We study the entangled states that can be generated using two species of atoms trapped in independently movable, two-dimensional optical lattices. We show that using two sets of measurements it is possible to measure a set of entanglement witness operators distributed over arbitrarily large regions of the lattice, and use these witnesses to produce two-dimensional plots of the entanglement content of these states. We also discuss the influence of noise on the states and on the witnesses, as well as connections to ongoing experiments.
Entanglement, quantum steering and nonlocality are distinct quantum correlations which are the resources behind various of quantum information and quantum computation applications. However, a central question of determining the precise quantitative relation among them is still unresolved. Here, we present a mapping criterion between Bell nonlocality and quantum steering in bipartite qudit-qubit system, as well as a mapping criterion between quantum steering and quantum entanglement in bipartite qubit-qudit system, starting from the fundamental concepts of quantum correlations. Precise quantita…
We investigate how the dynamical production of quantum entanglement for weakly coupled mapping systems is influenced by the chaotic dynamics of the corresponding classical system. We derive a general perturbative formula for the entanglement production rate which is defined by using the linear entropy of the subsystem. This formula predicts that {\it the increment of the strength of chaos does not enhance the production rate of entanglement} when the coupling is weak enough and the subsystems are strongly chaotic. The prediction is confirmed by numerical experiments for coupled kicked tops and…
We describe the transfer of quantum information and correlations from an entangled tripartite bosonic system to three separate qubits through their local environments also in the presence of various dissipative effects. Optimal state mapping and entanglement transfer are shown in the framework of optical cavity quantum electrodynamics involving qubit-like radiation states and two-level atoms via the mediation of cavity modes. For an input GHZ state mixed with white noise we show the occurrence of sudden death and birth of entanglement that is discontinuously exchanged among the tripartite subs…
Recently, we have shown how the phase of an electromagnetic field can be determined by measuring the population of either of the two states of a two-level atomic system excited by this field, via the so-called Bloch-Siegert oscillation resulting from the interference between the co- and counter-rotating excitations. Here, we show how a degenerate entanglement, created without transmitting any timing signal, can be used to teleport this phase information. This phase-teleportation process may be applied to achieve absolute phase-mapping of remote clocks.
We establish a general operational one-to-one mapping between coherence measures and entanglement measures: Any entanglement measure of bipartite pure states is the minimum of a suitable coherence measure over product bases. Any coherence measure of pure states, with extension to mixed states by convex roof, is the maximum entanglement generated by incoherent operations acting on the system and an incoherent ancilla. Remarkably, the generalized CNOT gate is the universal optimal incoherent operation. In this way, all convex-roof coherence measures, including the coherence of formation, are end…
We show that the general Heisenberg Hamiltonian with non-uniform couplings can be characterised by mapping the entanglement it generates as a function of time. Identification of the Hamiltonian in this way is possible as the coefficients of each operator control the oscillation frequencies of the entanglement function. The number of measurements required to achieve a given precision in the Hamiltonian parameters is determined and an efficient measurement strategy designed. We derive the relationship between the number of measurements, the resulting precision and the ultimate discrete error pro…
We show that any multi-qudit entanglement witness leads to a non-separability indicator for quantum optical fields, which involves intensity correlations. We get, e.g., necessary and sufficient conditions for intensity or intensity-rate correlations to reveal polarization entanglement. We also derive separability conditions for experiments involving multiport interferometers, now feasible with integrated optics. We show advantages of using intensity rates rather than intensities, e.g., a mapping of Bell inequalities to ones for optical fields. The results have implication for studies of non-cl…
Entanglement in the symmetric subspace: mapping multipartite to bipartite states
Istituto Nazionale di Ottica - Consiglio Nazionale delle Ricerche (INO-CNR)（意大利国家光学研究所-意大利国家研究委员会）；University of Tsukuba（筑波大学）；Center for Artificial Intelligence Research (C-AIR)（人工智能研究中心）；LENS（欧洲纳米科学实验室）
Recent developments of quantum information science critically rely on entanglement, an intriguing aspect of quantum mechanics where parts of a composite system can exhibit correlations stronger than any classical counterpart. In particular, scalable quantum networks require capabilities to create, store, and distribute entanglement among distant matter nodes via photonic channels. Atomic ensembles can play the role of such nodes. So far, in the photon counting regime, heralded entanglement between atomic ensembles has been successfully demonstrated via probabilistic protocols. However, an inhe…
An obvious way to entangle two atoms located at remote locations is to produce a pair of entangled photons half-way between the two atoms, to sent one photon to each location and to subsequently map the photonic entanglement into the atoms. The efficiency of this process is, however, fundamentally limited due to overall transmission losses. We propose a method to herald the success of the mapping operation in free space without destroying nor revealing the stored quantum state. Interestingly for a Bell test, the heralding signal does not open the detection loophole provided the measurement cho…
Recently, we have shown how the phase of an electromagnetic field can be determined by measuring the population of either of the two states of a two-level atomic system excited by this field, via the so-called Bloch-Siegert oscillation resulting from the interference between the co- and counter-rotating excitations. Here, we show how a degenerate entanglement, created without transmitting any timing signal, can be used to teleport this phase information. This phase-teleportation process may be applied to achieve frequency-locking of remote oscillators, thereby facilitating the process of synch…
We present a new approach to the analysis of entanglement in smooth bipartite continuous-variable states. One or both parties perform projective filterings via preliminary measurements to determine whether the system is located in some region of space; we study the entanglement remaining after filtering. For small regions, a two-mode system can be approximated by a pair of qubits and its entanglement fully characterized, even for mixed states. Our approach may be extended to any smooth bipartite pure state or two-mode mixed state, leading to natural definitions of concurrence and negativity de…
We describe the transfer of quantum information and entanglement from three flying (radiation) to three localized (atomic) qubits via cavity modes resonantly coupled to the atoms, in the presence of a common reservoir. Upon addressing the full dynamics of the resulting nine-qubit open system, we find that once the cavities are fed, fidelity and transferred entanglement are optimal, while their peak values exponentially decrease due to dissipative processes. The external radiation is then turned off and quantum correlations oscillate between atomic and cavity qubits. For a class of mixtures of …
This note shows how quantum entanglement may be simulated in classical computing. The simulated entanglement protocol is implemented using oblivious transfer in the simplest case and other many-to-one mappings in more general cases. For the case where the mapping is of order k, we prove a theorem that gives us the function of the entangled state. A useful generalization beyond this research will be the implementation of quantum states with arbitrary probability amplitude functions.
The production of quantum entanglement between weakly coupled mapping systems, whose classical counterparts are both strongly chaotic, is investigated. In the weak coupling regime, it is shown that time correlation functions of the unperturbed systems determine the entanglement production. In particular, we elucidate that the increment of nonlinear parameter of coupled kicked tops does not accelerate the entanglement production in the strongly chaotic region. An approach to the dynamical inhibition of entanglement is suggested.
Polarization-entangled photon pairs can be efficiently prepared into pure Bell states with a high fidelity via type-II spontaneous parametric down-conversion (SPDC) of narrow-band pump light. However, the use of femtosecond pump pulses to generate multi-photon states with precise timing often requires spectral filtering to maintain a high quality of polarization entanglement. This typically reduces the efficiency of photon pair collection. We experimentally map the polarization correlations of photon pairs from such a source over a range of down-converted wavelengths with a high spectral resol…
Parametric down-conversion can produce photons that are entangled both in polarization and in space. Here we show how the spatial entanglement can be used to purify the polarization entanglement using only linear optical elements. Spatial entanglement as an additional resource leads to a substantial improvement in entanglement output compared to a previous scheme. Interestingly, in the present context the thermal character of down-conversion sources can be turned into an advantage. Our scheme is realizable with current technology.
We present a simple model together with its physical implementation which allows one to generate multipartite entanglement between several spatial modes of the electromagnetic field. It is based on parametric down-conversion with N pairs of symmetrically-tilted plane waves serving as a pump. The characteristics of this spatial entanglement are investigated in the cases of zero as well as nonzero phase mismatch. Furthermore, the phenomenon of entanglement localization in just two spatial modes is studied in detail and results in an enhancement of the entanglement by a factor square root of N.
We present an efficient entanglement purification protocol with hyperentanglement in which additional spatial entanglement is utilized to purify the two-particle polarization-entangled state. The bit-flip error and phase-flip error can be corrected and eliminated in one step. Two remote parties can obtainmaximally entangled polarization states deterministically and only passive linear optics are employed. We also discuss the protocol with practical quantum source and noisy channel.
Most investigations of multipartite entanglement have been concerned with temporal modes of the electromagnetic field, and have neglected its spatial structure. We present a simple model which allows to generate tripartite entanglement between spatial modes by parametric down-conversion with two symmetrically-tilted plane waves serving as a pump. The characteristics of this entanglement are investigated. We also discuss the generalization of our scheme to 2N+1-partite entanglement using 2N symmetrically-tilted plane pump waves. Another interesting feature is the possibility of entanglement loc…
We propose a novel scheme to generate polarization entanglement from spatially-correlated photon pairs. We experimentally realized a scheme by means of a spatial correlation effect in a spontaneous parametric down-conversion and a modified Michelson interferometer. The scheme we propose in this paper can be interpreted as a conversion process from spatial correlation to polarization entanglement.
We investigate spatial entanglement in a system of spinless non-interacting bosons in a one dimensional harmonic trap described by the grand canonical ensemble. We find the lower bound for the amount of spatial entanglement contained in the average state of two bosons in the trap and show that it exists for arbitrary high temperatures.
We investigate spatial entanglement - particle-number entanglement between regions of space - in an ideal Bosonic gas. We quantify the amount spatial entanglement around the transition temperature for condensation by probing the gas with two localised two-level systems. We show that spatial entanglement in the gas is directly related to filling of the ground state energy level and therefore to the off-diagonal long-range order of the system and the onset of condensation.
In \cite{CMW19}, the authors introduced $k$-entanglement breaking linear maps to understand the entanglement breaking property of completely positive maps on taking composition. In this article, we do a systematic study of $k$-entanglement breaking maps. We prove many equivalent conditions for a $k$-positive linear map to be $k$-entanglement breaking, thereby study the mapping cone structure of $k$-entanglement breaking maps. We discuss examples of $k$-entanglement breaking maps and some of their significance. As an application of our study, we characterize completely positive maps that reduce…
The vacuum state of a relativistic quantum field contains entanglement between regions separated by spacelike intervals. Such spatial entanglement can be revealed using an operational method introduced in Ann. Phys. 351, 112 (2014), Phys. Rev. D 91, 016005 (2014). In this approach, a cavity is instantaneously divided into halves by an introduction of an extra perfect mirror. Causal separation of the two regions of the cavity reveals nonlocal spatial correlations present in the field, which can be quantified by measuring particles generated in the process. We use this method to study spatial en…
We present a novel scheme to generate entanglement between two spatially separated systems. The scheme makes use of spatial entanglement generated by a single-particle quantum walk which is used to entangle two spatially separated, not necessarily correlated, systems. This scheme can be used to entangle any two systems which can interact with the spatial modes entangled during the quantum walk evolution. A notable feature is that we can control the quantum walk dynamics and its ability to localize leads to a substantial control and improvement in the entanglement output.
The evolution of a many-particle system on a one-dimensional lattice, subjected to a quantum walk can cause spatial entanglement in the lattice position, which can be exploited for quantum information/communication purposes. We demonstrate the evolution of spatial entanglement and its dependence on the quantum coin operation parameters, the number of particles present in the lattice and the number of steps of quantum walk on the system. Thus, spatial entanglement can be controlled and optimized using a many-particle discrete-time quantum walk.
We discuss dynamical enhancement of entanglement in a driven Bose-Hubbard model and find an enhancement of two orders of magnitude which is robust against fluctuations in experimental parameters.
We investigate the mapping between magnetic susceptibility and entanglement in the metallic, insulating, conventional and exotic polarized superfluid phases of one-dimensional fermionic lattice systems as described by the Hubbard model. Motivated by recent proposals for determining and quantifying entanglement via magnetic susceptibility measurements, we numerically study the intrinsic relationship between the two quantities at zero temperature. We find signatures of the metal-insulator transition and of the BCS-BEC crossover, but the most relevant result is that for conventional and exotic su…
By making use of the spatial shape of paired photons, parametric down-conversion allows the generation of two-photon entanglement in a multidimensional Hilbert space. How much entanglement can be generated in this way? In principle, the infinite-dimensional nature of the spatial degree of freedom renders unbounded the amount of entanglement available. However, in practice, the specific configuration used, namely its geometry, the length of the nonlinear crystal and the size of the pump beam, can severely limit the value that could be achieved. Here we show that the use of quasi-phase-matching …
Cold atomic ensembles can mediate the generation of entanglement between pairs of photons. Photons with specific directions of propagation are detected, and the entanglement can reside in any of the degrees of freedom that describe the whole quantum state of the photons: polarization, spatial shape or frequency. We show that the direction of propagation of the generated photons determines the spatial quantum state of the photons and therefore, the amount of entanglement generated. When photons generated in different directions are combined, this spatial distinguishing information can degrade t…
The photon-pairs generated through spontaneous parametric down-conversion (SPDC) possess strong correlations in their transverse position and momentum degrees of freedom. For such photon pairs, the transverse position correlation length depends on the crystal thickness and the pump wavelength, whereas the transverse momentum correlation length depends on the beam waist and spatial coherence length of the pump. By controlling these parameters, it is possible to engineer the spatial entanglement. Here, we utilize the circular asymmetry of the pump by using an elliptical-Gaussian beam to change t…
A light beam is said to be position squeezed if its position can be determined to an accuracy beyond the standard quantum limit. We identify the position and momentum observables for bright optical beams and show that position and momentum entanglement can be generated by interfering two position, or momentum, squeezed beams on a beam splitter. The position and momentum measurements of these beams can be performed using a homodyne detector with local oscillator of an appropriate transverse beam profile. We compare this form of spatial entanglement with split detection-based spatial entanglemen…
The transverse spatial attributes of an optical beam can be decomposed into the position, momentum and orbital angular momentum observables. The position and momentum of a beam is directly related to the quadrature amplitudes, whilst the orbital angular momentum is related to the polarization and spin variables. In this paper, we study the quantum properties of these spatial variables, using a representation in the Stokes-operator basis. We propose a spatial detection scheme to measure all three spatial variables and consequently, propose a scheme for the generation of spatial Stokes operator …
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 present a one-step deterministic entanglement purification protocol with linear optics and postselection. Compared with the Simon-Pan protocol (Phys. Rev. Lett. 89, 257901 (2002)), this one-step protocol has some advantages. First, it can get a maximally entangled pair with only one step, not only improve the fidelity of less-entangled photon pairs by performing the protocol repeatedly. Second, it works in a deterministic way, not a probabilistic one, which will reduce a great deal of entanglement resources. Third, it does not require the polarization state be entangled, only spatial entang…
We consider how to generate and detect Einstein-Podolsky-Rosen (EPR) entanglement and the steering paradox between groups of atoms in two separated potential wells in a Bose-Einstein condensate (BEC). We present experimental criteria for this form of entanglement, and propose experimental strategies for detecting entanglement using two or four mode ground states. These approaches use spatial and/or internal modes. We also present higher order criteria that act as signatures to detect the multiparticle entanglement present in this system. We point out the difference between spatial entanglement…
The effects of an intracavity photonic crystal in a multimode optical parametric oscillator are studied, with a special focus on quantum fluctuations. The capability to either stimulate or inhibit the spatial instability, lowering or increasing the parametric threshold, allows to control the intensity fluctuations and correlations. A significative quantum noise reduction and an increase of the range of squeezed quadratures are found above threshold where spatial Einstein-Podolsky-Rosen entanglement and inseparability are found.
Optical entanglement is a key requirement for many quantum communication protocols. Conventionally entanglement is formed between two distinct beams, with the quantum correlations being measured at separate locations. We show entanglement between the modes within one beam. Our technique is particularly elegant and a major advance towards practical systems with minimum complexity. We demonstrate three major experimental achievements: (i) only one source is required to produce squeezed light in two orthogonal spatial modes, (ii) the entanglement is formed through lenses and beam rotation, withou…
We report an experiment in which the moments of spatial coordinates of down-converted photons are measured directly, without having to reconstruct any marginal probability distributions. We use a spatial light modulator to couple the spatial degrees of freedom and the polarization of the fields, which act as ancilla systems. Information about the spatial correlations is obtained via measurements on the ancilla qubits. Among other applications, this new method provides a more efficient technique to identify continuous variable entanglement.
We propose a scheme for generating three-dimensional entanglement between two atoms trapped in two spatially separated cavities reapectively via shortcuts to adi- abatic passage based on the approach of Lewis-Riesenfeld invariants in cavity quan- tum electronic dynamics. By combining Lewis-Riesenfeld invariants with quantum Zeno dynamics, we can generate three dimensional entanglement of the two atoms with high fidelity. The Numerical simulation results show that the scheme is ro- bust against the decoherences caused by the photon leakage and atomic spontaneous emission.
We extend a procedure exploiting spatial indistinguishability of identical particles to recover the spoiled entanglement between two qubits interacting with Markovian noisy environments. Here, the spatially localized operations and classical communication (sLOCC) operational framework is used to activate the entanglement restoration from the indistinguishable constituents. We consider the realistic scenario where noise acts for the whole duration of the process. Three standard types of noises are considered: a phase damping, a depolarizing, and an amplitude damping channel. Within this general…
Fiber transport of multi-dimensional photonic qudits promises high information capacity per photon without space restriction. This work experimentally demonstrates transmission of spatial qudits through multi-core optical fibers and measurement of the entanglement between two fibers with quantum state analyzers, each composed of a spatial light modulator and a single-mode fiber. Quantum state tomography reconstructs the four-dimension entangled state that verifies the non-locality through concurrences in two-dimensional subspaces and a high-dimensional Bell-type CGLMP inequality.
We show how to implement quantum transport, generate entangled state and achieve spatial entanglement distribution via topological Thouless pumping in one-dimensional disordered lattices. We introduce the on-site disorders to suppress the high-order resonant tunneling (which cause wavepacket dispersion) and realize dispersionless Thouless pumping. The interplay between the on-site disorders and the band topology enables robust unidirectional transport. We also demonstrate how to prepare spatially entangled two-particle NOON state via Hong-Ou-Mandel interference assisted by the Thouless pumping…
Multiparticle entangled states, essential ingredients for modern quantum technologies, are routinely generated in experiments of atomic Bose-Einstein condensates (BECs). However, the entanglement in ultracold interacting Fermi gases has not been yet exploited. In this work, by using an ansatz of composite bosons, we show that many-particle entanglement between two fermionic ensembles localized in spatially separated modes can be generated by splitting an ultracold interacting Fermi gas in the (molecular) BEC regime. This entanglement relies on the fundamental fermion exchange symmetry of molec…
Higher dimensional quantum systems are an important avenue for new explorations in quantum computing as well as quantum communications. One of the ubiquitous resources in quantum technologies is entanglement. However, so far, entanglement has been certified in higher dimensional systems through suitable bounds on known entanglement measures. In this work, we have, for the first time, quantified the amount of entanglement in bi-partite pure qutrit states by analytically relating statistical correlation measures and known measures of entanglement, and have determined the amount of entanglement i…
The time dependent quantum Monte Carlo method for fermions is introduced and applied for calculation of entanglement of electrons in one-dimensional quantum dots with several spin-polarized and spin-compensated electron configurations. The rich statistics of wave functions provided by the method allows one to build reduced density matrices for each electron and to quantify the spatial entanglement using measures such as quantum entropy by treating the electrons as identical or distinguishable particles. Our results indicate that the spatial entanglement in parallel-spin configurations is rathe…
We study the dynamics of the non-classical correlations for few atom systems in the presence of strong interactions for a number of recently developed adiabatic state preparation protocols. We show that entanglement can be created in a controlled fashion and can be attributed to two distinct sources, the atom-atom interaction and the distribution of atoms among different traps.
Initialization of composite quantum systems into highly entangled states is usually a must to allow their use for quantum technologies. However, the presence of unavoidable noise in the preparation stage makes the system state mixed, thus limiting the possibility of achieving this goal. Here we address this problem in the context of identical particle systems. We define the entanglement of formation for an arbitrary state of two identical qubits within the operational framework of spatially localized operations and classical communication (sLOCC). We then introduce an entropic measure of spati…
Remote spatial indistinguishability of identical subsystems as a direct controllable quantum resource at distant sites has not been yet experimentally proven. We design a setup capable to tune the spatial indistinguishability of two photons by independently adjusting their spatial distribution in two distant regions, which leads to polarization entanglement starting from uncorrelated photons. The amount of entanglement uniquely depends on the degree of remote spatial indistinguishability, quantified by an entropic measure $\mathcal{I}$, which enables teleportation with fidelities above the cla…
We consider the entanglement dynamics between two Unruh-DeWitt detectors at rest separated at a distance $d$. This simple model when analyzed properly in quantum field theory shows many interesting facets and helps to dispel some misunderstandings of entanglement dynamics. We find that there is spatial dependence of quantum entanglement in the stable regime due to the phase difference of vacuum fluctuations the two detectors experience, together with the interference of the mutual influences from the backreaction of one detector on the other. When two initially entangled detectors are still ou…
We investigate the practicality of the method proposed by Maciel et al. [Phys. Rev. A. 80, 032325(2009)] for detecting the entanglement of two spatial qutrits (3-dimensional quantum systems), which are encoded in the discrete transverse momentum of single photons transmitted through a multi-slit aperture. The method is based on the acquisition of partial information of the quantum state through projective measurements, and a data processing analysis done with semi-definite programs. This analysis relies on generating gradually an optimal entanglement witness operator, and numerical investigati…
In this paper we discuss the entanglement properties of a thermal non-relativistic free bosonic field. We demonstrate how to formally construct spatial modes in order to use a continuous variable separability criterion and show that the construction of the modes plays a significant role in the way the entanglement manifests itself. For instance the presence of entanglement depends on the shape of the modes and their separation. The temperature of the gas is another crucial factor and for one choice of modes we show that entanglement can be found at arbitrarily high temperatures as long as we c…
Nonclassical beams in high order spatial modes have attracted much interest but they exhibit much less squeezing and entanglement than the fundamental spatial modes, limiting their applications. We experimentally demonstrate the relation between pump modes and entanglement of first-order HG modes (HG10 entangled states) in a type II OPO and show that the maximum entanglement of high order spatial modes can be obtained by optimizing the pump spatial mode. To our knowledge, this is the first time to report this. Utilizing the optimal pump mode, the HG10 mode threshold can be reached easily witho…
We report on a new technique for entanglement distillation of the bipartite continuous variable state of spatially correlated photons generated in the spontaneous parametric down-conversion process (SPDC), where tunable non-Gaussian operations are implemented and the post-processed entanglement is certified in real-time using a single-photon sensitive electron multiplying CCD (EMCCD) camera. The local operations are performed using non-Gaussian filters modulated into a programmable spatial light modulator and, by using the EMCCD camera for actively recording the probability distributions of th…
