Using angular position-orbital angular momentum entangled photons, we propose an experiment to generate maximally entangled states of $D$-dimensional quantum systems, the so called qudits, by exploiting correlations of parametric down-converted photons. Angular diffraction masks containing $N$-slits in the arms of each twin photon define a qudit space of dimension $N^2$, spanned by the alternative pathways of the photons. Due to phase-matching conditions, the twin photons will pass only by symmetrically opposite angular slits, generating maximally entangled states between these different paths, which can be detected by high-order two-photon interference fringes via coincidence counts. Numerical results for $N$ angular slits with $N = 2, 4, 5, 6, 10$ are reported, corresponding to qudit Hilbert spaces of dimension $D=N^2=4,16,25, 36,100$, respectively. We discuss relevant experimental parameters for an experimental implementation of the proposed scheme using Spatial Light Modulators (SLMs), and twin-photons produced by Spontaneouos Parametric Down Conversion (SPDC). The entanglement of the qudit state can be quantified in terms of the Concurrence, which can be expressed in terms of the visibility of the interference fringes, or by using Entanglement Witnesses. These results provide an additional means for preparing entangled quantum states in high-dimensions, a fundamental resource for quantum simulation and quantum information protocols.
We propose a method to perform the quantum state tomography (QST) of an $\mathit{n}$-partite qudit state embedded in single photons using the Hong-Ou-Mandel (HOM) interference between the target state and probe state. This method requires only passive beam splitters for the HOM interference and removes all active optical devices in the target modes to control the measurement bases needed in conventional QST. Hence, it is applicable to various degree of freedom of the target state without altering the measurement setup. Moreover, a faithful estimation is realized even with classical probe light…
Conventional Glauber coherent states (CS) can be defined in several equivalent ways, e.g., by displacing the vacuum or, explicitly, by their infinite Poissonian expansion in Fock states. It is well known that these definitions become inequivalent if applied to finite $d$-level systems (qudits). We present a comparative Wigner-function description of the qudit CS defined (i) by the action of the truncated displacement operator on the vacuum and (ii) by the Poissonian expansion in Fock states of the Glauber CS truncated at $(d-1)$-photon Fock state. These states can be generated from a classical…
We prepare qudits based on angular multimode biphoton states by modulating the pump angular spectrum. The modes are prepared in the Schmidt basis and their intensity distributions do not overlap in space. This allows one to get rid of filtering losses while addressing single modes and to realize a single-shot qudit readout.
We analytically derive a decoherence model for orbital angular momentum states of a photon in a multimode optical fiber and show that rate of decoherence scales exponentially with $l^2$, where $l$ is the azimuthal mode order.~We also show numerically that for large values of $l$ the orbital angular momentum photon state completely dephases.~However for lower values of $l$ the decoherence can be minimized by using dynamical decoupling to allow for qudit high-bandwidth quantum communication and similar applications. \end{abstract}
Spin bases of relevance for quantum systems with cyclic symmetry as well as for quantum information and quantum computation are constructed from the theory of angular momentum. This approach is connected to the use of generalized Pauli matrices (in dimension d) arising from a polar decomposition of the group SU(2). Numerous examples are given for d=2, 3 and 4.
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…
Discussions about quantum interference, indistinguishability and superpostion between quantum states goes back to the beginning of quantum mechanics, but the theoretical problem concerning quantitative measures for quantum coherence was only recently solved by Baumgratz et al. [Phys. Rev. Lett. 113 140401 (2014)]. Since then many works have explored one of the possible coherence measures, the l1 norm, which has not yet been experimentally obtained for spatial photonic states in high dimensions. In this article we study states prepared with photons crossing multiple slits and implement the theo…
Geometrical and topological phases play a fundamental role in quantum theory. Geometric phases have been proposed as a tool for implementing unitary gates for quantum computation. A fractional topological phase has been recently discovered for bipartite systems. The dimension of the Hilbert space determines the topological phase of entangled qudits under local unitary operations. Here we investigate fractional topological phases acquired by photonic entangled qudits. Photon pairs prepared as spatial qudits are operated inside a Sagnac interferometer and the two-photon interference pattern reve…
The spatial modes of photons are one realization of a QuDit, a quantum system that is described in a D-dimensional Hilbert space. In order to perform quantum information tasks with QuDits, a general class of D-dimensional unitary transformations is needed. Among these, cyclic transformations are an important special case required in many high-dimensional quantum communication protocols. In this paper, we experimentally demonstrate a cyclic transformation in the high-dimensional space of photonic orbital angular momentum (OAM). Using simple linear optical components, we show a successful four-f…
The application of high-dimensional quantum systems (qudits) in quantum computing and communications seems to be a promising avenue due to the possibility of increasing the amount of information encoded in one physical carrier. In this work, we propose a method for implementing single-qudit gates for qudits based on light modes with orbital angular momentum. Method for logical qudits encoding, which ensures the quasi-cyclicity of operations, is introduced. Based on the protocol for converting the orbital angular momentum of light in the Raman quantum memory scheme [Vashukevich E.A. et. al. PRA…
Implementation and representation of qudit multi-controlled unitaries and hypergraph states by N-body angular momentum couplings
Universidade Federal de Mato Grosso, Faculdade de Ciência e Tecnologia（马托格罗索联邦大学科学技术学院）；Centro Internacional de Física, Instituto de Física, Universidade de Brasília（巴西利亚大学物理研究所国际物理中心）
A simple argument is presented that explicitly shows how to construct an arbitrary quantum gate acting on orbital angular momentum (OAM) of single photons. The scheme can be applied to implement subspace multiplexing, where a single high-dimensional OAM qudit represents effectively a stack of multiple independent lower-dimensional qudits. A special subclass of unitaries composed of single-photon controlled gates is studied in detail and notable examples of the general approach are discussed. The generalization of the simple argument leads to the parallelization scheme, which results in the sav…
Demonstration of subspace-search variational quantum eigensolver using photonic orbital angular momentum qudits
Korea Institute of Science and Technology (KIST)（韩国科学技术院）；Korea University of Science and Technology（韩国科学技术院）；Kyung Hee University（庆熙大学）；Gwangju Institute of Science and Technology（光州科学技术院）
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…
We discuss the appearance of fractional topological phases on cyclic evolutions of entangled qudits. The original result reported in Phys. Rev. Lett. \textbf{106}, 240503 (2011) is detailed and extended to qudits of different dimensions. The topological nature of the phase evolution and its restriction to fractional values are related to both the structure of the projective space of states and entanglement. For maximally entangled states of qudits with the same Hilbert space dimension, the fractional geometric phases are the only ones attainable under local SU($d$) operations, an effect that c…
Teleportation plays an important role in the communication of quantum information between the nodes of a quantum network and is viewed as an essential ingredient for long-distance Quantum Cryptography. We describe a method to teleport the quantum information carried by a photon in a superposition of a number $d$ of light modes (a "qudit") by the help of $d$ additional photons based on transcription. A qudit encoded into a single excitation of $d$ light modes (in our case Laguerre-Gauss modes which carry orbital angular momentum) is transcribed to $d$ single-rail photonic qubits, which are spat…
High-dimensional degrees of freedom of photons can encode more quantum information than their two-dimensional counterparts. While the increased information capacity has advantages in quantum applications (such as quantum communication), controlling and manipulating these systems has been challenging. Here we show a method to perform lossless arbitrary high-dimensional Pauli-X gates for single photon. The X-gate consists of a cyclic permutation of qudit basis vectors, and, together with the Z gate, forms the basis for performing arbitrary transformations. We propose an implementation of such ga…
Quantum state tomography (QST) is an essential tool for characterizing an unknown quantum state. Recently, QST has been performed for entangled qudits based on orbital angular momentum, time-energy uncertainty, and frequency bins. Here, we propose a QST for time-bin qudits, with which the number of measurement settings scales linearly with dimension $d$. Using the proposed scheme, we performed QST for a four-dimensional time-bin maximally entangled state with 16 measurement settings. We successfully reconstructed the density matrix of the entangled qudits, with which the average fidelity of th…
A new version of quantum hashing technique is developed wherein a quantum hash is constructed as a sequence of single-photon high-dimensional states (qudits). A proof-of-principle implementation of the high-dimensional quantum hashing protocol using orbital-angular momentum encoding of single photons is implemented. It is shown that the number of qudits decreases with increase of their dimension for an optimal ratio between collision probability and decoding probability of the hash. Thus, increasing dimension of information carriers makes quantum hashing with single photons more efficient.
We demonstrate a novel scheme to generate frequency-entangled qudits with dimension number higher than 10 and to distribute them over optical fibers of 15 km in total length. This scheme combines the technique of spectral engineering of biphotons generated by spontaneous parametric down-conversion and the technique of spectrally resolved Hong-Ou-Mandel interference. We characterized the comb-like spectral correlation structures of the qudits by time of arrival measurement and correlated spectral intensity measurement. The generation and distribution of the distinct entangled frequency modes ma…
Single photon interference due to passage through a periodic grating is considered in a novel proposal for processing D-dimensional quantum systems (quDits) encoded in the spatial degrees of freedom of light. We show that free space propagation naturally implements basic single quDit gates by means of the Talbot effect: an intricate time-space carpet of light in the near field diffraction regime. Adding a diagonal phase gate, we show that a complete set of single quDit gates can be implemented. We then introduce a spatially-dependent beam splitter that allows implementation of controlled opera…
The Fraunhofer diffraction intensity distribution of Laguerre-Gaussian beam is studied in an angular-double-slit interferometer. We demonstrate that the spiral phase structure of vortex light can be clearly revealed in this interference geometry, and it gives us an efficient way to distinguish different order of Laguerre-Gaussian beams. This angular-double-slit interference gives us a better understanding to the nature of orbital angular momentum and the interpretation of vortex beams interference phenomenon.
The codification in higher dimensional Hilbert Spaces (whose logical basis states are dubbed qudits in analogy with bidimensional qubits) presents various advantages both for Quantum Information applications and for studies on Foundations of Quantum Mechanics. Purpose of this review is to introduce qudits, to summarize their application to Quantum Communication and researches on Local Realism and, finally, to describe some recent experiment for realizing them. A Little more in details: after a short introduction, we will consider the advantages of testing local realism with qudits, discussing …
Google recently announced that they had achieved quantum supremacy with 53 qubits (base-2 binaries or radix-2), corresponding to a computational state-space of dimension 253 (about 1016). Google claimed to perform computations that took 200 seconds on their quantum processor that would have taken 10,000 years to accomplish on a classical supercomputer [1]. However, achieving superposition and entanglement of 53 qubits is not an easy task given the environmental noise that decoheres the qubits. In this paper, we claim that one can potentially achieve a similar computational dimension with fewer…
We develop a distillation protocol for multilevel qubits (qudits) using generalized beam splitters like in the proposal of Pan et al. for ordinary qubits. We find an acceleration with respect to the scheme of Bennet et al. when extended to qudits. It is also possible to distill entangled pairs of photons carrying orbital angular momenta (OAM) states that conserves the total angular momenta as those produced in recent experiments.
High-dimensional quantum states, or qudits, represent a promising resource in the quantum information field. Here we present the experimental generation of four-dimensional quantum states, or ququarts, encoded in the polarization and orbital angular momentum of a single photon. Our novel technique, based on the q-plate device, allows to prepare and measure the ququart in all five mutually unbiased bases. We report the reconstruction of the four dimensional density matrix through the tomographic procedure for different ququart states.
We study the statistics of quantum interference for completely positive maps. We calculate analytically the mean interference and its second moment for finite dimensional quantum systems interacting with a simple environment consisting of one or several spins (qudits). The joint propagation of the entire system is taken as unitary with an evolution operator drawn from the Circular Unitary Ensemble (CUE). We show that the mean interference decays with a power law as function of the dimension of the Hilbert space of the environment, with a power that depends on the temperature of the environment…
One of the main challenges in quantum technologies is the ability to control individual quantum systems. This task becomes increasingly difficult as the dimension of the system grows. Here we propose a general setup for cyclic permutations $X_d$ in $d$ dimensions, a major primitive for constructing arbitrary qudit gates. Using orbital angular momentum states as a qudit, the simplest implementation of the $X_d$ gate in $d$ dimensions requires a single quantum sorter $S_d$ and two spiral phase plates. We then extend this construction to a generalised $X_d(p)$ gate to perform a cyclic permutation…
We demonstrate the capability to discretize the frequency spectrum of broadband energy-time entangled photons by means of a spatial light modulator to encode qudits in various bases. Exemplarily, we implement three different discretization schemes, namely frequency bins, time bins and Schmidt modes. Entangled qudits up to dimension $d=4$ are then revealed by two-photon interference experiments with visibilities violating a $d$-dimensional Bell inequality.
The capability to generate and manipulate quantum states in high-dimensional Hilbert spaces is a crucial step for the development of quantum technologies, from quantum communication to quantum computation. One-dimensional quantum walk dynamics represents a valid tool in the task of engineering arbitrary quantum states. Here we affirm such potential in a linear-optics platform that realizes discrete-time quantum walks in the orbital angular momentum degree of freedom of photons. Different classes of relevant qudit states in a six-dimensional space are prepared and measured, confirming the feasi…
Quantum Teleportation, the transfer of the state of one quantum system to another without direct interaction between both systems, is an important way to transmit information encoded in quantum states and to generate quantum correlations (entanglement) between remote quantum systems. So far, for photons, only superpositions of two distinguishable states (one ``qubit'') could be teleported. Here we show how to teleport a ``qudit'', i.e. a superposition of an arbitrary number $d$ of distinguishable states present in the orbital angular momentum of a single photon using $d$ beam splitters and $d$…
Spectral-angular parametrization of open qudit dynamics: gap coordinates, Cartan structure, and GKLS decoupling
Université Paris Cité, CNRS（巴黎西岱大学，法国国家科学研究中心）；Faculty of Mathematics, University of Białystok（比亚韦斯托克大学数学系）；ESMaR, Faculty of Sciences at Univ. Mohammed V - Rabat（穆罕默德五世大学理学院ESMaR）
In high dimensional quantum communication networks, quantum frequency convertor (QFC) is indispensable as an interface in the frequency domain. For example, many QFCs have been built to link atomic memories and fiber channels. However, almost all of QFCs work in a two-dimensional space. It is still a pivotal challenge to construct a high-quality QFC for some complex quantum states, e.g., a high dimensional single-photon state that refers to a qudit. Here, we firstly propose a high-dimensional QFC for an orbital angular momentum qudit via sum frequency conversion with a flat top beam pump. As a…
Building an efficient quantum memory in high-dimensional Hilbert spaces is one of the fundamental requirements for establishing high-dimensional quantum repeaters, where it offers many advantages over two-dimensional quantum systems, such as a larger information capacity and enhanced noise resilience. To date, there have been no reports about how to achieve an efficient high-dimensional quantum memory. Here, we experimentally realize a quantum memory that is operational in Hilbert spaces of up to 25 dimensions with a storage efficiency of close to 60%. The proposed approach exploits the spatia…
The generation and control of quantum correlations in high-dimensional systems is a major challenge in the present landscape of quantum technologies. Achieving such non-classical high-dimensional resources will potentially unlock enhanced capabilities for quantum cryptography, communication and computation. We propose a protocol that is able to attain entangled states of $d$-dimensional systems through a quantum-walk-based {\it transfer \& accumulate} mechanism involving coin and walker degrees of freedom. The choice of investigating quantum walks is motivated by their generality and versatili…
We analyze Weyl algebra of quantum angular momentum system and construct qubit subalgebra out of it. We show that the commutant of this qubit subalgebra is isomorphic to the original algebra and prove the tensor product structure between qubit subalgebra and its commutant. This construction can be iterated to construct arbitrary number of qubit subalgebras from a single quantum system. We show a simple experimental realization of this proposed scheme using orbital angular momentum of single photons. We briefly discuss about construction of qudit subalgbra and generalization to other infinite d…
Spatial qudit states can be realized by using multi-slits to discretize the transverse momentum of a photon. The merit of this kind of spatial qudit states is that the implementation of higher dimensional qudits is relatively easy. As we have recently shown, the quantum states of these spatial qudits can be analyzed by scanning a single interference pattern. This method of single scan tomography can also be applied at higher dimensions, but the reconstruction becomes more sensitive to smaller details of the scanned patterns as the dimensions increase. In this paper, we investigate the effect o…
We report the possibility of completely destructive interference of three indistinguishable photons on a three port device providing a generalisation of the well known Hong-Ou-Mandel interference of two indistinguishable photons on a two port device. Our analysis is based on the underlying mathematical framework of SU(3) transformations rather than SU(2) transformations. We show the completely destructive three photon interference for a large range of parameters of the three port device and point out the physical origin of such interference in terms of the contributions from different quantum …
Bases of finite-dimensional Hilbert spaces (in dimension d) of relevance for quantum information and quantum computation are constructed from angular momentum theory and su(2) Lie algebraic methods. We report on a formula for deriving in one step the (1+p)p qupits (i.e., qudits with d = p a prime integer) of a complete set of 1+p mutually unbiased bases in C^p. Repeated application of the formula can be used for generating mutually unbiased bases in C^d with d = p^e (e > or = 2) a power of a prime integer. A connection between mutually unbiased bases and the unitary group SU(d) is briefly disc…
We introduce a matter wave interference scheme based on the quantization of orbital angular momentum in a ring trap. It operates without beam splitters, is sensitive to geometric phases induced by external gauge fields, and allows measuring interatomic scattering lengths. We argue that orbital angular momentum interferometry offers a versatile platform for quantum coherent experiments with cold atoms and Bose-Einstein condensates using state-of-the-art technology.
The ability to completely characterize the state of a quantum system is an essential element for the emerging quantum technologies. Here, we present a compressed-sensing inspired method to ascertain any rank-deficient qudit state, which we experimentally encode in photonic orbital angular momentum. We efficiently reconstruct these qudit states from a few scans with an intensified CCD camera. Since it requires only a few intensity measurements, our technique would provide an easy and accurate way to identify quantum sources, channels, and systems.
High-dimensional quantum entanglement is currently one of the most prolific fields in quantum information processing due to its high information capacity and error resilience. A versatile method for harnessing high-dimensional entanglement has long been hailed as an absolute necessity in the exploration of quantum science and technologies. Here we exploit Hong-Ou-Mandel interference to manipulate discrete frequency entanglement in arbitrary-dimensional Hilbert space. The generation and characterization of two-, four- and six-dimensional frequency entangled qudits are theoretically and experime…
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…
It is said about quantum interference that "In reality, it contains the only mystery". Indeed, together with non-locality it is often considered as the characteristic feature of quantum theory which can not be explained in any classical way. In this work we are concerned with a restricted setting of a single particle propagating in multi-path interferometric circuits, that is physical realisation of a qudit. It is shown that this framework, including collapse of the wave function, can be simulated with classical resources without violating the locality principle. We present a local ontological…
