems (A) and (B) and introduce two copies of subsystem (B), denoted (B_1) and (B_2). By evolving (A) together with (B_2) forward in time under a unitary (U_{A,B_2}(t)) and then evolving (A) together with (B_1) backward in time under (U_{A,B_1}^\dagger(t)), they define a “projected Loschmidt echo”

\