The definition of entanglement in identical-particle system is introduced. The separability criterion in two-identical particle system is given. The physical meaning of the definition is analysed. Applications to two-boson and two-fermion systems are made. It is found new entanglement and correlation phenomena in identical-boson systems exist, and they may have applications in the field of quantum information.
Identical particles and entanglement are both fundamental components of quantum mechanics. However, when identical particles are condensed in a single spatial mode, the standard notions of entanglement, based on clearly identifiable subsystems, break down. This has led many to conclude that such systems have limited value for quantum information tasks, compared to distinguishable particle systems. To the contrary, we show that any entanglement formally appearing amongst the identical particles, including entanglement due purely to symmetrization, can be extracted into an entangled state of ind…
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…
We suggest a formalism to illustrate the entanglement of identical particles in the first quantization language (1QL). Our 1QL formalism enables one to exploit all the well-established quantum information tools to understand the indistinguishable ones, including the reduced density matrix and familiar entanglement measures. The rigorous quantitative relation between the amount of entanglement and the spatial coherence of particles is possible in this formalism. Our entanglement detection process is a generalization of the entanglement extraction protocol for identical particles with mode split…
It was shown recently that entanglement of identical particles has a feature called dualism [Phys. Rev. Lett. 110, 140404 (2013)], which is fundamentally connected with quantum indistinguishability. Here we report an experiment that observes the entanglement duality for the first time with two identical photons, which manifest polarization entanglement when labeled by different paths or path entanglement when labeled by polarization states. By adjusting the mismatch in frequency or arrival time of the entangled photons, we tune the photon indistinguishability from quantum to classical limit an…
It has been shown that the vacuum state of a free quantum field is entangled and such vacuum entanglement can be harvested by a pair of initially uncorrelated detectors interacting locally with the vacuum field for a finite time. In this paper, we examine the entanglement harvesting phenomenon of two non-identical inertial detectors with different energy gaps locally interacting with massless scalar fields via a Gaussian switching function. We focus on how entanglement harvesting depends on the energy gap difference from two perspectives: the amount of entanglement harvested and the harvesting…
We introduce detector-level entanglement, a unified entanglement concept for identical particles that takes into account the possible deletion of many-particle which-way information through the detection process. The concept implies a measure for the effective indistinguishability of the particles, which is controlled by the measurement setup and which quantifies the extent to which the (anti-)symmetrization of the wave-function impacts on physical observables. Initially indistinguishable particles can gain or loose entanglement on their transition to distinguishability, and their quantum stat…
We identify a general criterion for detecting entanglement of pure bipartite quantum states describing a system of two identical particles. Such a criterion is based both on the consideration of the Slater-Schmidt number of the fermionic and bosonic analog of the Schmidt decomposition and on the evaluation of the von Neumann entropy of the one-particle reduced statistical operators.
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…
The robustness properties of bipartite entanglement in systems of N bosons distributed in M different modes are analyzed using a definition of separability based on commuting algebras of observables, a natural choice when dealing with identical particles. Within this framework, expressions for the robustness and generalized robustness of entanglement can be explicitly given for large classes of boson states: their entanglement content results in general much more stable than that of distinguishable particles states. Using these results, the geometrical structure of the space of N boson states …
We study the role of entanglement and non-locality in quantum protocols that make use of systems of identical particles. Unlike in the case of distinguishable particles, the notions of entanglement and non-locality for systems whose constituents cannot be distinguished and singly addressed are still debated. We clarify why the only approach that avoids incongruities and paradoxes is the one based on the second quantization formalism whereby it is the entanglement of the modes that can be populated by the particles what really matters and not the particles themselves. Indeed, by means of a metr…
We investigate the separability of arbitrary $n$-dimensional multipartite identical bosonic systems. An explicit relation between the dimension and the separability is presented. In particular, for $n=3$, it is shown that the property of PPT (positive partial transpose) and the separability are equivalent for tripartite systems.
We address the problem of entanglement protection against surrounding noise by a procedure suitably exploiting spatial indistinguishability of identical subsystems. To this purpose, we take two initially separated and entangled identical qubits interacting with two independent noisy environments. Three typical models of environments are considered: amplitude damping channel, phase damping channel and depolarizing channel. After the interaction, we deform the wave functions of the two qubits to make them spatially overlap before performing spatially localized operations and classical communicat…
We investigate the effects of fuzzy measurements on spin entanglement for identical particles, both fermions and bosons. We first consider an ideal measurement apparatus and define operators that detect the symmetry of the spatial and spin part of the density matrix as a function of particle distance. Then, moving on to realistic devices that can only detect the position of the particle to within a certain spread, it was surprisingly found that the entanglement between particles increases with the broadening of detection.
Quantum information and communication processing within quantum networks usually employs identical particles. Despite this, the physical role of quantum statistical nature of particles in large-scale networks remains elusive. Here, we show that just the indistinguishability of fermions makes it possible a new mechanism of entanglement transfer in many-node quantum networks. This process activates remote entanglement among distant sites, which do not share a common past, by only locally counting identical particles and classical communication. These results constitute the key achievement of the…
We have recently introduced a measure of the bipartite entanglement of identical particles, E_P, based on the principle that entanglement should be accessible for use as a resource in quantum information processing. We show here that particle entanglement is limited by the lack of a reference phase shared by the two parties, and that the entanglement is constrained to reference-phase invariant subspaces. The super-additivity of E_P results from the fact that this constraint is weaker for combined systems. A shared reference phase can only be established by transferring particles between the pa…
A detailed treatment of the entanglement dynamics of two distant but non-identical systems is presented. We study the entanglement evolution of two remote atoms interacting independently with a cavity field, as in the double Jaynes-Cummings (JC) model. The four-qubit pairwise concurrences are studied, allowing for asymmetric atom-cavity couplings and off-resonant ineractions. Counter to intuition, imperfect matching can prove advantageous to entanglement creation and evolution. For two types of initial entanglement, corresponding to spin correlated and anti-correlated Bell states Φand Ψ, a ful…
We review two general criteria for deciding whether a pure bipartite quantum state describing a system of two identical particles is entangled or not. The first one considers the possibility of attributing a complete set of objective properties to each particle belonging to the composed system, while the second is based both on the consideration of the Slater-Schmidt number of the fermionic and bosonic analog of the Schmidt decomposition and on the evaluation of the von Neumann entropy of the one-particle reduced statistical operators.
A new uncertainty relation (UR) is obtained for a system of N identical pure entangled particles if we use symmetrized observables when deriving the inequality. This new expression can be written in a form where we identify a term which explicitly shows the quantum correlations among the particles that constitute the system. For the particular cases of two and three particles, making use of the Schwarz inequality, we obtain new lower bounds for the UR that are different from the standard one.
Quantum correlations of identical particles are important for quantum-enhanced technologies. The recently introduced non-standard approach to treat identical particles [G. Compagno et al., Phil. Trans. R. Soc. A 376, 20170317 (2018)] is here exploited to show the effect of particle indistinguishability on the characterization of entanglement of three identical qubits. We show that, by spatially localized measurements in separated regions, three independently-prepared separated qubits in a pure elementary state behave as distinguishable ones, as expected. On the other hand, delocalized measurem…
Quantum correlations between identical particles are at the heart of quantum technologies. Several studies with two identical particles have shown that the spatial overlap and indistinguishability between the particles are necessary for generating bipartite entanglement. On the other hand, researches on the extension to more than two-particle systems are limited by the practical difficulty to control multiple identical particles in laboratories. In this work, we propose schemes to generate two fundamental classes of genuine tripartite entanglement, i.e., GHZ and W classes, which are experiment…
These two accompanying papers treat two mode entanglement for systems of identical massive bosons and the relationship to spin squeezing and other quantum correlation effects. Entanglement is a key quantum feature of composite systems where the probabilities for joint measurements on the composite sub-systems are no longer determined from measurement probabilities on the separate sub-systems. We focus on the meaning of entanglement, the quantum paradoxes associated with entangled states, and important tests that allow an experimentalist to determine whether a quantum state - in particular, one…
Recent results, extending the Schmidt decomposition theorem to wavefunctions of identical particles, are reviewed. They are used to give a definition of reduced density operators in the case of two identical particles. Next, a method is discussed to calculate time averaged entanglement. It is applied to a pair of identical electrons in an otherwise empty band of the Hubbard model, and to a pair of bosons in the the Bose-Hubbard model with infinite range hopping. The effect of degeneracy of the spectrum of the Hamiltonian on the average entanglement is emphasised.
We consider entanglement in a system of fixed number of identical particles. Since any operation should be symmetrized over all the identical particles and there is the precondition that the spatial wave functions overlap, the meaning of identical-particle entanglement is fundamentally different from that of distinguishable particles. The identical-particle counterpart of the Schmidt basis is shown to be the single-particle basis in which the one-particle reduced density matrix is diagonal. But it does not play a special role in the issue of entanglement, which depends on the single-particle b…
Entanglement is known to be a relative notion, defined with respect to the choice of physical observables to be measured (i.e., the measurement setup used). This implies that, in general, the same state can be both separable and entangled for different measurement setups, but this does not exclude the existence of states which are separable (or entangled) for all possible setups. We show that for systems of bosonic particles there indeed exist such universally separable states: they are i.i.d. pure states. In contrast, there is no such state for fermionic systems with a few exceptional cases. …
Entanglement of identical massive particles recently gained attention, because of its relevance in highly controllable systems, e.g. ultracold gases. It accounts for correlations among modes instead of particles, providing a different paradigm for quantum information. We prove that the entanglement of almost all states rarely vanishes in the presence of noise, and analyse the most relevant noise in ultracold gases: dephasing and particle losses. Furthermore, when the particle number increases, the entanglement decay can turn from exponential into algebraic.
We have studied the dynamics of quantum correlations such as entanglement, Bell-nonlocality and quantum discord between identical as well as unidentical atoms interacting with a single-mode cavity field and subject to cavity decay. The effect of single atom detuning, cavity decay rate and initial preparation of the atoms on the corresponding correlation measures have been investigated. It is found that even under strong dissipation, time evolution can create high quantum discord while entanglement and Bell nonlocality stay zero for an initially separable state. Quantum discord increases while …
The study of the entanglement properties of systems of N fermions has attracted considerable interest during the last few years. Various separability criteria for pure states of N identical fermions have been recently discussed but, excepting the case of two-fermions systems, these criteria are difficult to implement and of limited value from the practical point of view. Here we advance simple necessary and sufficient separability criteria for pure states of N identical fermions. We found that to be identified as separable a state has to comply with one single identity involving either the pur…
Quantum entanglement of identical particles is essential in quantum information theory. Yet, its correct determination remains an open issue hindering the general understanding and exploitation of many-particle systems. Operator-based methods have been developed that attempt to overcome the issue. We introduce a state-based method which, as second quantization, does not label identical particles and presents conceptual and technical advances compared to the previous ones. It establishes the quantitative role played by arbitrary wave function overlaps, local measurements and particle nature (bo…
For any quantum state representing a physical system of identical particles, the density operator must satisfy the symmetrisation principle (SP) and for massive particles also conform to super-selection rules (SSR) that prohibit coherences between differing total particle numbers. Here we consider bi-partitite states for systems of massive bosons, where both the system and subsystems are modes (or sets of modes), particles being associated with differing mode occupancies. To define non-entangled or separable states the subsystem density operators are also required to satisfy the SP and conform…
By identifying non-local effects in systems of identical Bosonic qubits through correlations of their commuting observables, we show that entanglement is not necessary to violate certain squeezing inequalities that hold for distinguishable qubits and that spin squeezing may not be necessary to achieve sub-shot noise accuracies in ultra-cold atom interferometry.
We argue that in the case of identical particles the most natural identification of separability, that is of absence of non-classical correlations, is via the factorization of mean values of commuting observables. It thus follows that separability and entanglement depend both on the state and on the choice of observables and are not absolute notions. We compare this point of view with a recent novel approach to the entanglement of identical particles, which allows for the definition of an entanglement entropy from a suitably defined reduced particle density matrix, without the need of labeling…
We explore the connection between quantum entanglement and the exchange symmetry of the states of N identical particles. Each particle has n-levels. The N particles span the nN dimensional Hilbert space. We shall call the general state of the particle as a qunit. The direct product of the N qunit space is given a decomposition in terms of states with definite permutation symmetry that are found to have a measure of entanglement which is related to the representation of the permutation group. The maximally entangled states are generated from the linear combinations of fully correlated but unent…
These two accompanying papers treat two mode entanglement for systems of identical massive bosons and the relationship to spin squeezing and other quantum correlation effects. Entanglement is a key quantum feature of composite systems where the probabilities for joint measurements on the composite sub-systems are no longer determined from measurement probabilities on the separate sub-systems. We focus on the meaning of entanglement, the quantum paradoxes associated with entangled states, and important tests that allow an experimentalist to determine whether a quantum state - in particular, one…
Here we discuss a particle-based approach to deal with systems of many identical quantum objects (particles) which never employs labels to mark them. We show that it avoids both methodological problems and drawbacks in the study of quantum correlations associated to the standard quantum mechanical treatment of identical particles. The core of this approach is represented by the multiparticle probability amplitude whose structure in terms of single-particle amplitudes we here derive by first principles. To characterise entanglement among the identical particles, this new method utilises the sam…
We provide a group-theoretical classification of the entangled states of N identical particles. The connection between quantum entanglement and the exchange symmetry of the states of N identical particles is made explicit using the duality between the permutation group and the simple unitary group. Each particle has n-levels and spans the n-dimensional Hilbert space. We shall call the general state of the particle as a qunit. The direct product of the N qunit space is given a decomposition in terms of states with definite permutation symmetry. The nature of fundamental entanglement of a state …
The concept of entangled quantum states is considered in the context of systems of identical particles, based on the requirement that in order to represent physical states both for the overall system and the sub-systems which may be entangled, the density operators must satisfy the symmetrisation principle and global and local super-selection rules that prohibit states in which there are coherences between differing particle numbers. These requirements and their justification are fully discussed. In the second quantisation approach used, both the system and the sub-systems are modes (or sets o…
We point out an earlier unnoticed implication of quantum indistinguishability, namely, a property which we call `dualism' that characterizes the entanglement of two identical particles (say, two ions of the same species) -- a feature which is absent in the entanglement of two non-identical particles (say, two ions of different species). A crucial application of this property is that it can be used to test quantum indistinguishability without bringing the relevant particles together, thereby avoiding the effects of mutual interaction. This is in contrast to the existing tests of quantum indisti…
We investigate the simultaneous estimation of $n$ not necessarily identical unitary channels using multi-partite entanglement. We examine whether it is possible for the rate at which the mean square error decreases to be greater than that using the channels individually. For a reasonably general situation, in which there is no functional dependence between the channels, we show that this is not possible. We look at a case in which the channels are not necessarily identical but depend on a common variable. In this case, the mean square error decreases more rapidly using multi-partite entangleme…
Two-photon interference effects, such as the Hong-Ou-Mandel (HOM) effect, can be used to characterize to what extent two photons are identical. Identical photons are necessary for both linear optics quantum computing and single-photon quantum cryptography. We study here how storage and delay of photons in coupled cavity arrays, which inevitably will lead to changes in the photons' spectral and temporal profiles, affects their HOM interference. In addition we consider various types of entanglement that occur naturally in such a context.
We study bipartite entanglement in systems of N identical bosons distributed in M different modes. For such systems, a definition of separability not related to any a priori Hilbert space tensor product structure is needed and can be given in terms of commuting subalgebras of observables. Using this generalized notion of separability, we classify the states for which partial transposition turns out to be a necessary and sufficient condition for entanglement detection.
We study the problem of detecting multipartite entanglement among indistinguishable fermionic particles. A multipartite concurrence for pure states of $N$ identical fermions, each one having a $d$-dimensional single-particle Hilbert space, is introduced. Such entanglement measure, in particular, is optimized for maximally entangled states of three identical fermions that play a role analogous to the usual (qubit) Greenberger-Horne-Zeilinger state. In addition, it is shown that the fermionic multipartite concurrence can be expressed as the mean value of an observable, provided two copies of the…
Quantum particles are known to be faster than classical when they propagate stochastically on certain graphs. A time needed for a particle to reach a target node on a distance, the hitting time, can be exponentially less for quantum walks than for classical random walks. It is however not known how fast would interacting quantum particles propagate on different graphs. Here we present our results on hitting times for quantum walks of identical particles on cycle graphs, and relate the results to our previous findings on the usefulness of identical interacting particles in quantum information t…
Particle identity and entanglement are two fundamental quantum properties that work as major resources for various quantum information tasks. However, it is still a challenging problem to understand the correlation of the two properties in the same system. While recent theoretical studies have shown that the spatial overlap between identical particles is necessary for nontrivial entanglement, the exact role of particle indistinguishability in the entanglement of identical particles has never been analyzed quantitatively before. Here, we theoretically and experimentally investigate the behavior…
Progress in the reliable preparation, coherent propagation and efficient detection of many-body states has recently brought collective quantum phenomena of many identical particles into the spotlight. This tutorial introduces the physics of many-boson and many-fermion interference required for the description of current experiments and for the understanding of novel approaches to quantum computing. The field is motivated via the two-particle case, for which the uncorrelated, classical dynamics of distinguishable particles is compared to the quantum behaviour of identical bosons and fermions. B…
A system of two two-level atoms interacting with a squeezed vacuum field can exhibit stationary entanglement associated with nonclassical two-photon correlations characteristic of the squeezed vacuum field. The amount of entanglement present in the system is quantified by the well known measure of entanglement called concurrence. We find analytical formulas describing the concurrence for two identical and nonidentical atoms and show that it is possible to obtain a large degree of steady-state entanglement in the system. Necessary conditions for the entanglement are nonclassical two-photon corr…
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…
The existence of fundamentally identical particles represents a foundational distinction between classical and quantum mechanics. Due to their exchange symmetry, identical particles can appear to be entangled -- another uniquely quantum phenomenon with far-reaching practical implications. However, a long-standing debate has questioned whether identical particle entanglement is physical or merely a mathematical artefact. In this work, we provide such particle entanglement with a consistent theoretical description as a quantum resource in processes frequently encountered in optical and cold atom…
We consider the bipartite spin entanglement between two identical fermions generated in spin-independent scattering. We show how the spatial degrees of freedom act as ancillas for the creation of entanglement to a degree that depends on the scattering angle, $θ$. The number of Slater determinants generated in the process is greater than 1, corresponding to genuine quantum correlations between the identical fermions. The maximal entanglement attainable of 1 ebit is reached at $θ=π/2$. We also analyze a simple $θ$ dependent Bell's inequality, which is violated for $π/4<θ\leqπ/2$. This phenomenon…
This work provides a comprehensive approach to analyze the entanglement between subsystems generated by identical particles, based on the symmetric/exterior algebra (SEA) and microcausality. Our method amends the no-labeling approach (NLA) to quantify any type of identical particles' entanglement, especially fermions with the parity superselection rule. We can analyze the non-local properties of identical particles' states in a fundamentally equivalent way to those for non-identical particles, which is achieved by the factorizability of the total Hilbert space of identical particles. This form…
We provide a general framework which allows one to obtain the dynamics of $N$ noninteracting spatially indistinguishable particles locally coupled to separated environments. The approach is universal, being valid for both bosons and fermions and for any type of system-environment interaction. It is then applied to study the dynamics of two identical qubits under paradigmatic Markovian noises, such as phase damping, depolarizing and amplitude damping. We find that spatial indistinguishability of identical qubits is a controllable intrinsic property of the system which protects quantum entanglem…
Unlike for bipartite states consisting of distinguishable particles, in the case of identical parties the notion of entanglement is still under debate. In the following, we review two different approaches to the entanglement of systems consisting of two Bosons or Fermions; the first approach is based on the particle aspect typical of first quantization and identifies separable pure states as those that allow to assign two orthogonal single particle vector states to both parties. The second approach makes full use of the mode aspect of second quantization whereby separability can be formulated …
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…
Estimation of physical parameters is a must in almost any part of science and technology. The enhancement of the performances in this task, e.g., beating the standard classical shot-noise limit, using available physical resources is a major goal in metrology. Quantum metrology in closed systems has indicated that entanglement in such systems may be a useful resource. However, it is not yet fully understood whether in open quantum systems such enhancements may still show up. Here, we consider a dissipative (open) quantum system of identical particles in which a parameter of the open dynamics it…
Theoretical considerations of Bell-inequality experiments usually assume identically prepared and independent pairs of particles. Here we consider pairs that exhibit both intra- and inter-pair entanglement. The pairs are taken from a large many-body system where all the pairs are generally entangled with each other. Using an explicit example based on single mode entanglement and an ancillary Bose-Einstein condensate, we show that the Bell-inequality violation in such systems can display statistical properties that are remarkably different from those obtained using identically prepared, indepen…
We reconsider the effect of indistinguishability on the reduced density operator of the internal degrees of freedom (tracing out the spatial degrees of freedom) for a quantum system composed of identical particles located in different spatial regions. We explicitly show that if the spin measurements are performed in disjoint spatial regions then there are no constraints on the structure of the reduced state of the system. This implies that the statistics of identical particles has no role from the point of view of separability and entanglement when the measurements are spatially separated. We …
