We introduce a general bipartite-like representation and Schmidt decomposition of an arbitrary pure state of $N$ indistinguishable fermions, based on states of $M<N$ and $(N-M)$ fermions. It is directly connected with the reduced $M$- and $(N-M)$-body density matrices (DMs), which have the same spectrum in such states. The concept of $M$-body entanglement emerges naturally in this scenario, generalizing that of one-body entanglement. Rigorous majorization relations satisfied by the normalized $M$-body DM are then derived, which imply that the associated entropy will not increase, on average, under a class of operations which have these DMs as post-measurement states. Moreover, such entropy is an upper bound to the average bipartite entanglement entropy generated by a class of operations which map the original state to a bipartite state of $M$ and $N-M$ effectively distinguishable fermions. Analytic evaluation of the spectrum of $M$-body DMs in some strongly correlated fermionic states is also provided.
While single boson entanglement is known to be equivalent to standard non-identical particle entanglement, the use of fermionic entanglement is constrained by a parity superselection rule. Nevertheless we show that within the framework of second quantization, where local modes play the role of qubits, the analogy with ordinary entanglement holds: two entangled spinless fermion modes can be used to teleport a single local fermion mode and to perform standard dense coding. Entanglement concentration and dilution are possible for fermions. We clarify the meaning of our results by discussing the c…
We have studied the entanglement of identical fermions in two spatial regions in terms of the Berry phase acquired by their spins. The analysis is done from the viewpoint of the geometrical interpretation of entanglement, where a fermion is visualized as a scalar particle attached with a magnetic flux quantum. The quantification of spin entanglement in terms of their Berry phases is novel and generalises the relationship between the entanglement of distinguishable spins and that of delocalised fermions.
We discuss the relation between fermion entanglement and bipartite entanglement. We first show that an exact correspondence between them arises when the states are constrained to have a definite local number parity. Moreover, for arbitrary states in a four dimensional single-particle Hilbert space, the fermion entanglement is shown to measure the entanglement between two distinguishable qubits defined by a suitable partition of this space. Such entanglement can be used as a resource for tasks like quantum teleportation. On the other hand, this fermionic entanglement provides a lower bound to t…
We examine the fermionic entanglement in the ground state of the fermionic Lipkin model and its relation with bipartite entanglement. It is first shown that the one-body entanglement entropy, which quantifies the minimum distance to a fermionic Gaussian state, behaves similarly to the mean-field order parameter and is essentially proportional to the total bipartite entanglement between the upper and lower modes, a quantity meaningful only in the fermionic realization of the model. We also analyze the entanglement of the reduced state of four single-particle modes (two up-down pairs), showing t…
We investigate the generation of entanglement in systems of identical fermions through a process involving particle detection, focusing on the implications that this kind of processes have for the concept of entanglement between fermionic particles. As a paradigmatic example we discuss in detail a scheme based on a splitting-plus-detection operation. This scheme generates states with accessible entanglement starting from an initial pure state of two indistinguishable fermions exhibiting correlations due purely to antisymmetrization. It is argued that the proposed extraction of entanglement doe…
In our two preceding papers we studied bipartite composite boson (or quasiboson) systems through their realization in terms of deformed oscillators. Therein, the entanglement characteristics such as the entanglement entropy and purity were found and expressed, for both one-quasiboson and more complex states, through the parameter of deformation. In this work we initiate an analogous study of composite fermions for two major cases: (i) "boson + fermion" composites; (ii) "deformed-boson + fermion" composites. Both the entanglement entropy and purity of composite fermions are dealt with, their de…
We study pairwise quantum entanglement in systems of fermions itinerant in a lattice from a second-quantized perspective. Entanglement in the grand-canonical ensemble is studied, both for energy eigenstates and for the thermal state. Relations between entanglement and superconducting correlations are discussed in a BCS-like model and for $η$-pair superconductivity.
In this work we have revisited a few principal formulae about one-tangle of multipartite entanglement of fermionic systems in noninertial frames calculated in the paper [Phys. Rev. A 83, 022314(2011)] and given their correct expressions.
This is a thesis submitted to university of Pune, India, for the Ph.D. degree. This work deals with entanglement production in two qubit, two qutrit and three qubit systems, entanglement in indistinguishable fermionic systems, quantum discord in a Heisenberg chain and geometric measure of quantum discord in an arbitrary state of a bipartite quantum system.
We investigate quantum entanglement between two (spin-1/2) fermions inside a cylindrical harmonic trap, making use of the von Neumann entropy for the reduced single particle density matrix as the pure state entanglement measure. We explore the dependence of pair entanglement on the geometry and strength of the trap and on the strength of the pairing interaction over the complete range of the effective BCS to BEC crossover. Our result elucidates an interesting connection between our model system of two fermions and that of two interacting bosons.
A class of squeezed boson and fermion states is studied with particular emphasis on the nature of entanglement. We first investigate the case of bosons, considering two-mode squeezed states. Then we construct the fermion version to show that such states are maximum entangled, for both bosons and fermions. To achieve these results, we demonstrate some relations involving squeezed boson states. The generalization to the case of fermions is made by using Grassmann variables.
We investigate multipartite entanglement in a non-interacting fermion gas, as a function of fermion separation, starting from the many particle fermion density matrix. We prove that all multiparticle entanglement can be built only out of two-fermion entanglement. Although from the Pauli exclusion principle we would always expect entanglement to decrease with fermion distance, we surprisingly find the opposite effect for certain fermion configurations. The von Neumann entropy is found to be proportional to the volume for a large number of particles even when they are arbitrarily close to each o…
In this four-part prospectus, we first give a brief introduction to the motivation for studying entanglement entropy and some recent development. Then follows a summary of our recent work about entanglement entropy in states with traditional long-range order. After that we demonstrate calculation of entanglement entropy in both one-dimensional spin-less fermionic systems as well as bosonic systems via different approaches, and connect them using one-dimensional bosonization. In the last part, we briefly sketch the idea of bosonization in high-dimensions, and discuss the possibility and advanta…
We study multipartite entanglement in a system consisting of indistinguishable fermions. Specifically, we have proposed a geometric entanglement measure for N spin-1/2 fermions distributed over 2L modes (single particle states). The measure is defined on the 2L qubit space isomorphic to the Fock space for 2L single particle states. This entanglement measure is defined for a given partition of 2L modes containing m >= 2 subsets. Thus this measure applies to m <= 2L partite fermionic system where L is any finite number, giving the number of sites. The Hilbert spaces associated with these subsets…
Since Fermions are based on anti-commutation relations, their entanglement can not be studied in the usual way, such that the available theory has to be modified appropriately. Recent publications consider in particular the structure of separable and of maximally entangled states. In this talk we want to discuss local operations and entanglement distillation from bipartite, Fermionic systems. To this end we apply an algebraic point of view where algebras of local observables, rather than tensor product Hilbert spaces play the central role. We apply our scheme in particular to Fermionic Gaussia…
Entanglement criteria for general (pure or mixed) states of systems consisting of two identical fermions are introduced. These criteria are based on appropriate inequalities involving the entropy of the global density matrix describing the total system, on the one hand, and the entropy of the one particle reduced density matrix, on the other one. A majorization-related relation between these two density matrices is obtained, leading to a family of entanglement criteria based on Rényi's entropic measure. These criteria are applied to various illustrative examples of parametrized families of mix…
We study genuine tripartite entanglement shared among the spins of three localized fermions in the non-interacting Fermi gas at zero temperature. Firstly, we prove analytically with the aid of entanglement witnesses that in a particular configuration the three fermions are genuinely tripartite entangled. Then various three-fermion configurations are investigated in order to quantify and calculate numerically the amount of genuine tripartite entanglement present in the system. Further we give a lower and an upper limit to the maximum diameter of the three-fermion configuration below which genui…
We demonstrate, in the context of quadratic fermion lattice models in one and two spatial dimensions, the potential of entanglement renormalization (ER) to define a proper real-space renormalization group transformation. Our results show, for the first time, the validity of the multi-scale entanglement renormalization ansatz (MERA) to describe ground states in two dimensions, even at a quantum critical point. They also unveil a connection between the performance of ER and the logarithmic violations of the boundary law for entanglement in systems with a one-dimensional Fermi surface. ER is reca…
We describe an efficient theoretical criterion suitable for the evaluation of the tripartite entanglement of any mixed three-boson or -fermion state, based on the notion of the entanglement of particles for bipartite systems of identical particles. Our approach allows one to quantify the accessible amount of quantum correlations in the systems without any violation of the local particle number superselection rule. A generalization of the tripartite negativity is here applied to some correlated systems including the continuous-time quantum walks of identical particles (both for bosons and fermi…
The fermionization regime and entanglement correlations of two distinguishable harmonically confined fermions interacting via a zero-range potential is addressed. We present two alternative representations of the ground state that we associate with two different types of one-dimensional spaces. These spaces, in turn, induce different correlations between particles and thus require a suitable definition of entanglement. We find that the entanglement of the ground state is strongly conditioned by those one-dimensional space features. We also find that in the strongly attractive regime the relati…
We are motivated by tripartite entanglement for fermions. While GHZ or W states involve 3-fold intrication, we consider here piecewise intrication of 3 fermions in ${\bf C}^2$, namely of type $ab+bc+ca$. Before interaction with Stern-Gerlach apparatus, qu-bits are distinguishable; at the output however they turn into un-distinguishable particles, whose anti-symmetric wave function is of the form $\det(b-a,c-a)$ (affine determinant). More generally, $d+1$ intricated fermions in ${\bf C}^d$ can be represented by the anti-symmetric wave function $\det(a_1-a_0,a_2-a_0,\cdots,a_d-a_0)$. We investig…
For both cases with and without interactions, bipartite entanglement of two-fermions from a Fermi gas in a trap is investigated. We show how the entanglement depends on the locations of the two fermions and the total number of particles. Fermions at the verge of trap have longer entanglement distance (beyond it, the entanglement disappears) than those in the center. We derive a lower limitation to the average overlapping for two entangled fermions in the BCS ground state, it is shown to be $\sqrt{Q/2M}$, a function of Cooper pair number $Q$ and total number of occupied energy levels $M$.
We show that one-body entanglement, which is a measure of the deviation of a pure fermionic state from a Slater determinant (SD) and is determined by the mixedness of the single-particle density matrix (SPDM), can be considered as a quantum resource. The associated theory has SDs and their convex hull as free states, and number conserving fermion linear optics operations (FLO), which include one-body unitary transformations and measurements of the occupancy of single-particle modes, as the basic free operations. We first provide a bipartitelike formulation of one-body entanglement, based on a …
Free fermions on Hamming graphs $H(d,q)$ are considered and the entanglement entropy for two types of subsystems is computed. For subsets of vertices that form Hamming subgraphs, an analytical expression is obtained. For subsets corresponding to a neighborhood, i.e. to a set of sites at a fixed distance from a reference vertex, a decomposition in irreducible submodules of the Terwilliger algebra of $H(d,q)$ also yields a closed formula for the entanglement entropy. Finally, for subsystems made out of multiple neighborhoods, it is shown how to construct a block-tridiagonal operator which commut…
The pairwise entanglement, measured by concurrence and geometric phase in $d=2,3$-dimensional free-fermion lattice systems have been studied for the ground state in this paper. Their derivation with respect to the external parameter show the singularity closed to the phase transition points, and can be used to detect the phase transition in this model. Furthermore our studies show for the free-fermion model that both concurrence and geometric phase shows the intimate connection with the correlation functions. The possible relation between concurrence and geometric phase has been also discussed…
Information that is stored in quantum-mechanical systems can be easily lost because of the interaction with the environment in a process known as decoherence. Possible physical implementations of many processes in quantum information theory involve systems of identical particles, whence comprehension of the dynamics of entanglement induced by decoherence processes in identical-particle open systems becomes relevant. Here we study the effects, and concomitant entanglement evolution, arising from the interaction between a system of two identical fermions and the environment, for two paradigmatic…
Quantum cluster states and entangled state analyzers are essential to measurement-based quantum computing. We propose to generate a quantum cluster-state and to make multipartite entanglement analyzer by using noninteracting free electrons or conduction electrons in quantum dots, based on polarizing beam splitters, charge detectors and single-spin rotations. Our schemes are deterministic without the need of qubit-qubit interaction.
In this work, we report that the Hawking radiation effect on fermions is fundamentally different from the case of scalar particles. Intrinsic properties of fermions (exclusion principle and spin) affect strongly the interaction of fermions with both Hawking radiation and metric of the spacetime. In particular we have found the following: first, while the fermion vacuum state seen by the Rindler observer is an entangled state in which the right and left Rindler wedge states appear in correlated pairs as in the case of the scalar particles, the entanglement disappears in the excited state due to…
The present paper is both a review on the Feynman problem, and an original research presentation on the relations between Fermionic theories and qubits theories, both regarded in the novel framework of operational probabilistic theories. The most relevant results about the Feynman problem of simulating Fermions with qubits are reviewed, and in the light of the new original results the problem is solved. The answer is twofold. On the computational side the two theories are equivalent, as shown by Bravyi and Kitaev (Ann. Phys. 298.1 (2002): 210-226). On the operational side the quantum theory of…
We analyze the problem of quantifying entanglement in pure and mixed states of fermionic systems with fixed number parity yet not necessarily fixed particle number. The "mode entanglement" between one single-particle level and its orthogonal complement is first considered, and an entanglement entropy for such a partition of a particular basis of the single-particle Hilbert space $\mathcal H$ is defined. The sum over all single-particle modes of this entropy is introduced as a measure of the total entanglement of the system with respect to the chosen basis and it is shown that its minimum over …
Under the historical definition of entanglement, namely that encountered in Einstein--Podolski--Rosen (EPR)--type experiments, it is shown that a particular Slater determinant is entangled. Thus a definition which holds that any Slater determinant is unentangled, found in the literature, is inconsistent with the historical definition. A generalization of the historical definition that embodies its meaning as quantum correlation of observables, in the spirit of the work of many others, but is much simpler and more physically transparent, is presented.
The problem of the choice of tensor product decomposition in a system of two fermions with the help of Bogoliubov transformations of creation and annihilation operators is discussed. The set of physical states of the composite system is restricted by the superselection rule forbidding the superposition of fermions and bosons. It is shown that the Wootters concurrence is not proper entanglement measure in this case. The explicit formula for the entanglement of formation is found and its dependence on tensor product decompositions of the Hilbert space is discussed. It is shown that the set of se…
The concepts of concurrence and mode concurrence are the measures of entanglement for spin-1/2 and spinless fermion systems respectively. Based on the Jordan-Wigner transformation, any spin-1/2 system is always associated with a fermion system (called counterpart system). The comparison of concurrence and mode concurrence can be made with the aid of the Marshall's sign rule for the ground states of spin-1/2 $XXZ$ and spinless fermion chain systems. We observe that there exists an inequality between concurrence and mode concurrence for the ground states of the two corresponding systems. The spi…
Non-equilibrium quantum dynamics represents an emerging paradigm for condensed matter physics, quantum information science, and statistical mechanics. Strongly interacting Rydberg atoms offer an attractive platform to study driven-dissipative dynamics of quantum spin models with long-range order. Here, we explore the conditions under which stationary many-body entanglement persists with near-unit fidelity and high scalability. In our approach, coherent many-body dynamics is driven by Rydberg-mediated laser transitions, while atoms at the lattice boundary reduce the entropy of the many-body sta…
The time dependent quantum Monte Carlo method for fermions is introduced and applied for calculation of entanglement of electrons in one-dimensional quantum dots with several spin-polarized and spin-compensated electron configurations. The rich statistics of wave functions provided by the method allows one to build reduced density matrices for each electron and to quantify the spatial entanglement using measures such as quantum entropy by treating the electrons as identical or distinguishable particles. Our results indicate that the spatial entanglement in parallel-spin configurations is rathe…
We show how to detect entanglement with criteria built from simple two-body correlation terms. Since many natural Hamiltonians are sums of such correlation terms, our ideas can be used to detect entanglement by energy measurement. Our criteria can straightforwardly be applied for detecting different forms of multipartite entanglement in familiar spin models in thermal equilibrium.
In this paper we describe a method for finding polynomial invariants under Stochastic Local Operations and Classical Communication (SLOCC), for a system of delocalized fermions shared between different parties, with global particle number conservation as the only constraint. These invariants can be used to construct entanglement measures for different types of entanglement in such a system. It is shown that the invariants, and the measures constructed from them, take a nonzero value only if the state of the system allows for the observation of Bell-nonlocal correlations. Invariants of this kin…
With an easily applicable criterion based on permutation symmetries of (identically prepared) replicas of quantum states we identify distinct entanglement classes in high-dimensional multi- partite systems. The different symmetry properties of inequivalent states provide a rather intuitive picture of the otherwise very abstract classification of many-body entangled states.
We study the properties of entanglement entropy among scattering particles as observed from different inertial moving frames, based on an exemplary QED process $e^+e^-\rightarrowμ^+μ^-$. By the explicit calculation of the Wigner rotation, the entanglement entropy of scattering particles is found to be Lorentz invariant. We also study the behavior of the entanglement between spin degrees of freedom for scattering particles in moving frames. This quantity, being found to change with different inertial reference frames, does not exhibit as a Lorentz invariant.
Unlike bosons, fermions always have a non-trivial entanglement. Intuitively, Slater determinantal states should be the least entangled states. To make this intuition precise we investigate entropy and entanglement of fermionic states and prove some extremal and near extremal properties of reduced density matrices of Slater determinantal states.
We show that with respect to any bipartite division of modes, pure fermion gaussian states display the same type of structure in its entanglement of modes as that of the BCS wave function, namely, that of a tensor product of entangled two-mode squeezed fermion states. We show that this structure applies to a wider class of "isotropic" mixed fermion states, for which we derive necessary and sufficient conditions for mode entanglement.
We present a new algorithm for calculating the Renyi entanglement entropy of interacting fermions using the continuous-time quantum Monte Carlo method. The algorithm only samples interaction correction of the entanglement entropy, which by design ensures efficient calculation of weakly interacting systems. Combined with Monte Carlo reweighting, the algorithm also performs well for systems with strong interactions. We demonstrate the potential of this method by studying the quantum entanglement signatures of the charge-density-wave transition of interacting fermions on a square lattice.
Entanglement plays a central role in numerous fields of quantum science. However, as one departs from the typical "Alice versus Bob" setting into the world of indistinguishable fermions, it is not immediately clear how the concept of entanglement is defined among these identical particles. Our endeavor to recover the notion of subsystems, or mathematically speaking, the tensor product structure of the Hilbert space, lead to two natural pictures of defining fermionic entanglement: the particle picture and the mode picture. In the particle picture, entanglement characterizes the deviation of a f…
We propose a new approach in the investigation and detection of axion and axion-like particles based on the study of the entanglement for two interacting fermions. We study a system made of two identical fermions with spin-1/2, and we show that fermion-fermion interaction mediated by axions leads to a non-zero entanglement between the fermions. An entanglement measurement can reveal the interaction, providing an indirect evidence of the existence of axions. We discuss how the other interactions affect the entanglement, and how to isolate the axion contribution. Particular care is devoted to th…
We show that the single-site entanglement of a generic spin-1/2 fermionic lattice system can be used as a reliable marker of a finite-order quantum phase transition, given certain provisos. We discuss the information contained in the single-site entanglement measure, and provide illustrations from the Mott-Hubbard metal-insulator transitions of the one-dimensional Hubbard model, and the Hubbard model with long-range hopping.
Entanglement in fermion many-body systems is studied using a generalized definition of separability based on partitions of the set of observables, rather than on particle tensor products. In this way, the characterizing properties of non-separable fermion states can be explicitly analyzed, allowing a precise description of the geometric structure of the corresponding state space. These results have direct applications in fermion quantum metrology: sub-shot noise accuracy in parameter estimation can be obtained without the need of a preliminary state entangling operation.
We propose a measure of interaction-induced ground state entanglement in many-fermion systems that is experimentally accessible. It is formulated in terms of cross-correlations of currents through resonant fermion levels weakly coupled to the probed system. The proposed entanglement measure vanishes in the absence of many-body interactions and it is related to measures of occupation number entanglement. We evaluate it for two examples of interacting electronic nanostructures.
The use of quantum entanglement to study condensed matter systems has been flourishing in critical systems and topological phases. Additionally, using real-space entanglement entropies and entanglement spectra one can characterize localized and delocalized phases of disordered fermion systems. Here we instead propose the momentum-space entanglement spectrum as a means of characterizing disordered models. We show that localization in 1D arises from the momentum space entanglement between left and right movers and illustrate our methods using explicit models with spatially correlated disorder th…
We present analytic and numerical calculations on the bipartite entanglement entropy in fractional quantum Hall states of the fermionic Laughlin sequence. The partitioning of the system is done both by dividing Landau level orbitals and by grouping the fermions themselves. For the case of orbital partitioning, our results can be related to spatial partitioning, enabling us to extract a topological quantity (the `total quantum dimension') characterizing the Laughlin states. For particle partitioning we prove a very close upper bound for the entanglement entropy of a subset of the particles with…
Nonlocal entanglement between two quantum dots can be generated through Majorana fermions. The two Majorana fermions at the ends of an one-dimensional topological superconductor form a nonlocal fermion level, coupling to the occupation states of two quantum dots put close to the two ends, and the entire system will come into an entangled state. After introducing a charging energy by a capacitor, entanglement of the entire system can manifest itself through the nonlocal entanglement between the two quantum dots. That is, when measuring the electron occupations of the quantum dots, the measureme…
When two identical fermions exchange their positions, their wave function gains phase factor $-1$. We show that this distance-independent effect can induce nonlocal entanglement in one-dimensional (1D) electron systems having Majorana fermions at the ends. It occurs in the system bulk and has nontrivial temperature dependence. In a system having a single Majorana at each end, the nonlocal entanglement has a Bell-state form at zero temperature and decays as temperature increases, vanishing suddenly at certain finite temperature. In a system having two Majoranas at each end, it is in a cluster-s…
We study quantum information aspects of the fermionic entanglement negativity recently introduced in [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 W and Greenberger-Horne-Zeilinger states and compare the results of bosonic and …
We analyze the operational meaning of the residual entanglement in non-inertial fermionic systems in terms of the achievable violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality. We demonstrate that the quantum correlations of fermions, which were previously found to survive in the infinite acceleration limit, cannot be considered to be non-local. The entanglement shared by an inertial and an accelerated observer cannot be utilized for the violation of the CHSH inequality in case of high accelerations. Our results are shown to extend beyond the single mode approximation commonly used …
Composite structure of particles somewhat modifies their statistics, compared to the pure Bose- or Fermi-ones. The spin-statistics theorem, so, is not valid anymore. Say, $π$-mesons, excitons, Cooper pairs are not ideal bosons, and, likewise, baryons are not pure fermions. In our preceding papers, we studied bipartite composite boson (i.e. quasiboson) systems via a realization by deformed oscillators. Therein, the interconstituent entanglement characteristics such as entanglement entropy and purity were found in terms of the parameter of deformation. Herein, we perform an analogous study of co…
