Robustness of Magic in the quantum Ising chain via Quantum Monte Carlo tomography

TL;DR

This work develops a hybrid quantum Monte Carlo tomography approach to quantify magic in mixed states of a quantum Ising chain, focusing on the Robustness of Magic $\mathcal{R}(\rho)$ and its bipartite mutual variant. By combining SSE-based QMC sampling of reduced density matrices with a column-generation convex optimization for RoM, the authors access partitions up to eight spins embedded in much larger systems and analyze behavior across the quantum critical point at $h/J=1$ and finite temperatures. They introduce the log-free robustness of magic $LR(\rho)$ and study its tri-partite partitioning, finding a power-law decay of mutual magic at criticality with exponents that depend on partition size, and an algebraic scaling of an effective temperature with system size, indicating magic does not exhibit sudden death. The results provide a concrete link between nonstabilizerness and quantum critical phenomena, offering a general numerical toolkit applicable to other many-body systems and guiding potential experimental investigations of magic correlations in mixed states.

Abstract

We study the behavior of magic as a bipartite correlation in the quantum Ising chain across its quantum phase transition, and at finite temperature. In order to quantify the magic of partitions rigorously, we formulate a hybrid scheme that combines stochastic sampling of reduced density matrices via quantum Monte Carlo, with state-of-the-art estimators for the robustness of magic - a {\it bona fide} measure of magic for mixed states. This allows us to compute the mutual robustness of magic for partitions up to 8 sites, embedded into a much larger system. We show how mutual robustness is directly related to critical behaviors: at the critical point, it displays a power law decay as a function of the distance between partitions, whose exponent is related to the partition size. Once finite temperature is included, mutual magic retains its low temperature value up to an effective critical temperature, whose dependence on size is also algebraic. This suggests that magic, differently from entanglement, does not necessarily undergo a sudden death.

Figures (6)

• Figure 1: (a) 1D spin chain with periodic boundary conditions. The system is transferred to (b) for the RDM calculation through QMC using SSE. We calculate the reduced density matrix by tracing out the $B$ part, leaving two disconnected subsystems $A_1$ and $A_2$.
• Figure 2: (a) LRoM for the full system of the 1D TFIM with periodic boundary conditions (PBC) as a function of $\beta$ for different values of $h$. The LRoM saturates and forms a stable plateau after a finite $\beta$. (b) LRoM for $L \leq 8$ and ground state approximation $\beta=2L$.
• Figure 3: LRoM ((a) size 2+2 and (c) 4+4) and MLRoM ((b) size 2+2 and (d) 4+4) with $h$ for the low temperature state $\beta=2L$.
• Figure 4: LRoM with system size for different values of $h$ for the partition size of (a) $2+2$ and (b) $4+4$. The power law scaling at $h=1$ is presented in the insets.
• Figure 5: MLRoM with system size for different values of $h$ for the partition size of (a) $2+2$ and (b) $4+4$. The power law scaling at the critical point $h=1$ is presented in the insets.