Rényi entropy of the permutationally invariant part of the ground state across a quantum phase transition

TL;DR

This work investigates the second-order Rényi entropy of the permutationally invariant part of the density matrix, $S_2(\Pi(\rho))$, as a scalable diagnostic of quantum phase transitions. Using the transverse-field Ising chain and DMRG, the authors show that $S_2(\Pi(\rho))$ vanishes in ferromagnetic limits but exhibits a broad peak near the quantum critical point that grows with system size, while in the antiferromagnetic case the peak disappears and $S_2(\Pi(\rho))$ scales linearly with system size in the ordered phase. They derive perturbative expressions for low- and high-field regimes and demonstrate distinct finite-size scaling laws for FM, AFM, and PM phases, suggesting $S_2(\Pi(\rho))$ as an experimentally accessible indicator of phase structure and criticality. The study highlights the potential of PIDM-based entropy as a practical tool for analyzing phase transitions in quantum many-body systems, with implications for experimental quantum state tomography and topological or nonlocal order detection. Overall, the work provides a concrete framework for connecting permutation symmetries in quantum states to phase-transition phenomena via scalable tomography and Renyi-entropy analysis.

Abstract

We investigate the role of the permutationally invariant part of the density matrix (PIDM) in capturing the properties of the ground state of the system during a quantum phase transition. In the context of quantum state tomography, PIDM is known to be obtainable with only a low number of measurement settings, namely $\mathcal{O}(L^2)$, where $L$ is the system size. Considering the transverse-field Ising chain as an example, we compute the second-order Rényi entropy of PIDM for the ground state by using the density matrix renormalization group algorithm. In the ferromagnetic case, the ground state is permutationally invariant both in the limits of zero and infinite field, leading to vanishing Rényi entropy of PIDM. The latter exhibits a broad peak as a function of the transverse field around the quantum critical point, which gets more pronounced for larger system size. In the antiferromagnetic case, the peak structure disappears and the Rényi entropy diverges like $\mathcal{O}(L)$ in the whole field range of the ordered phase. We discuss the cause of these behaviors of the Rényi entropy of PIDM, examining the possible application of this experimentally tractable quantity to the analysis of phase transition phenomena.

Figures (8)

• Figure 1: Schematic sketch of the linear spaces $\mathbb{D}$ and $\mathbb{P}$ spanned by the orthogonal operators $\{\bigotimes_{i=1}^L{\sigma}_{\kappa_i}\}$ and by the orthogonal PI operators $\{\mathcal{B}_d^{(n)}\}$ or the PI random operators $\{\mathcal{A}_d^{(n)}\}$, respectively. Any density matrix $\rho$ is an element of the space $\mathbb{D}$. The PI part of an arbitrary density matrix of an $L$-qubit system, $\mathit{\Pi}(\rho)$, exist in $\mathbb{P}\subset\mathbb{D}$.
• Figure 2: The protocols for evaluating the second-order Rényi entropy of PIDM in the experiments and in the numerical simulation (this work). In this work, we apply DMRG algorithm to obtain the ground state of a physical Hamiltonian and simulate the observations of $\mathcal{A}_\nu^{(n)}$ by calculating the expectation value of the matrix product operator $A_\nu$. The numerical calculations after the measurements are the same in the experimental protocol and on the simulations.
• Figure 3: (a) The purity and (b) the second-order Rényi entropy of PIDM of the ground state of the FM transverse field Ising model as a function of the transverse-field for various system sizes. The black dashed line represents the QCP ($\mathit{\Gamma}/|J|=1$).
• Figure 4: (a) The purity and (b) the second-order Rényi entropy of PIDM of the ground state of the AFM transverse-field Ising model as a function of the transverse field for various system sizes. The black dashed line represents the QCP ($\mathit{\Gamma}/J=1$).
• Figure 5: The second-order Rényi entropy of PIDM for the ground state of the transverse-field Ising chain as a function of $L$ in (a) FM case and (b) AFM cases. The dashed lines represent the fitting functions $f_{S_2}(L)$ defined by Eq. (\ref{['fS']}) with fitting parameters as shown in Figs. \ref{['fitpara_FM']} and \ref{['fitpara_AFM']}.