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arXiv:1707.05760 · Optica

Generation of the complete four-dimensional Bell basis

Feiran Wang, Manuel Erhard, Amin Babazadeh, Mehul Malik, Mario Krenn, Anton Zeilinger

Published Nov 27, 2017• 72 citations
arXivHTMLPDF

Abstract

The Bell basis is a distinctive set of maximally entangled two-particle quantum states that forms the foundation for many quantum protocols such as teleportation, dense coding and entanglement swapping. While the generation, manipulation, and measurement of two-level quantum states is well understood, the same is not true in higher dimensions. Here we present the experimental generation of a complete set of Bell states in a four-dimensional Hilbert space, comprising of 16 orthogonal entangled Bell-like states encoded in the orbital angular momentum of photons. The states are created by the application of generalized high-dimensional Pauli gates on an initial entangled state. Our results pave the way for the application of high-dimensional quantum states in complex quantum protocols such as quantum dense coding.

DOI
10.1364/optica.4.001462

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The Bell basis is a distinctive set of maximally entangled two-particle quantum states that forms the foundation for many quantum protocols such as teleportation, dense coding and entanglement swapping. While the generation, manipulation, and measurement of two-level quantum states is well understood, the same is not true in higher dimensions. Here we present the experimental generation of a complete set of Bell states in a four-dimensional Hilbert space, comprising of 16 orthogonal entangled Bell-like states encoded in the orbital angular momentum of photons. The states are created by the application of generalized high-dimensional Pauli gates on an initial entangled state. Our results pave the way for the application of high-dimensional quantum states in complex quantum protocols such as quantum dense coding.
The Bell basis, a set of maximally entangled biphoton states, is a critical prerequisite towards quantum information processing, and many quantum applications have highlighted the requirement for manipulation of the high-dimensional Bell basis. While Bell states can be created by using ingenious single-photon quantum gates, their implementation complexity in higher dimensions is significantly increased. Here we present an elaborate approach to show that the adaptive pump modulation enables the efficient preparation of the Bell basis in arbitrary-dimensional Hilbert space. A complete set of four-dimensional orbital angular momentum Bell states is experimentally created, yielding high fidelities for certifying the entanglement dimensionality. Our strategy can be simply generalized to prepare more complex forms of quantum states, even exploiting other physical degrees of freedom. Also, it can facilitate the use of high-dimensional entanglement in a variety of quantum protocols, in particular those requiring quantum dense coding.
Bell-state measurements are essential ingredients in many protocols for quantum information processing, ranging from quantum teleportation and dense coding to entanglement distribution in quantum networks. Their power relies on the fact that they are measurements in an entangled basis of a two-particle system and that the used Bell-state basis can be generated from a single Bell state by local unitary transformations. How can these measurements be generalized to more particles? We develop a general framework for this state-to-measurement problem: We introduce a hierarchy of classes of measurement bases, distinguished by the local transformations the parties may use for their generation from a single state. This leads to a generalization of the concept of maximally entangleable (or weighted hypergraph) states and the identification of a novel maximally entangled basis of four qubits, being a candidate for data-hiding tasks or distillation protocols. Finally, we prove that not all forms of entanglement can be encoded in an entire measurement basis.
ObjectiveQuantum communication is an emerging mode of communication with broad application prospects, combining with classical information science and technology. It offers unconditional security and efficient transmission of information, making it a hot research topic in todayƳs communication field. High-dimensional quantum entanglement often exhibits strong nonlocality, enhancing the security in quantum communication processes and playing a crucial role in quantum communication and computing. To meet the demand for high-dimensional entangled states in quantum information science, this paper proposes a method for generating high-dimensional Bell states based on the Orbital Angular Momentum (OAM) of photons, capable of producing the complete set of Bell state basis vectors with high fidelity.MethodsInitially, the Spontaneous Parametric Down-Conversion (SPDC) process involving photon OAM is studied. Furthermore, a Genetic Algorithm (GA) is employed to optimize the quantum state of the pump light, aiming to obtain the required high-dimensional maximally entangled states. We analyze the characteristics of Laguerre-Gaussian (LG) beams in detail and explore their application in the SPDC process. By optimizing the parameters of the pump light, maximally entangled states are generated in three-dimensional, four-dimensional, and five-dimensional spaces.ResultsA comparison of the spatial mode spectral distribution shows that the optimized entangled states possess higher fidelity to the target states. Additionally, by simulating the generation of 16 Bell states, it is verified that the produced quantum states form a complete basis for a four-dimensional Hilbert space.ConclusionThe proposed scheme not only effectively generates high-dimensional entangled states but also enables longer-distance quantum communication and potentially global quantum communication systems. Moreover, it provides new avenues for the application of high-dimensional entangled states in the fields of quantum communication and computing, significantly contributing to the advancement of quantum information science.
Transformations on quantum states form a basic building block of every quantum information system. From photonic polarization to two-level atoms, complete sets of quantum gates for a variety of qubit systems are well known. For multilevel quantum systems beyond qubits, the situation is more challenging. The orbital angular momentum modes of photons comprise one such high-dimensional system for which generation and measurement techniques are well studied. However, arbitrary transformations for such quantum states are not known. Here we experimentally demonstrate a four-dimensional generalization of the Pauli X gate and all of its integer powers on single photons carrying orbital angular momentum. Together with the well-known Z gate, this forms the first complete set of high-dimensional quantum gates implemented experimentally. The concept of the X gate is based on independent access to quantum states with different parities and can thus be generalized to other photonic degrees of freedom and potentially also to other quantum systems.
Quantum protocols require access to large-scale entangled quantum states, due to the requirement of channel capacity. As a promising candidate, the high-dimensional orbital angular momentum (OAM) entangled states have been implemented, but only one of four OAM Bell states in each individual subspace can be distinguished. Here we demonstrate the first realization of complete OAM Bell-state measurement (OAM-BSM) in an individual subspace, by seeking the suitable unitary matrix performable using only linear optics and breaking the degeneracy of four OAM Bell states in ancillary polarization dimension. We further realize the superdense coding via our complete OAMBSM with the average success probability of ~82% and the channel capacity of ~1.1(4) bits. This work opens the window for increasing the channel capacity and extending the applications of OAM quantum states in quantum information in future.
Orbital angular momentum (OAM) entanglement states have great potential in enhancing the information capacity in quantum communications. To make full use of the OAM degree of freedom, the measurement and distinction of OAM Bell states become a key step that needs to be accomplished. However, there is still no solution to achieve complete distinction of OAM Bell states at the single-photon level. Here we demonstrate the complete OAM Bell-state measurement without auxiliary photons by using linear optics. Furthermore, we propose a quantum communication network via OAM Bell-state multiplexing, perform a proof-of-principle experiment in the first- and second-order OAM subspaces for two users, and achieve average success probabilities of obtaining information of 91.6% for User-I and 89.1% for User-II. In principle, our scheme allows more users to be added by using high-order OAM Bell states. Our research extends the manipulation method for high-dimensional quantum states and paves the way for the application of OAM multiplexing in quantum communications.