We calculate the entanglement of formation and the entanglement of distillation for arbitrary mixtures of the zero spin states on an arbitrary-dimensional bipartite Hilbert space. Such states are relevant to quantum black holes and to decoherence-free subspaces based communication. The two measures of entanglement are equal and scale logarithmically with the system size. We discuss its relation to the black hole entropy law. Moreover, these states are locally distinguishable but not locally orthogonal, thus violating a conjecture that the entanglement measures coincide only on locally orthogonal states. We propose a slightly weaker form of this conjecture. Finally, we generalize our entanglement analysis to any unitary group.
We investigate the teleportation of the bipartite entangled states through two equally noisy quantum channels, namely mixture of Bell states. There is a particular mixed state channel for which all pure entanglement in a known Schmidt basis remain entangled after teleportation and it happens till the channel state remains entangled. Werner state channel lacks both these features. The relation of these noisy channels with violation of Bell's inequality and 2-E inequality is studied.
We present an entanglement purification protocol for a mixture of a pure entangled state and a pure product state, which are orthogonal to each other. The protocol is combination of bisection method and one-way hashing protocol. We give recursive formula for the rate of the protocol for different states, i.e. the number of maximally entangled two-qubit pairs obtained with the protocol per a single copy of the initial state. We also calculate numerically the rate for some states.
This paper is an appendix to a previous paper: quant-ph/0101123 ``Relaxation Method for Calculating Quantum Entanglement", by Robert Tucci. For certain mixtures of Bell basis states, namely the Werner States, we use the theoretical machinery of our previous paper to derive algebraic formulas for: the pure and mixed minimization entanglements (i.e., E_{pure} and E_{mixed}), their optimal decompositions and their entanglement operators. This complements and corroborates some results that were obtained numerically but not algebraically in our previous paper. Some of the algebraic formulas present…
Entanglement in incoherent mixtures of pure states of two qubits is considered via the concurrence measure. A set of pure states is optimal if the concurrence for any mixture of them is the weighted sum of the concurrences of the generating states. When two or three pure real states are mixed it is shown that 28.5% and 5.12% of the cases respectively, are optimal. Conditions that are obeyed by the pure states generating such optimally entangled mixtures are derived. For four or more pure states it is shown that there are no such sets of real states. The implications of these on superposition o…
We study the diffraction of two maximally entangled qubits encoded into spatial modes of light with orbital angular momenta $\pm \hbar l_0$, on angular apertures. Exploiting the uncertainty relation between angular position and momentum, we quantify the diffraction-induced entanglement loss, for different $l_0$, as a function of the angular uncertainty $Δϕ$ induced by the apertures. We demonstrate that the degradation of entanglement follows a universal law parametrised by the product $l_0Δϕ$ alone.} {We study the diffraction of two photonic qubits, which are initially maximally entangled in t…
Entanglement is the quintessential quantum mechanical phenomenon understood to lie at the heart of future quantum technologies and the subject of fundamental scientific investigations. Mixture, resulting from noise, is often an unwanted result of interaction with an environment, but is also of fundamental interest, and is proposed to play a role in some biological processes. Here we report an integrated waveguide device that can generate and completely characterize pure two-photon states with any amount of entanglement and arbitrary single-photon states with any amount of mixture. The device c…
We investigate the possibility to restore genuine tripartite entanglement under local amplitude damping. We show that it is possible to protect genuine entanglement using CNOT and Hadamard gates. We analyze several ordering of such recovery operations. We find that for recovery operations applied after exposing qubits to decoherence, there is no enhancement in lifetime of genuine entanglement. Actual retrieval of entanglement is only possible when reversal scheme is applied before and after the decoherence process. We find that retrieval of entanglement for mixture of $|\widetilde{W}\rangle$ s…
We investigate two-photon entangled states using two important degrees of freedom of the electromagnetic field, namely orbital angular momentum (OAM) and spin angular momentum. For photons propagating in the same direction we apply the idea of $\textit{entanglement duality}$ and develop schemes to do $\textit{entanglement sorting}$ based either on OAM or polarization. In each case the entanglement is tested using appropriate witnesses. We finally present generalizations of these ideas to three- and four-photon entangled states.
We demonstrate experimentally how orbital-angular-momentum entanglement of two photons evolves under influence of atmospheric turbulence. We find that the quantum channel capacity is surprisingly robust: Its typical horizontal decay distance is of the order of 2 kilometers, demonstrating the potential of photonic orbital angular momentum for free-space quantum communication in a metropolitan environment.
We consider the exchange of spin and orbital angular momenta between a circularly polarized Laguerre-Gaussian beam of light and a single atom trapped in a two-dimensional harmonic potential. The radiation field is treated classically but the atomic center-of-mass motion is quantized. The spin and orbital angular momenta of the field are individually conserved upon absorption, and this results in the entanglement of the internal and external degrees of freedom of the atom. We suggest applications of this entanglement in quantum information processing.
The Shannon dimensionality of orbital angular momentum (OAM) entanglement produced in spontaneous parametric down-conversion can be probed by using multi-sector phase analysers. We demonstrate a spatial light modulator-based implementation of these analysers, and use it to measure a Schmidt number of about 50.
We study the entanglement evolution of photonic orbital angular momentum qubit states with opposite azimuthal indices $l_0$, in a weakly turbulent atmosphere. Using asymptotic methods, we deduce analytical expressions for the amplitude of turbulence-induced crosstalk between the modes $l_0$ and $-l_0$. Furthermore, we analytically establish distinct, universal entanglement decay laws for Kolmogorov's turbulence model and for two approximations thereof.
Quantum protocols will be more efficient with high-dimensional entangled states. Photons carrying orbital angular momenta can be used to create a high-dimensional entangled state. In this paper we experimentally demonstrate the entanglement of the orbital angular momentum between the Stokes and anti-Stokes photons generated in a hot atomic ensemble using spontaneous four-wave-mixing. This experiment also suggests the existence of the entanglement concerned with spatial degrees of freedom between the hot atomic ensemble and the Stokes photon.
We provide an analytical expression for the entanglement decay of initially maximally entangled orbital angular momentum bi-photon states, when scattered off an obstruction. We show that the decay is controlled by the diffraction-induced mutual overlap between the diffracted field modes, and quantify its dependence on the size and position of the obstruction.
We examine the entanglement induced by an angular momentum coupling between two harmonic systems. The Hamiltonian corresponds to that of a charged particle in a uniform magnetic field in an anisotropic quadratic potential, or equivalently, to that of a particle in a rotating quadratic potential. We analyze both the vacuum and thermal entanglement, obtaining analytic expressions for the entanglement entropy and negativity through the gaussian state formalism. It is shown that vacuum entanglement diverges at the edges of the dynamically stable sectors, increasing with the angular momentum and sa…
We report the first quantum entanglement experiment in South Africa. The spatial modes of the entangled photon pair are investigated with their potential for high-dimensional entanglement. The generation, measurement and characterisation of the entangled states are examined in detail, where we show high-dimensional entanglement in a Hilbert space of dimension 25. We highlight the experimental challenges contained within each step and provide practical techniques for future experiments in the quantum regime.
Entanglement swapping generates remote quantum correlations between particles that have not interacted and is the cornerstone of long-distance quantum communication, quantum networks, and fundamental tests of quantum science. In the context of spatial modes of light, high-dimensional entanglement provides an avenue to increase the bandwidth of quantum communications and provides more stringent limits for tests of quantum foundations. Here we simultaneously swap the entanglement of multiple orbital angular momentum states of light. The system is based on a degenerate filter that cannot distingu…
We propose a method for generation of entangled photonic states in high dimensions, the so-called qudits, by exploiting quantum correlations of Orbital Angular Momentum (OAM) entangled photons, produced via Spontaneous Parametric Down Conversion. Diffraction masks containing $N$ angular slits placed in the path of twin photons define a qudit space of dimension $N^2$, spanned by the alternative pathways of OAM-entangled photons. We quantify the high-dimensional entanglement of path-entangled photons by the Concurrence, using an analytic expression valid for pure states. We report numerical resu…
So far experimental confirmation of entanglement has been restricted to qubits, i.e. two-state quantum systems including recent realization of three- and four-qubit entanglements. Yet, an ever increasing body of theoretical work calls for entanglement in quantum system of higher dimensions. Here we report the first realization of multi-dimensional entanglement exploiting the orbital angular momentum of photons, which are states of the electromagnetic field with phase singularities (doughnut modes). The properties of such states could be of importance for the efforts in the field of quantum com…
The singular nature of a non-integer spiral phase plate allows easy manipulation of spatial degrees of freedom of photon states. Using two such devices, we have observed very high dimensional (D > 3700) spatial entanglement of twin photons generated by spontaneous parametric down-conversion.
We show that the transfer of the plane wave spectrum of the pump beam to the fourth-order transverse spatial correlation function of the two-photon field generated by spontaneous parametric down-conversion leads to the conservation and entanglement of orbital angular momentum of light. By means of a simple experimental setup based on fourth-order (or two-photon) interferometry, we show that our theoretical model provides a good description for down-converted fields carrying orbital angular momentum.
Any practical experiment utilising the innate D-dimensional entanglement of the orbital angular momentum (OAM) state space of photons is subject to the modal capacity of the detection system. We show that given such a constraint, the number of measured, entangled OAM modes in photon pairs generated by spontaneous parametric down-conversion (SPDC) can be maximised by tuning the phase-matching conditions in the SPDC process. We demonstrate a factor of 2 increase on the half-width of the OAM-correlation spectrum, from 10 to 20, the latter implying \approx 50 -dimensional two-photon OAM entangleme…
We report the first experimental characterization of the first-order continuous variable orbital angular momentum states. Using a spatially non-degenerate optical parametric oscillator (OPO) we produce quadrature entanglement between the two first-order Laguerre-Gauss modes. The family of OAM modes is mapped on an orbital Poincare sphere, and the modes position on the sphere is spanned by the three orbital parameters. Using the non-degenerate OPO we produce squeezing of these parameters, and as an illustration, we reconstruct the "cigar-shaped" uncertainty volume on the orbital Poincare sphere…
The orbital angular momentum of light (OAM) provides a promising approach for the implementation of multidimensional states (qudits) for quantum information purposes. In order to characterize the degradation undergone by the information content of qubits encoded in a bidimensional subspace of the orbital angular momentum degree of freedom of photons, we study how the state fidelity is affected by a transverse obstruction placed along the propagation direction of the light beam. Emphasis is placed on the effects of planar and radial hard-edged aperture functions on the state fidelity of Laguerr…
Three-dimensional entanglement of orbital angular momentum states of an atomic qutrit and a single photon qutrit has been observed. Their full state was reconstructed using quantum state tomography. The fidelity to the maximally entangled state of Schmidt rank 3 exceeds the threshold 2/3. This result confirms that the density matrix cannot be decomposed into ensemble of pure states of Schmidt rank 1 or 2. That is, the Schmidt number of the density matrix must be equal to or greater than 3.
Recently, atomic ensemble and single photons were successfully entangled by using collective enhancement [D. N. Matsukevich, \textit{et al.}, Phys. Rev. Lett. \textbf{95}, 040405(2005).], where atomic internal states and photonic polarization states were correlated in nonlocal manner. Here we experimentally clarified that in an ensemble of atoms and a photon system, there also exists an entanglement concerned with spatial degrees of freedom. Generation of higher-dimensional entanglement between remote atomic ensemble and an application to condensed matter physics are also discussed.
Here I present a method how intersections of a certain density matrix of rank two with the zero-polytope can be calculated exactly. This is a purely geometrical procedure which thereby is applicable to obtaining the zeros of SL- and SU-invariant entanglement measures of arbitrary polynomial degree. I explain this method in detail for a recently unsolved problem. In particular, I show how a three-dimensional view, namely in terms of the Boch-sphere analogy, solves this problem immediately. To this end, I determine the zero-polytope of the three-tangle, which is an exact result up to computer ac…
The fact that the computational cost of simulating a many-body quantum system on a computer increases with the amount of entanglement has been considered as the major bottleneck for simulating its out-of-equilibrium dynamics. Some aspects of the dynamics are, nevertheless, robust under appropriately devised approximations. Here we present a possible algorithm that allows to systematically approximate the equilibration value of local operators after a quantum quench. At the core of our proposal there is the idea to transform entanglement between distant parts of the system into mixture, and at …
The Einstein Podolsky Rosen (EPR) entangled quantum state is of special importance not only for fundamental research in quantum mechanics, but also for information processing in the field of quantum information. Previous EPR entangled state demonstrations were constructed with photons of equal phase wave fronts. More complex scenarios with structured wave fronts have not been investigated. Here, we report the first experimental demonstration of EPR entanglement for photon pairs carrying orbital angular momentum (OAM) information, resulting in an OAM embedded EPR entangled state. We measured th…
Entangled quantum states play an important role in quantum information science and also in quantum mechanics fundamental investigations. Implementation and characterization of techniques allowing for easy preparation of entangled states are important steps in such fields. Here we generated entangled quantum states encoded in photons transversal paths, obtained by pumping a non-linear crystal with multiple transversal Gaussian beams. Such approach allows us to generate entangled states of two qubits and two qutrits encoded in Gaussian transversal path of twin photons. We make a theoretical anal…
Photons with a twisted phase front carry a quantized amount of orbital angular momentum (OAM) and have become important in various fields of optics, such as quantum and classical information science or optical tweezers. Because no upper limit on the OAM content per photon is known, they are also interesting systems to experimentally challenge quantum mechanical prediction for high quantum numbers. Here, we take advantage of a recently developed technique to imprint unprecedented high values of OAM, namely spiral phase mirrors (SPM), to generate photons with more than 10,000 quanta of OAM. More…
Here we present the quantum storage of three-dimensional orbital-angular-momentum photonic entanglement in a rare-earth-ion-doped crystal. The properties of the entanglement and the storage process are confirmed by the violation of the Bell-type inequality generalized to three dimensions after storage ($S=2.152\pm0.033$). The fidelity of the memory process is $0.993\pm0.002$, as determined through complete quantum process tomography in three dimensions. An assessment of the visibility of the stored weak coherent pulses in higher-dimensional spaces, demonstrates that the memory is highly reliab…
We use a recently proposed measure of quantum correlations (work deficit), to measure the strength of the nonlocality of an equal mixture of two bipartite, orthogonal, but locally indistinguishable separable states. This gives supporting evidence of nonzero value for a separable state, for this measure of nonlocality. We also show that this measure of quantum correlations places a different order on the set of states, than the good asymptotic measures of entanglement. And that such a different order imposed on two states by the work deficit and any entanglement measure, cannot be explained by …
The fundamental quantum dynamics of two interacting oscillator systems are studied in two different scenarios. In one case, both oscillators are assumed to be linear, whereas in the second case, one oscillator is linear and the other is a non-linear, angular-momentum oscillator; the second case is, of course, more complex in terms of energy transfer and dynamics. These two scenarios have been the subject of much interest over the years, especially in developing an understanding of modern concepts in quantum optics and quantum electronics. In this work, however, these two scenarios are utilized…
Different regimes of entanglement growth under measurement have been demonstrated for quantum many-body systems, with an entangling phase for low measurement rates and a disentangling phase for high rates (quantum Zeno effect). Here we study entanglement growth on a disordered Bose-Fermi mixture with the bosons playing the role of the effective self-induced measurement for the fermions. Due to the interplay between the disorder and a non-Abelian symmetry, the model features an entanglement growth resonance when the boson-fermion interaction strength is varied. With the addition of a magnetic f…
Orbital angular momentum (OAM) multiplexing provides an efficient method to improve data-carrying capacity in various quantum communication protocols. It is a precondition to distribute OAM multiplexed quantum resources in quantum channels for implementing quantum communication. However, quantum steering of OAM multiplexed optical fields and the effect of channel noise on OAM multiplexed quantum resources remain unclear. Here, we generate OAM multiplexed continuous-variable (CV) entangled states and distribute them in lossy or noisy channels. We show that the decoherence property of entangleme…
We establish a conceptual framework for the identification and the characterization of induced interactions in binary mixtures and reveal their intricate relation to entanglement between the components or species of the mixture. Exploiting an expansion in terms of the strength of the entanglement among the two species, enables us to deduce an effective single-species description. In this way, we naturally incorporate the mutual feedback of the species and obtain induced interactions for both species which are effectively present among the particles of same type. Importantly, our approach incor…
High-dimensional entanglement has demonstrated potential for increasing channel capacity and resistance to noise in quantum information processing. However, its distribution is a challenging task, imposing a severe restriction on its application. Here we report the first distribution of three-dimensional orbital angular momentum (OAM) entanglement via a 1-km-long optical fibre. Using an actively-stabilizing phase pre-compensation technique, we successfully transport one photon of a three-dimensional OAM entangled photon pair through the fibre. The distributed OAM entangled state still shows a …
The standard kicked top involves a periodically kicked angular momentum. By considering this angular momentum as a collection of entangled spins, we compute the bipartite entanglement dynamics as a function of the dynamics of the classical counterpart. Our numerical results indicate that the entanglement of the quantum top depends on the specific details of the dynamics of the classical top rather than depending universally on the global properties of the classical regime. These results are grounded on linking the entanglement rate to averages involving the classical angular momentum, thereby …
Converting long-range entanglement into mixture: tensor-network approach to local equilibration
Max-Planck-Institut für Quantenoptik（马克斯·普朗克量子光学研究所）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；Institute of Fundamental Physics IFF-CSIC（CSIC基础物理研究所）
Orbital angular momentum entanglement is one of the most intriguing topics in quantum physics. A broad range of research have been dedicated either to unravel its underlying physics or to expand the entanglement dimensions and degrees. In this paper, we present a theoretical study on the orbital angular momentum entanglement by employing the Heisenberg uncertainty principle to quantum position correlation within the azimuthal region. In this study, we decompose the pump light into a set of pump cone states characterized by their radii. The OAM entanglement can be manipulated by controlling the…
We introduce the concept of Shannon dimensionality D as a new way to quantify bipartite entanglement as measured in an experiment. This is applied to orbital-angular-momentum entanglement of two photons, using two state analyzers composed of a rotatable angular-sector phase plate that is lens-coupled to a single-mode fiber. We can deduce the value of D directly from the observed two-photon coincidence fringe. In our experiment, D varies between 2 and 6, depending on the experimental conditions. We predict how the Shannon dimensionality evolves when the number of angular sectors imprinted in th…
Constructing a quantum memory for a photonic entanglement is vital for realizing quantum communication and network. Besides enabling the realization of high channel capacity communication, entangled photons of high-dimensional space are of great interest because of many extended applications in quantum information and fundamental physics fields. Photons entangled in a two-dimensional space had been stored in different system, but there have been no any report on the storage of a photon pair entangled in a high-dimensional space. Here, we report the first experimental realization of storing an …
The turbulence induced decay of orbital angular momentum (OAM) entanglement between two photons is investigated numerically and experimentally. To compare our results with previous work, we simulate the turbulent atmosphere with a single phase screen based on the Kolmogorov theory of turbulence. We consider two different scenarios: in the first only one of the two photons propagates through turbulence, and in the second both photons propagate through uncorrelated turbulence. Comparing the entanglement evolution for different OAM values, we found the entanglement to be more robust in turbulence…
We study the evolution of an orbital angular momentum (OAM) entangled bipartite photonic state for the case where one of the photons propagates through Kolmogorov turbulence, using the concurrence as a measure of entanglement. Quantum state tomography was performed to reconstruct the two qubit density matrices for a range of scintillation strengths. Our results give the first direct experimental confirmation of the existing theories for decay of entanglement due to atmospheric turbulence. We also show how the modal scattering increases with increasing scintillation and we discuss the impact of…
Single photons with helical phase structures may carry a quantized amount of orbital angular momentum (OAM) and their entanglement is important for quantum information science and fundamental tests of quantum theory. Because there is no theoretical upper limit on how many quanta of OAM a single photon can carry, it is possible to create entanglement between two particles with an arbitrary high difference in the quantum number. By transferring polarization entanglement to OAM with an interferometric scheme, we generate and verify entanglement up to 600 quanta difference in the orbital angular m…
We experimentally demonstrate a quantum key distribution (QKD) protocol using photon pairs entangled in orbit angular momentum (OAM). In our protocol, Alice and Bob modulate their OAM states on each entangled pair with spatial light modulators (SLMs), respectively. Alice uses a fixed phase hologram in her SLM, while Bob designs $N$ different suitable phase holograms and uses them to represent his $N$-based information in his SLM. With coincidences, Alice can fully retrieve the key stream sent by Bob without information reconciliation or privacy amplification. We report the experiment results w…
We put forward an approach to estimate the amount of bipartite spatial entanglement of down-converted photon states correlated in orbital angular momentum and the magnitude of the transverse (radial) wave vectors. Both degrees of freedom are properly considered in our framework, which only requires azimuthal local linear optical transformations and mode selection analysis with two fiber detectors. The coincidence distributions predicted by our approach give an excellent fit to the distributions measured in a recent experiment aimed to show the very high-dimensional transverse entanglement of t…
We demonstrate entanglement between the orbital angular momentum (OAM) of two photoelectrons ionized via the strongly correlated process of non-sequential double ionization (NSDI). Due to the quantization of OAM, this entanglement is easily quantified and has a simple physical interpretation in terms of conservation laws. We explore detection by an entanglement witness, decomposable into local measurements, which strongly reduces the difficulty of experimental implementation. We compute the logarithmic negativity measure, which is directly applicable to mixed states, to demonstrate that the en…
We propose an interferometric method to investigate the non-locality of high-dimensional two-photon orbital angular momentum states generated by spontaneous parametric down conversion. We incorporate two half-integer spiral phase plates and a variable-reflectivity output beam splitter into a Mach-Zehnder interferometer to build an orbital angular momentum analyzer. This setup enables testing the non-locality of high-dimensional two-photon states by repeated use of the Clauser-Horne inequality.
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 …
