Evaluating Many-Body Stabilizer Rényi Entropy by Sampling Reduced Pauli Strings: Singularities, Volume Law, and Nonlocal Magic
Yi-Ming Ding, Zhe Wang, Zheng Yan
Abstract
We present a novel quantum Monte Carlo method for evaluating the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>α</a:mi> </a:math> -stabilizer Rényi entropy (SRE) for any integer <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>α</c:mi> <c:mo>≥</c:mo> <c:mn>2</c:mn> </c:math> . By interpreting the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>α</e:mi> </e:math> -SRE as partition-function ratios, we eliminate the sign problem in the imaginary-time path integral by sampling within a , which enables efficient classical computations of the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>α</g:mi> </g:math> -SRE and its derivatives to explore magic in previously inaccessible two- or higher-dimensional systems. We first isolate the free-energy part in <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mn>2</i:mn> </i:math> -SRE, which is a trivial term. Notably, at quantum critical points in one-dimensional or two-dimensional transverse-field Ising (TFI) models, we reveal nontrivial singularities associated with the contribution, directly tied to magic. Their interplay leads to complicated behaviors of <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mn>2</k:mn> </k:math> -SRE, avoiding extrema at critical points generally. In contrast, analyzing the volume-law correction to SRE reveals a discontinuity tied to criticalities, suggesting that it is more informative than the full-state magic. For conformal critical points, we claim that it could reflect nonlocal magic residing in correlations. Finally, we verify that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mn>2</m:mn> </m:math> -SRE fails to characterize magic in mixed states (e.g., Gibbs states), yielding nonphysical results. This work provides a powerful tool for exploring the roles of magic in large-scale many-body systems and reveals the intrinsic relation between magic and many-body physics.