Recurrence and explicit formulae for contractions (partial traces) of antisymmetric and symmetric products of identical trace class operators are derived. Contractions of product density operators of systems of identical fermions and bosons are proved to be asymptotically equivalent to, respectively, antisymmetric and symmetric products of density operators of a single particle, multiplied by a normalization integer. The asymptotic equivalence relation is defined in terms of the thermodynamic limit of expectation values of observables in the states represented by given density operators. For some weaker relation of asymptotic equivalence, concerning the thermodynamic limit of expectation values of product observables, normalized antisymmetric and symmetric products of density operators of a single particle are shown to be equivalent to tensor products of density operators of a single particle.
This paper presents the results of a part of the author's thesis [W. Radzki, "Kummer contractions of product density matrices of systems of $n$ fermions and $n$ bosons" (Polish), MS thesis, Institute of Physics, Nicolaus Copernicus University, Toruń, 1999].
We study the problem of particle indistinguishability for the three cases known in nature: identical classical particles, identical bosons and identical fermions. By exploiting the fact that different types of particles are associated with Hilbert space vectors with different symmetries, we establish some relations between the expectation value of several different operators, as the particle number one and the interparticle correlation one, evaluated for states of a pair of identical (a) classical particles, (b) bosons and (c) fermions. We find that the quantum behavior of a pair of identical …
The boson sampling problem has triggered a lot of interest in the scientific community because of its potential of demonstrating the computational power of quantum interference without the need of non-linear processes. However, the intractability of such a problem with any classical device relies on the realization of single photons approximately identical in their spectra. In this paper we discuss the physics of boson sampling with non-identical single photon sources, which is strongly relevant in view of scalable experimental realizations and triggers fascinating questions in complexity theo…
Efficiently simulating the work distribution of multiple identical bosons with boson sampling
Center for Quantum Technology Research and Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurements (MOE), School of Physics, Beijing Institute of Technology（北京理工大学物理学院量子技术研究中心及先进光电子量子架构与测量教育部重点实验室）
In this article combinatorial aspects of normal ordering annihilation and creation operators of a multi-mode boson system are discussed. The modes are assumed to be coupled since otherwise the problem of normal ordering is reduced to the corresponding problem of the single-mode case. To describe the normal ordering in the multi-mode case for each mode a colour is introduced and coloured contractions are considered. A depiction for coloured contractions via coloured linear representations is given. In analogy to the single-mode case associated coloured Stirling numbers are defined as coefficien…
We first devise a scheme to perform a universal entangling gate via controlled collisions between pairs of atomic qubits trapped with optical tweezers. Second, we present a modification to this scheme to allow the preparation of atomic Bell pairs via selective excitation, suitable for quantum information processing applications that do not require universality. Both these schemes are enabled by the inherent symmetries of identical composite particles, as originally proposed by Hayes et al. Our scheme provides a technique for producing weighted graph states, entangled resources for quantum comm…
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…
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.
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 …
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…
How do indistinguishable identical bosons manage to obey Bose-Einstein statistics---and hence be correlated---even when they do not interact with each other? Part of the answer is that the bosons have to interact indirectly with each other by interacting with the same environment. A joint measurement interaction provides a good example. Thermalization occurs whenever there are two competing processes, one diagonal in the energy basis (namely, reversible Hamiltonian evolution), the other irreversible and diagonal in a complementary basis (for example, a measurement in a spatially localized basi…
In this paper we define generalizations of boson normal ordering. These are based on the number of contractions whose vertices are next to each other in the linear representation of the boson operator function. Our main motivation is to shed further light onto the combinatorics arising from algebraic and Fock space properties of boson operators.
In a mixture of two kinds of identical bosons there are two types of pairs, identical bosons' pairs, of either species, and pairs of distinguishable bosons. In the present work fragmentation of pairs in a trapped mixture of Bose-Einstein condensate is investigated using a solvable model, the symmetric harmonic-interaction model for mixtures. The natural geminals for pairs made of identical or distinguishable bosons are explicitly contracted by diagonalizing the intra-species and inter-species reduced two-particle density matrices, respectively. Properties of pairs' fragmentation in the mixture…
Quantum correlations and entanglement in identical-particle systems have been a puzzling question which has attracted vast interest and widely different approaches. A novel approach is introduced by Kraus \emph{et al.}, [Phys. Rev. A \textbf{79}, 012306 (2009)] based on pairing correlations in fermionic systems and the use of witness formalism to detect pairing. In this contribution, this approach has been extended to bosonic systems and separability bounds based on pairing correlations for fermions and bosons have been obtained. A two-particle annihilation operator is used for constructing a …
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…
The envelope theory, also known as the auxiliary field method, is a simple technique to compute approximate solutions of Hamiltonians for $N$ identical particles in $D$-dimension. The accuracy of this method is tested by computing the ground state of $N$ identical bosons for various systems. A method is proposed to improve the quality of the approximations by modifying the characteristic global quantum number of the method.
In this work, dissipative quantum backflow is studied for a superposition of two stretched Gaussian wave packets and two identical spinless particles within the Caldirola-Kanai framework. Backflow is mainly an interference process and dissipation is not able to suppress it in the first case. For two identical spinless particles, apart from interference terms, the symmetry of the wave function seems to be crucial in this dynamics. The combined properties make bosons display this effect, even for the dissipative regime but, for fermions, backflow is not exhibited in any regime, dissipative and n…
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…
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…
Non-linear dynamics in the quantum random walk setting have been shown to enable conditional speedup of Grover's algorithm. We examine the mean field approximation required for the use of the Gross-Pitaevskii equation on identical bosons evolving on the complete graph. We show that the states of such systems are parameterized by the basis of Young diagrams and determine their one- and two-party marginals. We find that isolated particles are required for good agreement with the mean field approximation, proving that without isolated particles the matrix fidelity agreement is bounded from above …
We calculate the three-body spectrum for identical bosons interacting via attractive $1/r^2$ potentials. We have found an infinite number of three-body states even when the pair interactions are too weak to support any two-body states. These new states thus share this surprising scenario with the Efimov effect, but are not themselves Efimov states. Our effect occurs for both identical bosons and identical fermions, and it persists in the presence of two-body bound states.
The celebrated Hong-Ou-Mandel effect is the paradigm of two-particle quantum interference. It has its roots in the symmetry of identical quantum particles, as dictated by the Pauli principle. Two identical bosons impinging on a beam splitter (of transmittance 1/2) cannot be detected in coincidence at both output ports, as confirmed in numerous experiments with light or even matter. Here, we establish that partial time reversal transforms the beamsplitter linear coupling into amplification. We infer from this duality the existence of an unsuspected two-boson interferometric effect in a quantum …
Path integral molecular dynamics (PIMD) has been successfully applied to perform simulations of large bosonic systems in a recent work (Hirshberg et al., PNAS, 116, 21445 (2019)). In this work we extend PIMD techniques to study Green's function for bosonic systems. We demonstrate that the development of the original PIMD method enables us to calculate Green's function and extract momentum distribution from our simulations. We also apply our method to systems of identical interacting bosons to study Berezinskii-Kosterlitz-Thouless transition around its critical temperature.
For systems of identical Bosons, it is necessary to reformulate the notions of separability and entanglement in algebraic terms shifting the emphasis from the particle aspect of first quantization to the mode description typical of second quantization. Within this new framework, we show that, unlike for systems consisting of distinguishable qubits, negativity is an exhaustive bipartite entanglement witness for systems with fixed number of Bosons; further, we investigate the impact of dephasing noise in relation to the use of such many-body Bosonic systems in metrological applications.
The effect of boson bunching is frequently mentioned and discussed in the literature. This effect is the manifestation of bosons tendency to "travel" in clusters. One of the core arguments for boson bunching was formulated by Feynman in his well-known lecture series and has been frequently used ever since. By comparing the scattering probabilities of two bosons and of two non-identical particles, Feynman concluded: "We have the result that it is twice as likely to find two identical Bose particles scattered into the same state as you would calculate assuming the particles were different." [1].…
In this work, our purpose is to show how the symmetry of identical particles can influence the time evolution of free particles in the nonrelativistic and relativistic domains. For this goal, we consider a system of either two distinguishable or indistinguishable (bosons and fermions) particles. Two classes of initial conditions have been studied: different initial locations with the same momenta, and the same locations with different momenta. The flight time distribution of particles arriving at a `screen' is calculated in each case. Fermions display broader distributions as compared with eit…
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…
It is pointed out that the cross section for the scattering of identical charged bosons is isotropic over a broad angular range around 90 degrees when the Sommerfeld parameter has a critical value, which depends exclusively on the spin of the particle. A discussion of systems where this phenomenon can be observed is presented.
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. …
Identical particles exhibit correlations even in the absence of inter-particle interaction, due to the exchange (anti)symmetry of the many-particle wavefunction. Two fermions obey the Pauli principle and anti-bunch, whereas two bosons favor bunched, doubly occupied states. Here, we show that the collective interference of three or more particles leads to a much more diverse behavior than expected from the boson-fermion dichotomy known from quantum statistical mechanics. The emerging complexity of many-particle interference is tamed by a simple law for the strict suppression of events in the Be…
I consider non-relativistic bosons interacting via pairwise potentials with infinite scattering length and supporting no two-body bound states. To lowest order in effective field theory, these conditions lead to non-interacting bosons, since the coupling constant of the Lieb-Liniger model vanishes identically in this limit. Since any realistic pairwise interaction is not a mere delta function, the non-interacting picture is an idealisation indicating that the effect of interactions is weaker than in the case of off-resonant potentials. I show that the leading order correction to the ground sta…
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.
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…
Quantum contextuality in systems of identical bosonic particles is explicitly exhibited via the maximum violation of a suitable inequality of Clauser-Horne-Shimony-Holt type. Unlike the approaches considered so far, which make use of single-particle observables, our analysis involves collective observables constructed using multi-boson operators. An exemplifying scheme to test this violation with a quantum optical setup is also discussed.
We identify the correlation in a state of two identical particles as the residual information beyond what is already contained in the 1-particle reduced density matrix, and propose a correlation measure based on the maximum entropy principle. We obtain the analytical results of the correlation measure, which make it computable for arbitrary two-particle states. We also show that the degrees of correlation in the same two-particle states with different particle types will decrease in the following order: bosons, fermions, and distinguishable particles.
In this work, momentum-space decoherence using minimum and nonminimum-uncertainty-product (stretched) Gaussian wave packets in the framework of Caldeira-Leggett formalism and under the presence of a linear potential is studied. As a dimensionless measure of decoherence, purity, a quantity appearing in the definition of the {\it linear entropy}, is studied taking into account the role of the stretching parameter. Special emphasis is on the open dynamics of the well-known cat states and bosons and fermions compared to distinguishable particles. For the cat state, while the stretching parameter s…
Interference and diffraction of two-identical-particles are considered in the context of open quantum systems. This theoretical study is carried out within two approaches, the effective time-dependent Hamiltonian due to Caldirola-Kanai (CK) and the Caldeira-Leggett (CL) one where a master equation for the reduced density matrix is used under the presence of dissipation and temperature of the environment. Two simple but very illustrative examples are considered, diffraction by a single and two Gaussian slits by analyzing the mean square separation between particles, single-particle probability …
It has been recently conjectured that the state-independency of quantum contextuality may be lost when the indistinguishability of identical particles is taken into account. Here, we show that quantum state-independent contextuality exists for any system of more than one identical bosonic qudits, and for most systems of fermionic qudits. The only exception is the case of d fermionic qudits, since there the dimension of the antisymmetric subspace is 1, which is insufficient for contextuality. For all the other cases, either the symmetry precludes the existence of physical states, or we provide …
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.
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.
Three-body systems that are continuously squeezed from a three-dimensional (3D) space into a two-dimensional (2D) space are investigated. Such a squeezing can be obtained by means of an external confining potential acting along a single axis. However, this procedure can be numerically demanding, or even undoable, especially for large squeezed scenarios. An alternative is provided by use of the dimension $d$ as a parameter that changes continuously within the range $2\leq d \leq 3$. The simplicity of the $d$-calculations is exploited to investigate the evolution of three-body states after progr…
The multiconfigurational time-dependent Hartree method (MCTDH) [Chem. Phys. Lett. {\bf 165}, 73 (1990); J. Chem. Phys. {\bf 97}, 3199 (1992)] is celebrating nowadays entering its third decade of tackling numerically-exactly a broad range of correlated multi-dimensional non-equilibrium quantum dynamical systems. Taking in recent years particles' statistics explicitly into account, within the MCTDH for fermions (MCTDHF) and for bosons (MCTDHB), has opened up further opportunities to treat larger systems of interacting identical particles, primarily in laser-atom and cold-atom physics. With the i…
A fermionic operator circuit is a product of fermionic operators of usually different and partially overlapping support. Further elements of fermionic operator circuits (FOCs) are partial traces and partial projections. The presented framework allows for the introduction of fermionic versions of known qudit operator circuits (QUOC), important for the simulation of strongly correlated d-dimensional systems: The multiscale entanglement renormalization ansatz (MERA), tree tensor networks (TTN), projected entangled pair states (PEPS), or their infinite-size versions (iPEPS etc.). After the definit…
We study diffraction and interference of indistinguishable particles. We consider some examples where the wavefunctions and detection probabilities can be evaluated in an analytical way. The diffraction pattern of a two-particle system shows notorious differences for the cases of bosons, fermions and distinguishable particles. In the example of near-field interferometry, the exchange effects for two-fermion systems lead to the existence of planes at which the probability of double detection is null. We also discuss the relation with the approach to systems of identical particles based on corre…
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 backflow is the classically-forbidden effect pertaining to the fact that a particle with a positive momentum may exhibit a negative probability current at some space-time point. We investigate how this peculiar phenomenon extends to many-particle systems. We give a general formulation of quantum backflow for systems formed of $N$ free nonrelativistic structureless particles, either identical or distinguishable. Restricting our attention to bosonic systems where the $N$ identical bosons are in the same one-particle state allows us in particular to analytically show that the maximum achi…
We investigate finite-range effects in systems with three identical bosons. We calculate recombination rates and bound state spectra using two different finite-range models that have been used recently to describe the physics of cold atomic gases near Feshbach resonances where the scattering length is large. The models are built on contact potentials which take into account finite range effects; one is a two-channel model and the other is an effective range expansion model implemented through the boundary condition on the three-body wave function when two of the particles are at the same point…
Quantum mechanical particles in a confining potential interfere with each other while undergoing thermodynamic processes far from thermal equilibrium. By evaluating the corresponding transition probabilities between many-particle eigenstates we obtain the quantum work distribution function, for identical Bosons and Fermions, which we compare with the case of distinguishable particles. We find that the quantum work distributions for Bosons and Fermions significantly differ at low temperatures, while, as expected, at high temperatures the work distributions converge to the classical expression. …
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.
The quon algebra gives a description of particles, ``quons,'' that are neither fermions nor bosons. The parameter $q$ attached to a quon labels a smooth interpolation between bosons, for which $q = +1$, and fermions, for which $q = -1$. Wigner and Ehrenfest and Oppenheimer showed that a composite system of identical bosons and fermions is a fermion if it contains an odd number of fermions and is a boson otherwise. Here we generalize this result to composite systems of identical quons. We find $q_{composite}=q_{constituent}^{n^2}$ for a system of $n$ identical quons. This result reduces to the …
A fundamental pillar of quantum mechanics concerns indistinguishable quantum particles. In three dimensions they may be classified into fermions or bosons, having, respectively, antisymmetric or symmetric wave functions under particle exchange. One of numerous manifestations of this quantum statistics is the tendency of fermions (bosons) to anti-bunch (bunch). In a two-particle scattering experiment with two possible outgoing channels, the probability of the two particles to arrive each at a different terminal is enhanced (with respect to classical particles) for fermions, and reduced for boso…
By generalizing the recently developed path integral molecular dynamics for identical bosons and fermions, we consider the finite-temperature thermodynamic properties of fictitious identical particles with a real parameter $ξ$ interpolating continuously between bosons ($ξ=1$) and fermions ($ξ=-1$). Through general analysis and numerical experiments we find that the average energy may have good analytical property as a function of this real parameter $ξ$, which provides the chance to calculate the thermodynamical properties of identical fermions by an extrapolation with a simple polynomial func…
We present the Schmidt decomposition for arbitrary wavefunctions of two indistinguishable bosons, extending the recent studies of entanglement or quantum correlations for two fermion systems [J. Schliemann et al., Phys. Rev. B {\bf 63}, 085311 (2001) and quant-ph/0012094]. We point out that the von Neumann entropy of the reduced single particle density matrix remains to be a good entanglement measure for two identical particles.
