**From left to right: Stefania Sciara, Rosario Lo Franco, Giuseppe Compagno**

**Authors: Stefania Sciara**

^{1,2}, Rosario Lo Franco^{2,3}, Giuseppe Compagno^{2}**Affiliation:**

^{1}INRS-EMT, Varennes, Québec J3X 1S2, Canada,

^{2}Dipartimento di Fisica e Chimica, Università di Palermo, Italy,

^{3}Dipartimento di Energia, Ingegneria dell'Informazione e Modelli Matematici, Università di Palermo, Italy.The Schmidt decomposition is an important mathematical tool which has been already utilized during the early stages of quantum theory by Schrödinger in the context of quantum measurements [1-3]. This tool allows to determine the set of measurements on one part of the system such that the measurement outcome on the other part is determined, in the sense that to each outcome of the first measurement it corresponds a unique outcome for the second measurement. Schmidt decomposition has been shown to be at the heart of quantum information theory, quantifying entanglement in bipartite systems. It has been also widely employed in the context of Einstein-Podolski-Rosen (EPR) paradox, Bell non-locality and black-hole physics [2-4].

Every element of this decomposition consists unavoidably of two independent subsystems. Application of the Schmidt decomposition to identical particles is therefore hindered by the fact that overlapping particles can never be considered independent. In fact, despite its wide utilization in systems of distinguishable particles, the Schmidt decomposition has remained debated for identical particles [5,6]. For instance, it is well known that for distinguishable particles this tool assesses the entanglement of the system by the von Neumann entropy of the reduced density matrix, whose eigenvalues are the squares of the Schmidt coefficients appearing in the decomposition [3]. Differently, in the case of identical particles, it has been claimed that the relationship between the Schmidt coefficients and the eigenvalues of the reduced density matrix breaks down [6]. In strict connection with this issue, the partial trace operation to get the reduced state has not been considered suitable for quantifying the entanglement of pure states of identical particles [6-8].

We recall that, in Nature, particles are of different types, all particles of each type (electrons, protons, photons and so on) being identical. In the quantum world, the identity of particles gives rise to a new characteristic with respect to the classical world, that is the indistinguishability among particles of the same type [9,10]. This exclusive quantum trait leads to fundamental properties of matter such that particles can be of two classes, named bosons and fermions. Moreover, at variance with the case of distinguishable particles, when identical particles have wave functions that spatially overlap, they can never be taken as independent of each other [11,12].

The latter behavior is an essential requisite in determining features like quantum correlations (e.g., entanglement) among the particles themselves and in the theory of measurement [1]. Nevertheless, identical particles constitute the building blocks of quantum information and computation theory, being present in Bose-Einstein condensates [13,14], quantum dots [15,16], superconducting circuits [17] and optical setups [18,19]. It is thus important to have trustable methods and tools to characterize the quantum features of composite systems of identical particles under these general conditions.

A first step towards this goal has been provided by a recent non-standard approach [12] which deals with systems of identical particles within a particle-based description (that is, in terms of particle states) without resorting to the usual practice to assign fictitious labels to the particles [9,10], which render the latter distinguishable removing their indistinguishability. In this way, the ambiguity arising from the introduction of these labels in evaluating quantum correlations in identical particle systems is avoided. This method has, in fact, provided a way to calculate partial trace and von Neumann entropy for identical particles. Using this new approach, in a recent work [20] we have been able to demonstrate that the Schmidt decomposition is universal, meaning that it is also obtainable for an arbitrary state of indistinguishable particles (bosons or fermions) under general conditions of spatial overlap. Thanks to this achievement, the amount of entanglement present in identical particle systems in pure states can be evaluated by the von Neumann entropy of the reduced density matrix, as occurs in the case of distinguishable particles.

We have used the Schmidt decomposition to analyze some paradigmatic states of two identical particles to find the suitable measurement basis for unveiling their entanglement properties. These systems are illustrated in Figure 1 and represent simple yet effective examples which make it emerge the effect of particle identity. The first one is the well-known condition of two identical qubits with opposite pseudospins in spatially separated locations, denoted with left (L) and right (R) (see Fig. 1a).

**Figure 1. (a) Two identical qubits in two spatially separated places with opposite pseudospins. (b) Two identical qubits in the same spatial mode with arbitrary pseudospins. (c) Two identical qutrits (three-level quantum systems) in the same spatial mode. The shaded ellipses indicate that the particles are entangled. Figure from Ref. [20].**

Indeed, we have found that the Schmidt decomposition and the corresponding von Neumann entropy supply the results which are physically expected in this situation, such as zero entanglement for a product (separable) state of the two particles and maximal entanglement for a Bell state. We have also explicitly shown that nonlocal measurements induce entanglement in a state of distant identical particles. This means that, in principle, identical particles cannot be considered completely independent even when they are spatially separated. Application of the Schmidt decomposition to a system of two boson qubits in the same site (see Fig. 1b) has evidenced that the entanglement increases as the two internal states tend to be orthogonal, as displayed in the plot of Fig. 2. This finding is a generalization of a previous result obtained only for two identical qubits in the same spatial mode with orthogonal internal states (or pseudospins, that is θ = π in the plot of Fig. 2) [12].

**Figure 2. Entanglement between the pseudospins of two identical particles in the same site, quantified by the von Neumann entropy, as a function of the angle θ between the directions of the two pseudospins (internal states). Figure from Ref. [20].**

Two identical (boson) qutrits, that is three-level quantum systems, constitute a system of interest being promising candidates for quantum processors thanks to their good capacity to store quantum information [21,22]. We consider the qutrits in the same site, that is under the condition of complete spatial overlap, as depicted in Fig. 1c. Our method easily determines the two-qutrit entanglement for given combinations of their internal states and provides a physical interpretation. We remark that the entanglement found for the two qutrits is different from that obtained for the same system by an alternative approach, based on a so-called subalgebra technique [8]. The origin of this difference in the entanglement measure requires further investigation, for instance by comparing the two theoretical approaches or by experimental verifications.

We now briefly discuss the practical aspect concerning the possibility to exploit the entanglement of identical particles, as identified by the Schmidt decomposition we have introduced. In the context of quantum information processing with distinguishable particles, a well-established resource theory is based on local operations and classical communication (LOCC), where each particle is individually addressed [1-3]. On the other hand, a peculiar trait of indistinguishable particles is that it is not possible to operate on a given individual particle. This characteristic in general hinders the manipulation of composite systems of identical particles for desired tasks of quantum information and computation. Nevertheless, it is known that the entanglement quantified by the von Neumann entropy can be extracted from the state of identical particles and then utilized in a conditional way by LOCC [23].

In conclusion, the universal character of the Schmidt decomposition shown in our work allows a faithful treatment of the entanglement of composite systems of identical particles and opens the way to more general investigations of collective properties of such systems.

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