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A simple expression to compute the quantum discord between two orbitals in fermion systems is derived using the parity superselection rule. As the correlation between orbitals depends on the basis chosen, we discuss a special orbital basis, the natural one. We show that quantum correlations between natural orbital pairs disappear when the pairing tensor is zero, i.e. the particle number symmetry is preserved. The Hartree-Fock orbitals within a Slater determinant state, a Hartree-Fock-Bogoliubov quasiparticle orbitals in a quasiparticle vacuum, or the ground state of a Hamiltonian with particle symmetry and their corresponding natural orbitals are some relevant examples of natural basis and their corresponding states. Since natural orbitals have that special property, we seek for the quantum discord in non-natural orbital basis. We analyze our findings in the context of the Lipkin-Meshkov-Glick and Agassi models.Quantum correlations have been a central field of research since the inception of quantum mechanics Einstein et al. (1935); Schrödinger (1935). They are a fundamental feature of the quantum theory and give rise to many interesting phenomena like those observed in the fields of quantum cryptography Nielsen and Chuang (2011); Ekert (1991); Holevo (2012); Watrous (2018), quantum teleportation Bennett et al. (1993); Braunstein and Kimble (1998); Pirandola et al. (2015), quantum phase transitions in many body systems Sachdev (2001); Vojta (2003); Allegra et al. (2011), etc.They can be studied from different points of view. For instance, from a many body perspective, it is known that, if we solve a many body Hamiltonian through mean field techniques, such as the Hartree-Fock or Hartree-Fock-Bogoliubov method, the ground state will, in general, not preserve the symmetries of the Hamiltonian Ring and Schuck (2004). This spontaneous breaking of symmetries is interpreted as a way to catch important correlations of the exact ground state while preserving the simple mean field picture. From a quantum information point of view, quantum correlations show up when we analyze the state of a given partition in a quantum system. For instance, if we have a pure state living in a Hilbert space with some tensor product structure, we can use the von Neumann entropy of one of the marginals in order to quantify the amount of entanglement between parties Nielsen and Chuang (2011). However, when we try to join both perspectives, some subtleties arise. For example, when we try to quantify quantum correlations through partial traces, we usually need a Hilbert space with a tensor product structure. Nonetheless, in dealing with many body systems it is customary to deal with identical fermionic particles, and due to the antisymmetrization principle, the tensor structure is lost Friis (2016). Moreover, as it is physically impossible to distinguish among identical particles, it is inconsistent to compute correlations between them through, for example, the von Neumann entropy of the marginals. While it’s true that those subtleties are solved by defining a fermionic partial trace Friis et al. (2013), or quantifying the particle entanglement through the one body entropy Kanada-En’yo (2015), in general quantum information concepts are not directly applicable in fermionic many body systems.A very powerful measure of quantum correlations is the so-called quantum discord Ollivier and Zurek (2001). It quantifies all quantum correlations beyond entanglement and allows us to differentiate between classical and quantum correlations Ollivier and Zurek (2001). However, its calculation requires to perform arbitrary projective measurements in one of the subsystems and therefore it implicitly requires the existence of a tensor product structure. Moreover, it is in general hard to compute, both analytically and numerically, due to the variational process implicit in its definition Girolami and Adesso (2011); Huang (2014).In this work, we aim to study correlations in many body fermionic systems from a quantum information perspective through the analysis of the quantum discord. Specifically, we derive a very simple expression in order to compute the quantum discord between two fermionic orbitals in a arbitrary mixed state, and we apply it the characterization of some well known benchmarking models. The paper is structured as follows. In Sec. II we briefly introduce the concept of quantum discord. In Sec. III we derive an expression to compute the quantum discord between pair of orbitals in fermion systems. In IV and V we discuss some properties of quantum discord related to the orbital basis. In VI we compute it in the context of the Agassi and Lipkin-Meshkov-Glick models. Finally, in Sec. VII and the appendix we summarize the results obtained and we discuss some connections with other results in the literature, respectively.Given the Hilbert space ℋℋ\mathcalHcaligraphic_H of a quantum system, let us assume there exists a bi-partition ℋ=ℋ(
