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arXiv:1611.08007 · Physical review. B./Physical review. B

Entanglement negativity bounds for fermionic Gaussian states

Jens Eisert, Viktor Eisler, Zoltán Zimborás

Published Apr 13, 2018• 85 citations
arXivHTMLPDF

Abstract

The entanglement negativity is a versatile measure of entanglement that has numerous applications in quantum information and in condensed matter theory. It can not only efficiently be computed in the Hilbert space dimension, but for noninteracting bosonic systems, one can compute the negativity efficiently in the number of modes. However, such an efficient computation does not carry over to the fermionic realm, the ultimate reason for this being that the partial transpose of a fermionic Gaussian state is no longer Gaussian. To provide a remedy for this state of affairs, in this work, we introduce efficiently computable and rigorous upper and lower bounds to the negativity, making use of techniques of semidefinite programming, building upon the Lagrangian formulation of fermionic linear optics, and exploiting suitable products of Gaussian operators. We discuss examples in quantum many-body theory and hint at applications in the study of topological properties at finite temperature.

DOI
10.1103/physrevb.97.165123

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The entanglement negativity is a versatile measure of entanglement that has numerous applications in quantum information and in condensed matter theory. It can not only efficiently be computed in the Hilbert space dimension, but for noninteracting bosonic systems, one can compute the negativity efficiently in the number of modes. However, such an efficient computation does not carry over to the fermionic realm, the ultimate reason for this being that the partial transpose of a fermionic Gaussian state is no longer Gaussian. To provide a remedy for this state of affairs, in this work, we introduce efficiently computable and rigorous upper and lower bounds to the negativity, making use of techniques of semidefinite programming, building upon the Lagrangian formulation of fermionic linear optics, and exploiting suitable products of Gaussian operators. We discuss examples in quantum many-body theory and hint at applications in the study of topological properties at finite temperature.
We study quantum information aspects of the fermionic entanglement negativity recently introduced [H. Shapourian et al., Phys. Rev. B 95, 165101 (2017)] based on the fermionic partial transpose. In particular, we show that it is an entanglement monotone under the action of local quantum operations and classical communications, which preserves the local fermion-number parity, and satisfies other common properties expected for an entanglement measure of mixed states. We present fermionic analogs of tripartite entangled states such as WWW and Greenberger-Horne-Zeilinger states and compare the results of bosonic and fermionic partial transpose in various fermionic states, where we explain why the bosonic partial transpose fails in distinguishing separable states of fermions. Finally, we explore a set of entanglement quantities which distinguish different classes of entangled states of a system with two and three fermionic modes. In doing so, we prove that vanishing entanglement negativity is a necessary and sufficient condition for separability of $N\ensuremath{\ge}2$ fermionic modes with respect to the bipartition into one mode and the rest. We further conjecture that the entanglement negativity of inseparable states which mix local fermion-number parity is always nonvanishing.
We discuss the behavior of positive linear maps in fermionic systems and then propose the phase partial transpose and the phase entanglement negativity. We show that every fermionic state which mixes local fermion-number parity must have nonvanishing nontrivial phase entanglement negativity, which gives an affirmative answer to a conjecture proposed by Shapourian and Ryu [Phys. Rev. A 99, 022310 (2019)]. In addition, we prove that the phase entanglement negativity is an entanglement monotone and establish some equalities and inequalities related to the phase entanglement negativity which, particularly, provide some upper bounds and lower bounds of the fermionic entanglement negativity. A more detailed discussion of the (1+M)(1+M)(1+M)-mode case is also presented, and our results generalize some known findings.
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.
We consider Gaussian states of fermionic systems and study the action of the partial transposition on the density matrix. It is shown that, with a suitable choice of basis, these states are transformed into a linear combination of two Gaussian operators that are uniquely defined in terms of the covariance matrix of the original state. In case of a reflection symmetric geometry, this result can be used to efficiently calculate a lower bound for a well-known entanglement measure, the logarithmic negativity. Furthermore, exact expressions can be derived for traces involving integer powers of the partial transpose. The method can also be applied to the quantum Ising chain and the results show perfect agreement with the predictions of conformal field theory.
We consider Gaussian states of fermionic systems and study the action of the partial transposition on the density matrix. It is shown that, with a suitable choice of basis, these states are transformed into a linear combination of two Gaussian operators that are uniquely defined in terms of the covariance matrix of the original state. In case of a reflection symmetric geometry, this result can be used to efficiently calculate a lower bound for a well-known entanglement measure, the logarithmic negativity. Furthermore, exact expressions can be derived for traces involving integer powers of the partial transpose. The method can also be applied to the quantum Ising chain and the results show perfect agreement with the predictions of conformal field theory.
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.