Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1906.04211 · SciPost Physics

Twisted and untwisted negativity spectrum of free fermions

Hassan Shapourian, Paola Ruggiero, Shinsei Ryu, Pasquale Calabrese

Published Sep 25, 2019• 95 citations
arXivHTMLPDF

Abstract

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.

DOI
10.21468/scipostphys.7.3.037

Related papers

Negativity spectrum of one-dimensional conformal field theories

Paola Ruggiero, Vincenzo Alba, Pasquale Calabrese

Logarithmic negativity is a proper measure of quantum entanglement in mixed states. Originally proposed in the context of quantum information, in recent years it is attracting interest in quantum field theory, condensed matter physics, and statistical physics. The key object to construct the negativity is the partial transpose of the reduced density matrix. Here, focusing on two adjacent intervals, the authors provide a complete study of the distribution of the eigenvalues of the partial transpose (negativity spectrum) for gapless one-dimensional systems described by a conformal field theory (CFT). Similar to the entanglement spectrum, the distribution of the negativity spectrum is universal, and it depends only on the central charge of the underlying CFT. The precise form of the spectrum depends on whether the two intervals are in a pure or mixed state, and in both cases, a dependence on the sign of the eigenvalues is found. This dependence is weak for bulk eigenvalues, whereas it is strong at the spectrum edges. The authors also investigate the scaling of the smallest (negative) and largest (positive) eigenvalues of the partial transpose. The analytical results are thoroughly checked against numerical DMRG simulations as well as in exactly solvable models.

arXiv:1607.02992•Physical review. B./Physical review. B•Nov 10, 2016•74 citations

Anyonic Partial Transpose I: Quantum Information Aspects

Hassan Shapourian, Roger S. K. Mong, Shinsei Ryu

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.

arXiv:2012.02222•Dec 3, 2020•5 citations

Finite temperature negativity Hamiltonians of the massless Dirac fermion

Federico Rottoli, Sara Murciano, Pasquale Calabrese

Abstract The negativity Hamiltonian, defined as the logarithm of a partially transposed density matrix, provides an operatorial characterisation of mixed-state entanglement. However, so far, it has only been studied for the mixed-state density matrices corresponding to subsystems of globally pure states. Here, we consider as a genuine example of a mixed state the one-dimensional massless Dirac fermions in a system at finite temperature and size. As subsystems, we consider an arbitrary set of disjoint intervals. The structure of the corresponding negativity Hamiltonian resembles the one for the entanglement Hamiltonian in the same geometry: in addition to a local term proportional to the stress-energy tensor, each point is non-locally coupled to an infinite but discrete set of other points. However, when the lengths of the transposed and non-transposed intervals coincide, the structure remarkably simplifies and we retrieve the mild non-locality of the ground state negativity Hamiltonian. We also conjecture an exact expression for the negativity Hamiltonian associated to the twisted partial transpose, which is a Hermitian fermionic matrix. We finally obtain the continuum limit of both the local and bi-local operators from exact numerical computations in free-fermionic chains.

arXiv:2304.09906•Journal of High Energy Physics•Jun 23, 2023•8 citations

Characterizing the entanglement of anyonic systems using the anyonic partial transpose

Nico Kirchner, Wonjune Choi, Frank Pollmann

Entanglement of mixed quantum states can be quantified using the partial transpose and its corresponding entanglement measure, the logarithmic negativity. Recently, the notion of partial transpose has been extended to systems of anyons, which are exotic quasiparticles whose exchange statistics go beyond the bosonic and fermionic cases. Studying the fundamental properties of this anyonic partial transpose, we first reveal that when applied to the special case of fermionic systems, it can be reduced to the fermionic partial transpose or its twisted variant depending on whether or not a boundary Majorana fermion is present. Focusing on ground state properties, we find that the anyonic partial transpose captures both the correct entanglement scaling for gapless systems, as predicted by conformal field theory, and the phase transition between a topologically trivial and a nontrivial phase. For non-Abelian anyons and the bipartition geometry, we find a rich multiplet structure in the eigenvalues of the partial transpose, the so-called negativity spectrum, and reveal the possibility of defining both a charge- and an imbalance-resolved negativity.

arXiv:2403.12121•Physical review. B./Physical review. B•Aug 21, 2024•3 citations

Fermionic partial transpose in the overlap matrix framework for entanglement negativity

Jun Qi Fang, Xiao Yan Xu

Over the past two decades, the overlap matrix approach has been developed to compute quantum entanglement in free-fermion systems, particularly to calculate entanglement entropy and entanglement negativity. This method involves the use of partial trace and partial transpose operations within the overlap matrix framework. However, in previous studies, only the conventional partial transpose in fermionic systems has been considered, which does not account for fermionic anticommutation relations. Although the concept of a fermionic partial transpose was introduced by Shapourian et al. [Phys. Rev. B 95, 165101 (2017)], it has not yet been systematically incorporated into the overlap matrix framework. In this paper, we introduce the fermionic partial transpose into the overlap matrix approach, provide a systematic analysis of the validity of partial trace and partial transpose operations, and derive an explicit formula for calculating entanglement negativity in bipartite systems. Additionally, we numerically compute the logarithmic negativity of two lattice models to verify the Gioev-Klich-Widom scaling law. For tripartite geometries, we uncover limitations of the overlap matrix method and demonstrate that the previously reported logarithmic negativity result for a homogeneous one-dimensional chain in a disjoint interval geometry exceeds its theoretical upper bound.

arXiv:2503.07742•Physical review. B./Physical review. B•Aug 13, 2025•2 citations

Estimation for entanglement negativity of free fermions

Christopher P. Herzog, Yihong Wang

Abstract In this letter we study the negativity of one dimensional free fermions. We derive the general form of the Z N symmetric term in moments of the partial transposed (reduced) density matrix, which is an algebraic function of the end points of the system. Such a path integral turns out to be a convenient tool for making estimations for the negativity.

arXiv:1601.00678•Journal of Statistical Mechanics Theory and Experiment•Jul 1, 2016•43 citations
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.
Logarithmic negativity is a proper measure of quantum entanglement in mixed states. Originally proposed in the context of quantum information, in recent years it is attracting interest in quantum field theory, condensed matter physics, and statistical physics. The key object to construct the negativity is the partial transpose of the reduced density matrix. Here, focusing on two adjacent intervals, the authors provide a complete study of the distribution of the eigenvalues of the partial transpose (negativity spectrum) for gapless one-dimensional systems described by a conformal field theory (CFT). Similar to the entanglement spectrum, the distribution of the negativity spectrum is universal, and it depends only on the central charge of the underlying CFT. The precise form of the spectrum depends on whether the two intervals are in a pure or mixed state, and in both cases, a dependence on the sign of the eigenvalues is found. This dependence is weak for bulk eigenvalues, whereas it is strong at the spectrum edges. The authors also investigate the scaling of the smallest (negative) and largest (positive) eigenvalues of the partial transpose. The analytical results are thoroughly checked against numerical DMRG simulations as well as in exactly solvable models.
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.
Abstract The negativity Hamiltonian, defined as the logarithm of a partially transposed density matrix, provides an operatorial characterisation of mixed-state entanglement. However, so far, it has only been studied for the mixed-state density matrices corresponding to subsystems of globally pure states. Here, we consider as a genuine example of a mixed state the one-dimensional massless Dirac fermions in a system at finite temperature and size. As subsystems, we consider an arbitrary set of disjoint intervals. The structure of the corresponding negativity Hamiltonian resembles the one for the entanglement Hamiltonian in the same geometry: in addition to a local term proportional to the stress-energy tensor, each point is non-locally coupled to an infinite but discrete set of other points. However, when the lengths of the transposed and non-transposed intervals coincide, the structure remarkably simplifies and we retrieve the mild non-locality of the ground state negativity Hamiltonian. We also conjecture an exact expression for the negativity Hamiltonian associated to the twisted partial transpose, which is a Hermitian fermionic matrix. We finally obtain the continuum limit of both the local and bi-local operators from exact numerical computations in free-fermionic chains.
Entanglement of mixed quantum states can be quantified using the partial transpose and its corresponding entanglement measure, the logarithmic negativity. Recently, the notion of partial transpose has been extended to systems of anyons, which are exotic quasiparticles whose exchange statistics go beyond the bosonic and fermionic cases. Studying the fundamental properties of this anyonic partial transpose, we first reveal that when applied to the special case of fermionic systems, it can be reduced to the fermionic partial transpose or its twisted variant depending on whether or not a boundary Majorana fermion is present. Focusing on ground state properties, we find that the anyonic partial transpose captures both the correct entanglement scaling for gapless systems, as predicted by conformal field theory, and the phase transition between a topologically trivial and a nontrivial phase. For non-Abelian anyons and the bipartition geometry, we find a rich multiplet structure in the eigenvalues of the partial transpose, the so-called negativity spectrum, and reveal the possibility of defining both a charge- and an imbalance-resolved negativity.
Over the past two decades, the overlap matrix approach has been developed to compute quantum entanglement in free-fermion systems, particularly to calculate entanglement entropy and entanglement negativity. This method involves the use of partial trace and partial transpose operations within the overlap matrix framework. However, in previous studies, only the conventional partial transpose in fermionic systems has been considered, which does not account for fermionic anticommutation relations. Although the concept of a fermionic partial transpose was introduced by Shapourian et al. [Phys. Rev. B 95, 165101 (2017)], it has not yet been systematically incorporated into the overlap matrix framework. In this paper, we introduce the fermionic partial transpose into the overlap matrix approach, provide a systematic analysis of the validity of partial trace and partial transpose operations, and derive an explicit formula for calculating entanglement negativity in bipartite systems. Additionally, we numerically compute the logarithmic negativity of two lattice models to verify the Gioev-Klich-Widom scaling law. For tripartite geometries, we uncover limitations of the overlap matrix method and demonstrate that the previously reported logarithmic negativity result for a homogeneous one-dimensional chain in a disjoint interval geometry exceeds its theoretical upper bound.