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arXiv:2503.07731 · Proceedings of the National Academy of Sciences

Twisted Rényi negativity as a reliable proxy for mixed-state entanglement in fermionic systems

Fo-Hong Wang, Xiao Yan Xu

Published Sep 29, 2026• 0 citations
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Abstract

Characterizing mixed-state entanglement in fermionic quantum matter represents a fundamental challenge at the intersection of condensed matter physics and quantum information. Unlike pure states, mixed states lack a universally accepted and easily computable entanglement measure, a difficulty compounded in fermionic systems by the ambiguity of the partial transpose operation. Here, motivated by the Hermiticity of the partially transposed density matrix and the analytic continuation of the Rényi negativity to logarithmic negativity, we address this puzzle by demonstrating the "twisted" Rényi negativity as a physically consistent proxy for mixed-state entanglement in the Hubbard and spinless [Formula: see text]-[Formula: see text] models. Through large-scale quantum Monte Carlo simulations of these paradigmatic correlated systems, we show that this indicator captures essential physical expectations-including the area law and monotonic thermal suppression-while excluding anomalous behavior present in alternative definitions. Our findings not only provide a robust theoretical framework for quantifying entanglement in strongly correlated electronic systems but also offer a practical tool for experimental detection in rapidly developing platforms, including ultracold atomic gases and programmable quantum simulators.

DOI
10.1073/pnas.2534602123

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Characterizing mixed-state entanglement in fermionic quantum matter represents a fundamental challenge at the intersection of condensed matter physics and quantum information. Unlike pure states, mixed states lack a universally accepted and easily computable entanglement measure, a difficulty compounded in fermionic systems by the ambiguity of the partial transpose operation. Here, motivated by the Hermiticity of the partially transposed density matrix and the analytic continuation of the Rényi negativity to logarithmic negativity, we address this puzzle by demonstrating the "twisted" Rényi negativity as a physically consistent proxy for mixed-state entanglement in the Hubbard and spinless [Formula: see text]-[Formula: see text] models. Through large-scale quantum Monte Carlo simulations of these paradigmatic correlated systems, we show that this indicator captures essential physical expectations-including the area law and monotonic thermal suppression-while excluding anomalous behavior present in alternative definitions. Our findings not only provide a robust theoretical framework for quantifying entanglement in strongly correlated electronic systems but also offer a practical tool for experimental detection in rapidly developing platforms, including ultracold atomic gases and programmable quantum simulators.
Many-body entanglement unveils additional aspects of quantum matter and offers insights into strongly correlated physics. While ground-state entanglement has received much attention in the past decade, the study of mixed-state quantum entanglement using negativity in interacting fermionic systems remains largely unexplored. We demonstrate that the partially transposed density matrix of interacting fermions, similar to their reduced density matrix, can be expressed as a weighted sum of Gaussian states describing free fermions, enabling the calculation of rank-n Rényi negativity within the determinant quantum Monte Carlo framework. We calculate the rank-two Rényi negativity for the half-filled Hubbard model and the spinless t-V model. Our calculation reveals that the area law coefficient of the Rényi negativity for the spinless t-V model has a logarithmic finite-size scaling at the finite-temperature transition point. Our work contributes to the calculation of entanglement and sets the stage for future studies on quantum entanglement in various fermionic many-body mixed states.
Quantum entanglement is a fundamental resource that gives rise to correlations with no classical counterpart in strongly correlated fermionic systems. However, its characterization in mixed quantum states remains challenging. Established entanglement diagnostics, such as logarithmic negativity, are computationally demanding in interacting systems, and their evaluation by exact diagonalization is restricted to small system sizes. Meanwhile, conventional Rényi proxies, including twisted and untwisted Rényi negativities, can be positive even for separable states. In this work, we introduce a family of \textit{relative Rényi negativities} that serve as entanglement witnesses. By relating fermionic partial time reversal to the size superoperator, we establish a path-integral representation for these Rényi negativities, enabling their evaluation in thermal states of the Sachdev-Ye-Kitaev (SYK) model and its generalizations via a large-NNN expansion. Applying this method, we investigate entanglement in both the original SYK model and the Maldacena-Qi coupled SYK model. We identify a special class of relative Rényi negativities which serve as plausible proxies for logarithmic negativity and uncover first-order entanglement transitions that are invisible to simple observables. These results highlight the utility of relative Rényi negativities as probes of mixed-state entanglement in strongly correlated fermionic systems and suggest the existence of a new class of transitions where these negativities serve as free energies.
Quantum systems are known to exhibit several types of exotic behavior beyond classical physics, including quantum entanglement, which describes a web of nonlocal correlations among the constituents of the system. Despite numerous advances in understanding and quantifying the bipartite entanglement entropy of a pure state, where the whole system is described by a wave function, less is understood about quantum entanglement of a mixed state, where the system is described by a density matrix. The partial transpose of the density matrix, in which one takes the transpose only for a subsystem, and its corresponding entanglement measure -- called the (logarithmic) negativity -- have been introduced as an effective probe for the quantum entanglement in bosonic mixed states. In this work, the authors present a scheme to compute an analog of the entanglement negativity in the fermionic mixed states using a partial time-reversal transformation. Various examples are investigated. In particular, it is shown that the partial time reversal is an intrinsically fermionic construction, in that it can capture the formation of the edge Majorana fermions, while the partial transpose obtained from the bosonic partial transpose through the Jordan-Wigner transformation fails.
A basic diagnostic of entanglement in mixed quantum states is known as the positive partial transpose (PT) criterion. Such criterion is based on the observation that the spectrum of the partially transposed density matrix of an entangled state contains negative eigenvalues, in turn, used to define an entanglement measure called the logarithmic negativity. Despite the great success of logarithmic negativity in characterizing bosonic many-body systems, generalizing the operation of PT to fermionic systems remained a technical challenge until recently when a more natural definition of PT for fermions that accounts for the Fermi statistics has been put forward. In this paper, we study the many-body spectrum of the reduced density matrix of two adjacent intervals for one-dimensional free fermions after applying the fermionic PT. We show that in general there is a freedom in the definition of such operation which leads to two different definitions of PT: the resulting density matrix is Hermitian in one case, while it becomes pseudo-Hermitian in the other case. Using the path-integral formalism, we analytically compute the leading order term of the moments in both cases and derive the distribution of the corresponding eigenvalues over the complex plane. We further verify our analytical findings by checking them against numerical lattice calculations.
A basic diagnostic of entanglement in mixed quantum states is known as the partial transpose and the corresponding entanglement measure is called the logarithmic negativity. Despite the great success of logarithmic negativity in characterizing bosonic many-body systems, generalizing the partial transpose to fermionic systems remained a technical challenge until recently when a new definition that accounts for the Fermi statistics was put forward. In this paper, we propose a way to generalize the partial transpose to anyons with (non-Abelian) fractional statistics based on the apparent similarity between the partial transpose and the braiding operation. We then define the anyonic version of the logarithmic negativity and show that it satisfies the standard requirements such as monotonicity to be an entanglement measure. In particular, we elucidate the properties of the anyonic logarithmic negativity by computing it for a toy density matrix of a pair of anyons within various categories. We conjecture that the subspace of states with a vanishing logarithmic negativity is a set of measure zero in the entire space of anyonic states, in contrast with the ordinary qubit systems where this subspace occupies a finite volume. We prove this conjecture for multiplicity-free categories.
Entanglement entropy has been a powerful tool for analyzing phases and criticality in pure ground states via quantum Monte Carlo (QMC). However, mixed-state entanglement, relevant to systems with dissipation, finite temperature, and disjoint regions, remains less explored due to the lack of efficient numerical methods. In this work, we present a practical and easy-to-implement QMC method within the reweight-annealing framework, enabling efficient computation of the entanglement R\'enyi negativity by tracking its variation along given parameter paths. This method is scalable, parallelizable, and well suited for high-dimensional and large-scale simulations. Applying it to diverse scenarios, including one- and two-dimensional systems, ground and thermal states, and bipartite and tripartite partitions, not only is the information of the underlying conformal field theory achieved, but the role of entanglement in quantum and thermal phase transitions is revealed.