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arXiv:0705.1103 · Physical Review A

Entanglement in fermionic systems

Mari Carmen Bañuls, J. I. Cirac, Michael M. Wolf

Published Aug 13, 2007• 150 citations
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Abstract

The anticommuting properties of fermionic operators, together with the presence of parity conservation, affect the concept of entanglement in a composite fermionic system. Hence different points of view can give rise to different reasonable definitions of separable and entangled states. Here we analyze these possibilities and the relationship between the different classes of separable states. The behavior of the various classes when taking multiple copies of a state is also studied, showing that some of the differences vanish in the asymptotic regime. In particular, in the case of only two fermionic modes all the classes become equivalent in this limit. We illustrate the differences and relations by providing a complete characterization of all the sets defined for systems of two fermionic modes. The results are applied to Gibbs states of infinite chains of fermions whose interaction corresponds to a $XY$ Hamiltonian with transverse magnetic field.

DOI
10.1103/physreva.76.022311

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The anticommuting properties of fermionic operators, together with the presence of parity conservation, affect the concept of entanglement in a composite fermionic system. Hence different points of view can give rise to different reasonable definitions of separable and entangled states. Here we analyze these possibilities and the relationship between the different classes of separable states. The behavior of the various classes when taking multiple copies of a state is also studied, showing that some of the differences vanish in the asymptotic regime. In particular, in the case of only two fermionic modes all the classes become equivalent in this limit. We illustrate the differences and relations by providing a complete characterization of all the sets defined for systems of two fermionic modes. The results are applied to Gibbs states of infinite chains of fermions whose interaction corresponds to a XYXYXY Hamiltonian with transverse magnetic field.
The characterization of entanglement is a fundamental issue for Quantum Information Theory. But the definition of entanglement depends on the notion of locality, and thus on the tensor product structure of the state space of the composite system. This notion is affected by the presence of superselection rules that restrict the accessible Hilbert space to a direct sum of subspaces. Indistinguishability of particles imposes one such restriction, namely to totally symmetric or totally antisymmetric states. The entanglement can in this case be defined with respect to partitions of modes in the second quantization formalism. For fermionic systems the Fock space of m modes is isomorphic to the space of m qubits, but the action of creation and annihilation operators is not local, due to their anticommutation. Conservation of the parity of fermion number imposes another relevant superselection rule. It requires that local physical observables commute with the local parity operator. Taking into account the considerations above, it is possible to define the set of separable states or equivalently the concept of entanglement for fermionic systems in a number of ways. Here we analyze systematically these possibilities and the relation among the various sets of separable states. We also discuss the behavior of the different classes when taking several copies of the state, as well as the characterization of the sets in terms of the usual criteria regarding the tensor product.
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.
Due to their great importance, both from the fundamental and from the practical points of view, it is imperative that the various facets of the concepts of information and entanglement are explored systematically in connection with diverse physical systems and processes. These concepts are at the core of the emerging field of the Physics of Information. In this Thesis I investigate some aspects of the dynamics of information in both classical and quantum mechanical systems and then move on to explore entanglement in fermion systems by searching for novel ways to classify and quantify entanglement in fermionic systems. In Chapter 1 a brief review of the different information and entropic measures as well as of the main evolution equations of classical dynamical and quantum mechanical systems is given. The conservation of information as a fundamental principle both at the classical and quantum levels, and the implications of Landauer's theorem are discussed in brief. An alternative and more intuitive proof of the no-broadcasting theorem is also provided. Chapter 2 is a background chapter on quantum entanglement, where the differences between the concept of entanglement in systems consisting of distinguishable subsystems and the corresponding concept in systems of identical fermions are emphasized. Different measures of entanglement and relevant techniques such as majorization, are introduced. To illustrate some of the concepts reviewed here I discuss the entanglement properties of an exactly soluble many-body model which was studied in paper (E) of the publication list corresponding to the present Thesis. An alternative approach to the characterization of quantum correlations, based on perturbations under local measurements, is also briefly reviewed. The use of uncertainty relations as entanglement indicators in composite systems having distinguishable subsystems is then examined in some detail. Chapter 3 is based on papers (A) and (B) of the list of publications. Extended Landauer-like principles are developed, based amongst others on the conservation of information of divergenceless dynamical systems. Conservation of information within the framework of general probabilistic theories, which include the classical and quantum mechanical probabilities as particular instances, is explored. Furthermore, Zurek's information transfer theorem and the no-deleting theorem are generalized. Chapter 4 is based on articles (C) and (D) mentioned in the publication list, and investigates several separability criteria for fermions. Criteria for the detection of entanglement are developed based either on the violation of appropriate uncertainty relations or on inequalities involving entropic measures. Chapter 5 introduces an approach for the characterization of quantum correlations (going beyond entanglement) in fermion systems based upon the state disturbances generated by the measurement of local observables. Chapter 6 summarizes the conclusions drawn in the previous chapters. The work leading up to this Thesis has resulted in five publications in peer reviewed science research journals.
We study characterization of separable (classically correlated) states for composite systems of distinguishable fermions. In the computation of entanglement formation for such systems where located subsystems are coupled by the canonical anticommutation relations (not by tensor product), the state decompositions to be taken should respect the univalence superselection rule. (The usual entanglement formation taking all the state decompositions measures non-separability of states between a given subsystem and its commutant and cannot detect non-separability between the CAR pair under our consideration properly.) We prove that any fermion hopping terms always induce non-separability. This feature contrasts with the case of tensor product systems where the states with bosonic hopping correlation may or may not be separable. If we transform a given bipartite fermion system into a tensor product system by Jordan-Klein-Wigner transformation, then any separable state for the former is also separable for the latter. We provide a class of U(1) gauge invariant mixed states that are non-separable for the former, however, separable for the latter. ∗ Mailing address:
We determine the conditions under which general dimer-type spin chains with XYZXYZXYZ couplings of arbitrary range in a general transverse field will exhibit an exactly separable parity-breaking eigenstate. We also provide sufficient conditions which ensure that it will be a ground state. We then examine the exact side limits at separability of the entanglement between any two spins in a finite chain, showing that in the vicinity of separability, the system will loose all signatures of dimerization, with pairwise entanglement approaching infinite range and becoming independent of separation and interaction range. The possibility of a nonuniform exactly separable ground state induced by an alternating field is also shown. As illustration, we examine the behavior of the pairwise entanglement in a finite XYXYXY dimer chain under a uniform as well as alternating field. Related aspects of the magnetization are also discussed.