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.
Highlights
A basic diagnostic of entanglement in mixed quantum states is known as the positive partial transpose (PT) criterion
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
This can be readily seen by comparing T −1 and T. These boundary conditions correspond to two replica-symmetric spin structures for the spacetime manifold. This is different from bosonic PT of fermionic systems [107, 110], where Rényi negativity (RN) is given by sum over all possible spin structures
Summary
Entanglement is an intrinsic property of quantum systems beyond classical physics. Having efficient frameworks to compute entanglement between two parts of a system is essential for fundamental interests such as characterizing phases of matter [1,2,3,4] and spacetime physics [5] and for application purposes such as identifying useful resources to implement quantum computing processes. A procedure for the PT of fermions based on the fermion-boson mapping (Jordan-Wigner transformation) was proposed [104] and was used in the subsequent studies [105,106,107,108,109,110,111] This definition turned out to cause certain inconsistencies within fermionic theories such as violating the additivity property and missing some entanglement in topological superconductors, and give rise to incorrect classification of time-reversal symmetric topological insulators and superconductors. According to this definition it is computationally hard to find the PT (and calculate the entanglement negativity) even for free fermions, since the PT of a fermionic Gaussian state is not Gaussian This motivates us to use another way of implementing a fermionic PT which was proposed recently by some of us in the context of time-reversal symmetric SPT phases of fermions [100,101,112]. We give further details of the analytical calculations and make connections with other related concepts
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
