Schmidt Decomposition Made Universal to Unveil the Entanglement of Identical Particles

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

Authors: Stefania Sciara^1,2, Rosario Lo Franco^2,3, Giuseppe Compagno²

Affiliation:
¹INRS-EMT, Varennes, Québec J3X 1S2, Canada,
²Dipartimento di Fisica e Chimica, Università di Palermo, Italy,
³Dipartimento 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.

References:
[1] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, Karol Horodecki, “Quantum entanglement”, Review of Modern Physics, 81, 865 (2009). Abstract.
[2] John Preskill, “Lectures notes for physics 229: Quantum information and computation” (1998) Link.
[3] Michael A. Nielsen and Isaac L. Chuang, “Quantum Computation and Quantum Information” (Cambridge University Press, Cambridge, 2000).
[4] E.D. Belokolos, M.V. Teslyk, “Scalar field entanglement entropy of a Schwarzschild black hole from the Schmidt decomposition viewpoint”, Classical and Quantum Gravity, 26, 235008 (2009). Abstract.
[5] R. Paškauskas, L. You, “Quantum correlations in two-boson wave functions”, Physical Review A, 64, 042310 (2001). Abstract.
[6] Malte C Tichy, Florian Mintert, Andreas Buchleitner, “Essential entanglement for atomic and molecular physics”, Journal of Physics B: Atomic, Molecular and Optical Physics. 44, 192001 (2011). Abstract.
[7] GianCarlo Ghirardi, Luca Marinatto, Tullio Weber, “Entanglement and properties of composite quantum systems: a conceptual and mathematical analysis”, Journal of Statistical Physics 108, 49 (2002). Abstract.
[8] A. Balachandran, T. Govindarajan, Amilcar R. de Queiroz, A. Reyes-Lega, “Entanglement and particle identity: A unifying approach”, Physical Review Letters, 110, 080503 (2013). Abstract.
[9] Asher Peres, “Quantum Theory: Concepts and Methods” (Springer, Dordrecht, The Netherlands, 1995).
[10] Claude Cohen-Tannoudji, Bernard Diu, Franck Laloe, “Quantum mechanics, Vol. 2” (Wiley-VCH, Paris, France, 2005).
[11] John Schliemann, J. Ignacio Cirac, Marek Kuś, Maciej Lewenstein, Daniel Loss, “Quantum correlations in two-fermion systems”, Physical Review A, 64, 022303 (2001). Abstract.
[12] Rosario Lo Franco, Giuseppe Compagno, “Quantum entanglement of identical particles by standard information-theoretic notions”, Scientific  Reports, 6, 20603 (2016). Abstract.
[13] Immanuel Bloch, Jean Dalibard, Wilhelm Zwerger, “Many-body physics with ultracold gases”, Review of Modern Physics, 80, 885 (2008). Abstract.
[14] Marco Anderlini, Patricia J. Lee, Benjamin L. Brown, Jennifer Sebby-Strabley, William D. Phillips,  J. V. Porto, “Controlled exchange interaction between pairs of neutral atoms in an optical lattice”, Nature 448, 452 (2007). Abstract.
[15] Michael H. Kolodrubetz, Jason R. Petta, “Coherent holes in a semiconductor quantum dot”, Science 325, 42–43 (2009). Abstract.
[16] Frederico Martins, Filip K. Malinowski, Peter D. Nissen, Edwin Barnes, Saeed Fallahi, Geoffrey C. Gardner, Michael J. Manfra, Charles M. Marcus, Ferdinand Kuemmeth, “Noise suppression using symmetric exchange gates in spin qubits”, Physical Review Letters, 116, 116801 (2016). Abstract.
[17] R. Barends, L. Lamata, J. Kelly, L. García-Álvarez, A. G. Fowler, A Megrant, E Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Yu Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill,  P. J. J. O’Malley,  C. Quintana,  P. Roushan, A. Vainsencher, J. Wenner, E. Solano, John M. Martinis, “Digital quantum simulation of fermionic models with a superconducting circuit”, Nature Communications, 6, 7654 (2015). Abstract.
[18] Andrea Crespi, Linda Sansoni, Giuseppe Della Valle, Alessio Ciamei, Roberta Ramponi, Fabio Sciarrino, Paolo Mataloni, Stefano Longhi, Roberto Osellame, “Particle statistics affects quantum decay and Fano interference”, Physical Review Letters, 114, 090201 (2015). Abstract.
[19] Christian Reimer, Michael Kues, Piotr Roztocki, Benjamin Wetzel, Fabio Grazioso, Brent E. Little, Sai T. Chu, Tudor Johnston, Yaron Bromberg, Lucia Caspani, David J. Moss, Roberto Morandotti, “Generation of multiphoton entangled quantum states by means of integrated fre-quency combs”, Science 351, 1176 (2016). Abstract.
[20] Stefania Sciara, Rosario Lo Franco, Giuseppe Compagno, “Universality of Schmidt decomposition and particle identity”, Scientific Reports, 7, 44675 (2017). Abstract.
[21] B.P. Lanyon, T.J. Weinhold, N.K. Langford, J.L. O’Brien, K.J. Resch, A. Gilchrist, A.G. White, “Manipulating biphotonic qutrits”, Physical Review Letters, 100, 060504 (2008). Abstract.
[22] K. S. Kumar, A. Vepsalainen, S. Danilin,  G.S. Paraoanu, “Stimulated Raman adiabatic passage in a three-level superconducting circuit”, Nature Communications, 7, 10628 (2016). Abstract.
[23] N. Killoran, M. Cramer, M. B. Plenio, “Extracting entanglement from identical particles”, Physical Review Letters, 112, 150501 (2014). Abstract.

A New Approach to Quantum Entanglement for Identical Particles

Rosario Lo Franco (right) and Giuseppe Compagno (left)

Authors: Rosario Lo Franco¹ and Giuseppe Compagno²

Affiliation:
¹Dipartimento di Energia, Ingegneria dell'Informazione e Modelli Matematici, Università di Palermo, Italy,
²Dipartimento di Fisica e Chimica, Università di Palermo, Italy.

Entanglement for distinguishable particles is well established from a conceptual point of view with standard tools capable to identify and quantify it [1, 2]. This is instead not the case for identical particles, bosons (e.g., photons, atoms) and fermions (e.g., electrons), where particle identity may give place to fictitious contributions to entanglement which has been the origin of a long-standing debate [2-4]. For all practical purposes, when two identical particles are spatially separated, as in experiments with photons in different optical modes or with strongly repelling trapped ions, no ambiguity is possible for which particle has a given property so they can be effectively treated as distinguishable objects [3]: in this case, no physical contribution to entanglement arises due to indistinguishability.

Figure 1: Asymmetric double-well configuration. One particle has a localized (orange) wave function A in the left well L, while one particle has a (blue) wave function B which overlaps with A, being nonzero in both the left well L and the right well R. This is a typical instance where one particle can tunnel from a site to the other one and indistinguishability counts.

This aspect comes from a natural requirement known as cluster decomposition principle stating that distant experiments are not influenced by each other [6]. Otherwise, quantum indistinguishability comes into play when the constituting particles are close enough to spatially overlap. This happens for all the applications of quantum information processing based, for instance, on quantum dot technology with electrons [7,8] or on Bose-Einstein condensates [9,10], where the particles have the possibility to tunnel from a location to the other (Fig. 1). Hence, correctly treating identical particle entanglement, besides its fundamental interest, is a central requirement in quantum information theory. Despite this, the analysis of identical particle entanglement has been suffering both conceptual and technical pitfalls [2-4].

The ordinary approach to deal with identical particles in quantum mechanics textbooks consists in assigning them unobservable labels which give rise to a new fictitious system of distinguishable particles [5]. In order that this new system behaves as the original (bosonic or fermionic) one, only symmetrized or anti-symmetrized states with respect to labels are allowed. The byproduct is that, according to the usual notion of non-separability employed in quantum information theory to define entanglement, such states entangled. Ordinary entanglement measures, such as the von Neumann entropy of the reduced state obtained by partial trace, fail then to be directly applied to these states. In particular, they witness entanglement even for independent separated particles which are clearly uncorrelated and also show contradictory results for bosons and fermions [3].

As a consequence, methods utilizing notions at variance with the ordinary ones adopted for distinguishable particles have been formulated to overcome this issue [3,4]. In any case, these alternative methods remain technically awkward and unsuited to quantify entanglement under general conditions of scalability and wave function overlap. The use of new notions to discuss quantum entanglement for identical particles looks surprising, not less than the introduction of unobservable labels which is in contrast with the quantum mechanical requirement that the state of any physical system is uni-vocally described by a complete set of commuting observables. So far, there has not been general agreement even whether the entanglement between two identical particles in the same site may exist [3, 11, 12]. The characterization of quantum entanglement for identical particles has thus remained problematic, jeopardizing the general understanding and exploitation of composite systems.

In our recent work [13], we provide a straightforward description of entanglement in systems of identical particles, based on simple physical concepts, which unambiguously answers the general question: when and at which degree the identity of quantum particles plays a physical role in determining the entanglement among the particles? This is achieved by introducing a novel approach for identical particles without resorting to fictitious labels, differently from the usual textbook practice. The core of this approach is that the state of several identical particles must be considered a whole entity while the transition probability amplitude between two of such states is expressible in terms of single-particle amplitudes by applying the basic quantum-mechanical superposition principle with no which-way information to alternative paths. Our approach enables the determination of entanglement for both bosons and fermions by the same notions usually adopted in entanglement theory for distinguishable particles, such as the von Neumann entropy of the reduced state. The latter is obtained through the partial trace defined by local single-particle measurements.

Figure 2: Panel A. Entanglement as a function of system parameters for a fixed degree of spatial overlap for bosons (blue dotted line) and fermions (orange dashed line), compared to the corresponding entanglement of nonidentical particles (red solid line). Panel B. Density plot of bosonic entanglement as a function of both relative phase in the system state and degree of spatial overlap.

We have analyzed a system of two identical qubits (two-level systems) with orthogonal internal states (opposite pseudospins). The qubits are supposed to have wave functions (spatial modes) which can overlap at an arbitrary extent. A simple system which realizes this condition is that of the asymmetric double-well configuration illustrated in Fig. 1. When the two particles partially overlap in a spatial region where local single-particle measurements can be done, entanglement depends on their overlap and an ordering emerges for different particle types, fermions or bosons (Fig. 2). Moreover, identical particles are found to be at least as entangled as non-identical ones placed in the same quantum state.

This result suggests that identical particles may be more efficient than distinguishable ones for entanglement-based quantum information tasks. The main findings of this analysis can be summarized as: (i) an absolute degree of entanglement for identical particles, independent of local measurements, can be assigned only when the particles are spatially separated or in the same site; (ii) the act of bringing identical particles into overlapping spatial modes creates an “entangling gate” whose effectiveness depends on the amount of overlap. Our results finally show that a natural creation of maximally entangled states is possible just by moving two identical particles with opposing pseudospin states into the same site, supplying theoretical support to recent observations in an experiment with ultracold atoms transported in an optical tweezer [14]. This gives a definitive positive answer whether identical particles in the same site can be entangled.

Our approach contributes, from a fundamental point of view, to clarify the relation between quantum entanglement and identity of particles. It remarkably allows the quantitative study of entanglement under completely general conditions of overlap and scalability, motivating studies on correlations other than entanglement [15] in the context of identical particle systems. Moreover, our study paves the way to interpret experiments which use quantum correlations in relevant scenarios where identical particles can overlap.

References:
[1] Luigi Amico, Rosario Fazio, Andreas Osterloh, Vlatko Vedral, “Entanglement in many-body systems”, Review of Modern Physics, 80, 517 (2008). Abstract.
[2] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, Karol Horodecki, “Quantum entanglement”, Review of Modern Physics, 81, 865 (2009). Abstract.
[3] Malte C. Tichy, Florian Mintert, Andreas Buchleitner, “Essential entanglement for atomic and molecular physics”, Journal of Physics B: Atomic, Molecular and Optical Physics, 44, 192001 (2011). Full Text.
[4] F. Benatti, R. Floreanini, K. Titimbo, “Entanglement of identical particles”, Open Systems & Information Dynamics, 21, 1440003 (2014). Abstract.
[5] Claude Cohen-Tannoudji, Bernard Diu, Franck Laloë, “Quantum mechanics. Vol. 2” (Willey-VCH, Paris, France, 2005).
[6] Asher Peres, “Quantum Theory: Concepts and Methods” (Kluwer Academic,1995).
[7] Michael H. Kolodrubetz, Jason R. Petta, “Coherent holes in a semiconductor quantum dot”, Science 325, 42 (2009). Abstract.
[8] Z.B. Tan, D. Cox, T. Nieminen, P. Lähteenmäki, D. Golubev, G.B. Lesovik, P.J. Hakonen, “Cooper pair splitting by means of graphene quantum dots”, Physical Review Letters, 114, 096602 (2015). Abstract.
[9] Immanuel Bloch, Jean Dalibard, Wilhelm Zwerger, “Many-body physics with ultracold gases”, Review of Modern Physics, 80, 885 (2008). Abstract.
[10] Marco Anderlini, Patricia J. Lee, Benjamin L. Brown, Jennifer Sebby-Strabley, William D. Phillips, J.V. Porto, “Controlled ex-change interaction between pairs of neutral atoms in an optical lattice”, Nature, 448, 452 (2007). Abstract.
[11] GianCarlo Ghirardi, Luca Marinatto, Tullio Weber, “Entanglement and properties of composite quantum sys-tems: a conceptual and mathematical analysis”, Journal of Statistical Physics, 108, 49 (2002). Abstract.
[12] N. Killoran, M. Cramer, M. B. Plenio, “Extracting entanglement from identical particles”, Physical Review Letters, 112, 150501 (2014). Abstract.
[13] Rosario Lo Franco, Giuseppe Compagno, “Quantum entanglement of identical particles by standard information-theoretic notions”, Scientific Reports, 6, 20603 (2016). Full Text.
[14] A.M. Kaufman, B.J. Lester, M. Foss-Feig, M.L. Wall, A.M. Rey,  C.A. Regal, “Entangling two transportable neutral atoms via local spin exchange”, Nature 527, 208 (2015). Abstract.
[15] Kavan Modi, Aharon Brodutch, Hugo Cable, Tomasz Paterek, Vlatko Vedral, “The classical-quantum boundary for corre-lations: Discord and related measures”, Review of Modern Physics, 84, 1655 (2012). Abstract.