PaperlayerView exact revision on arXiv

Source-linked AI summary

Advances in High Dimensional Quantum Entanglement

Manuel Erhard, Mario Krenn, Anton Zeilinger

arXiv:1911.10006v2quant-phphysics.app-phphysics.optics

TL;DR

High-dimensional entanglement raises fundamental and technological questions beyond qubit-based quantum information. This review synthesizes experimental approaches for generating and manipulating photonic entanglement across path, spatial-mode, and time/frequency degrees of freedom, reporting progress from multiphoton chips to 14-dimensional states while identifying scaling challenges.

  • Problem

    Qubit-based systems do not capture the full range of complex entangled systems involving many particles and multiple local dimensions.

  • Method

    The review compares experimental techniques for generating and manipulating high-dimensional photonic entanglement across path, spatial-mode, and time/frequency degrees of freedom.

  • Results

    The reviewed experiments include a 4-photon entangled state on a programmable chip, 14-dimensional path entanglement with fidelities from 96% at d = 4 to 81% at d = 12, and approximately 90% fidelity for three-dimensional OAM entanglement.

  • Takeaways & Limitations

    High-dimensional photonic entanglement provides experimentally demonstrated platforms for quantum information processing and applications across multiple discrete photonic degrees of freedom.

Abstract

from arXiv · show

Since its discovery in the last century, quantum entanglement has challenged some of our most cherished classical views, such as locality and reality. Today, the second quantum revolution is in full swing and promises to revolutionize areas such as computation, communication, metrology, and imaging. Here, we review conceptual and experimental advances in complex entangled systems involving many multilevel quantum particles. We provide an overview of the latest technological developments in the generation and manipulation of high-dimensionally entangled photonic systems encoded in various discrete degrees of freedom such as path, transverse spatial modes or time/frequency bins. This overview should help to transfer various physical principles for the generation and manipulation from one to another degree of freedom and thus inspire new technical developments. We also show how purely academic questions and curiosity led to new technological applications. Here fundamental research provides the necessary knowledge for coming technologies such as a prospective quantum internet or the quantum teleportation of all information stored in a quantum system. Finally, we discuss some important problems in the area of high-dimensional entanglement and give a brief outlook on possible future developments.

I. INTRODUCTION

The introduction frames higher-dimensional quantum systems as a natural extension beyond qubits, with advantages for information capacity, noise resistance, and foundational tests. It organizes the field by particle number and local dimensionality, connecting fundamental questions to technological applications.

  • Motivation: Qubits use two-state systems, whereas higher-dimensional systems offer a more complex information alphabet.The paper contrasts the qubit with DNA’s four-letter alphabet as motivation for studying larger state spaces.
  • Motivation: Higher-dimensional communication protocols provide higher information capacity and increased resistance to noise.
  • Scope: The review focuses on discrete photonic degrees of freedom, including path, transverse spatial modes, and time/frequency bins.Other physical platforms exist, while continuous-variable systems are outside this focus.
  • Scope: The review explains experimental methods for generating, manipulating, detecting, and distributing high-dimensional photonic entanglement across different degrees of freedom.It emphasizes analogies and synergies that may transfer laboratory techniques between degrees of freedom.
  • Fundamental and technological categories: The paper presents system size—particle number N and local dimensionality d—as a framework linking fundamental questions with technological applications.It highlights how initially academic questions can originate later quantum-technology applications.
  • Increasing system size: Increasing particle number expands Hilbert space exponentially and produces qualitatively new insights into classical–quantum relationships.For many qubits, GHZ states also support universal quantum computation and quantum error-correction schemes.

III. PHOTONIC CARRIERS OF HIGH-DIMENSIONAL ENTANGLEMENT

This section surveys photonic carriers and nonlinear processes used to generate and manipulate high-dimensional entangled photon pairs. It covers SPDC and SFWM, their phase-matching and emission behavior, and the practical relevance of source and detector technologies.

  • Photonic carriers: Photons encode high-dimensional quantum information in discrete degrees of freedom and can distribute it over long distances.The section emphasizes local measurements for both fundamental and application-oriented investigations.
  • Nonlinear generation: SPDC uses a χ(2) nonlinearity, while SFWM uses a χ(3) nonlinearity to generate photon pairs.In SFWM, two pump photons are converted into two output photons.
  • SPDC: SPDC phase matching combines energy and linear-momentum conservation, enabling probabilistic conversion of one pump photon into two photons.The frequencies and momenta identify pump, idler, and signal photons.
  • SPDC: The low-power SPDC state is a Fock-basis expansion containing vacuum, single-pair, and multipair terms.The pair-emission amplitude α controls the relative contribution of these terms.
  • Experimental conditions: Because multipair emission is possible, experiments typically keep the pair-emission amplitude much smaller than one.This suppresses simultaneous emission of two or more photon pairs.
  • Practical considerations: SPDC and SFWM offer different implementation and application advantages, while efficient sources and detectors remain important practical technologies.The section directs readers to broader reviews of single-photon sources and detectors.

A. Path

Path encoding supports arbitrary single-photon transformations and scalable on-chip generation of high-dimensional entanglement. Experiments have demonstrated certified entanglement, quantum processing, and transmission using path-based photonic systems.

  • A. Path: Beam splitters and phase-shifters implement arbitrary single-qudit transformations in path encoding.Implementations can use bulk optical elements or integrated optics, including silicon chips with high interferometric stability.
  • A. Path: Pixel entanglement combined with a deformable mirror and single-photon detector enabled certification of 8.4 bits per photon channel capacity.A single lens performs the position–momentum basis transformation needed for entanglement certification.
  • A. Path: 16 identical SFWM spiral-waveguide sources enabled 14-dimensional path entanglement on-chip using more than 550 optical components.Observed fidelities ranged from 96% for d = 4 to 81% for d = 12, with each source producing photon pairs at 2kHz.
  • A. Path: On-chip path qudits have supported a general 2-qubit processor and quantum Hamiltonian learning with a coupled nitrogen-vacancy center.Two pre-entangled ququarts were used probabilistically to implement arbitrary 2-qubit unitary operations, including CNOT gates.
  • A. Path: Multi-core fibres require interferometric phase stability, yet four-dimensional quantum-key distribution has been achieved over distances up to 0.3km.Six-dimensional entanglement has also been transmitted through a graded-index multimode fibre over 2m.

B. Transverse Spatial modes of photons

Transverse spatial modes, especially Laguerre-Gaussian OAM modes, provide high-dimensional photonic encodings with experimentally demonstrated generation, sorting, and transmission techniques. OAM entanglement is naturally produced by SPDC, although higher-dimensional manipulation remains difficult.

  • B. Transverse Spatial modes of photons: OAM entanglement can be generated in a single nonlinear crystal because OAM is conserved during SPDC.The resulting spiral spectrum and dimensionality depend on crystal length and beam-waist parameters.
  • B. Transverse Spatial modes of photons: Path-identity using coherently pumped indistinguishable crystals creates arbitrary d-dimensional OAM states, with three-dimensional fidelity of approximately F ≈90%.Relative pump powers and phases control the magnitudes and phases of the state coefficients.
  • B. Transverse Spatial modes of photons: Higher-dimensional OAM manipulation has been difficult despite established projective measurements and parity-based sorting methods.Cyclic transformations were only recently realized using the MELVIN computer algorithm.
  • B. Transverse Spatial modes of photons: Log-polar mode sorters demonstrated sorting for mode numbers up to 50 with approximately 2% next-mode overlap.The transformation converts OAM into linear momentum so distinct modes emerge at different positions.
  • B. Transverse Spatial modes of photons: Multi-plane light conversion sorted 210 Laguerre-Gaussian modes into spatial Gaussian spots.The device had a theoretical insertion loss of 2.5dB and a measured insertion loss of 6dB.
  • B. Transverse Spatial modes of photons: OAM systems have transmitted classical information over free-space, fibre, underwater, and 143km island links.OAM has also been used in demonstrations involving quantum-information transmission and fundamental tests with angular momenta up to 10.010ℏ.

C. Discretized Time and Frequency modes

Time-bin and frequency-bin encodings provide long-distance-compatible platforms for generating, manipulating, distributing, and testing high-dimensional photonic entanglement. Recent work combines scalable sources with high-fidelity transformations and noise-resilient demonstrations.

  • C. Discretized Time and Frequency modes: Sequences of d indistinguishable coherent pump pulses can scale time-bin entanglement to d dimensions, demonstrated for 11 dimensions.Keeping the pair-generation amplitude α small maintains the probability of simultaneous two-pair emission below 0.1 over Tcoh.
  • C. Discretized Time and Frequency modes: An experiment observed an 18-dimensionally entangled time-bin state using unbalanced interferometric measurements.These measurements use neighbouring time-bins to establish a lower bound on the generated entanglement dimensionality.
  • C. Discretized Time and Frequency modes: Integrated Kerr frequency-comb sources create anti-correlated frequency-bin qudits through energy-conserving SFWM in micro-ring cavities.The attainable dimensionality is bounded by the SFWM phase-matching range and the cavity’s allowed frequency modes.
  • C. Discretized Time and Frequency modes: An EOM–pulse-shaper–EOM architecture implements arbitrary frequency-bin unitaries, including a three-bin quantum Fourier transform with fidelity up to 99.9%.The pulse shaper applies phases across the complete output space before the final EOM recombines the modes.
  • C. Discretized Time and Frequency modes: Time-bin encoding achieved a 2.7 Mbit/second secure key rate over 20km of optical fibre, although intercept-and-resend attacks were not included in the security analysis.The protocol used Franson visibility measurements and included Gaussian attacks.
  • C. Discretized Time and Frequency modes: Noise-fraction certification reached 92% for 80-dimensional time-bin entanglement and 63% for seven-dimensional OAM encoding.The comparison demonstrates distinct experimentally measured noise robustness values across the two encodings.

D. Combining and converting different DoFs

Combining multiple photonic degrees of freedom produces higher-dimensional states, while mode-sorter interfaces can convert entanglement between encodings. These approaches support deterministic hybrid gates and connect distinct experimental platforms.

  • D. Combining and converting different DoFs: Joint polarization, OAM, and time-frequency encoding produces a 12-dimensional entangled state across three degrees of freedom.The dimension is 2 × 3 × 2, corresponding to 12 orthogonal states.
  • D. Combining and converting different DoFs: The three classes of multipartite entanglement distinguish separable multiphoton states, genuinely multiphoton entangled states, and genuinely high-dimensional multiphoton states.Examples include two time-entangled photon pairs, a four-photonic path graph state, and a three-photonic three-dimensional GHZ state.
  • D. Combining and converting different DoFs: Hybrid degrees of freedom permit deterministic CNOT gates, supporting applications such as superdense coding and superdense quantum teleportation.The stated advantage arises from combining the separately encoded degrees of freedom.
  • D. Combining and converting different DoFs: A mode sorter can convert path entanglement into OAM entanglement while maintaining the encoded entanglement.The conversion uses the sorter in reverse after a nonlinear crystal illuminates a three-slit mask.

A. Multi-Photon Entanglement in high Dimensions

The review classifies multipartite high-dimensional photonic entanglement using Schmidt-rank vectors and distinguishes three experimental categories by separability and local dimensionality.

  • Classification framework: The Schmidt-Rank-Vector collects Schmidt ranks across all bipartitions of an N-partite pure state.For N particles, the number of possible bipartitions is k = 2^N−1 −1.
  • Three experimental categories: Category I contains many-photon states in a high-dimensional degree of freedom that may be bi-separable.An example is |ψ⟩ = |φA,B⟩⊗|φC,D⟩, whose Schmidt-Rank-Vector includes entries equal to 1.
  • Three experimental categories: Category II contains genuine multiphoton entanglement encoded in a high-dimensional space, but with only two-dimensional entanglement.A 4-photon 2-dimensional GHZ state in the path degree of freedom is an example.
  • Three experimental categories: Category III contains genuine multiphotonic high-dimensional entanglement in which all photons are entangled in more than two dimensions.The review summarizes corresponding experiments in its table of multiphoton high-dimensional degrees of freedom.

1. Path

Path encoding supports complex multiphoton experiments through programmable integrated optics and reconfigurable entangling operations, progressing from class (II) states to genuine high-dimensional entanglement.

  • Early path experiments: BosonSampling experiments used path modes to study mode-occupation probabilities rather than produce or fully measure well-defined entangled states.Their motivation was the computational difficulty of linear-optical transformations for classical computers.
  • Integrated path photonics: 2019 marked the first genuine multiphoton entangled state generated and measured on a programmable chip.The chip demonstrated graph states including a 4-photon Star graph, locally equivalent to a 2-dimensional GHZ state.
  • Integrated path photonics: The programmable-chip experiment reached 78% fidelity with count rates of approximately 5.7 mHz.A reconfigurable postselected entangling gate used Hong-Ou-Mandel interference to remove which-crystal information between photon pairs.
  • Spatial-mode and higher-dimensional progress: 2016 produced the first class (II) multiphoton entangled state using spatial modes, while a later experiment introduced partial three-dimensional entanglement.The 2016 state used double-pair SPDC emissions probabilistically split by three beam splitters; the later state had fidelity 80.1% and a count rate of approximately 15 mHz.
  • Spatial-mode and higher-dimensional progress: 2018 demonstrated the first genuine high-dimensional multiparticle entanglement as a three-dimensional GHZ state.A multi-port coherently manipulated several photons and removed cross-correlation terms; fidelity was 75.2% with a count rate of 1.2 mHz.
  • Cross-degree-of-freedom transfer: The physical concepts are transferable across discrete photonic degrees of freedom, allowing the multi-port to be applied to path or time-bin encoding.This connects the demonstrated spatial-mode architecture to other high-dimensional photonic platforms.

3. Discretized Time and Frequency modes

Discretized time and frequency modes provide scalable photonic encodings for multiphoton experiments, including coherent four-photon states and a verified three-photon W state.

  • Frequency-comb experiments: A quantum frequency comb provides roughly 100 equally spaced frequency bins within 100 nm around a 1550 nm pump wavelength.Symmetric frequency-line pairs can be occupied by photon pairs and used to create time-bin entanglement.
  • Frequency-comb experiments: The frequency comb enabled coherent occupation of two frequency-bin pairs at the same time.The experiment generated a coherent four-photon state by controlling the relative phase.
  • Experimental overview: Table I summarizes multiphoton experiments in high-dimensional degrees of freedom.The table serves as the review’s compact overview of the reported experimental landscape.
  • Frequency-comb experiments: The coherent four-photon state had 64% fidelity and a count rate of roughly 0.17 Hz, but remained separable and therefore class (I).The authors argue that reducing losses could increase four-photon rates to the kHz range; genuine multiphoton entanglement requires removing which-frequency-bin information.
  • Three-photon frequency entanglement: A three-photon W state generated with discretized frequency was verified as class (II) entanglement with roughly 90% fidelity and a 20 mHz count rate.Two spontaneous-four-wave-mixing photon pairs were probabilistically split, with one trigger photon heralding the W state in three detectors.

4. Multiple degrees-of-freedom

High-dimensional photonic systems can combine multiple degrees of freedom to encode entanglement beyond one qubit per photon. The section reviews this strategy alongside three-dimensional teleportation, whose realization requires ancillary photons and fidelity above 2/3.

  • Multiple degrees-of-freedom: An 18-qubit entangled state was created using six photons and three degrees of freedom: polarization, path, and spatial modes.The reported fidelity was 70.8%, with a six-fold count rate of 55 mHz.
  • Multiple degrees-of-freedom: Each photon’s polarization, path, and orbital-angular-momentum parity were transformed jointly to produce the 18-qubit entangled GHZ state.The transformations map H and V polarization components onto path and parity labels.
  • Quantum teleportation in high dimensions: High-dimensional teleportation uses an entangled pair, a teleportee, an ancillary photon, and a three-dimensional Bell-state measurement.Linear optics requires ancillary particles for high-dimensional Bell-state measurements.
  • Quantum teleportation in high dimensions: Genuine three-dimensional teleportation requires fidelity above F = 2/3, while fidelities below 50% are achievable classically and 50%–66.6% with qubit systems.This threshold distinguishes the three-dimensional demonstration from lower-dimensional or classical strategies.
  • Quantum teleportation in high dimensions: Two 2019 experiments demonstrated three-dimensional path-encoded teleportation, including a quantum-Fourier-transform Bell measurement generalizable to arbitrary dimensions.The reported approaches also showed high-quality long-time stability; one retained interference visibilities beyond 98% for 48 hours.

1. Applications in Entanglement Swapping

High-dimensional entanglement swapping connects initially independent photons through Bell-state projections onto multiple two-dimensional subspaces. The resulting correlations support ghost imaging, while the demonstrated system remains not high-dimensionally entangled.

  • Applications in Entanglement Swapping: Entanglement swapping can entangle photons that never interacted or shared a common past, supporting long-distance quantum-network concepts.The process is also relevant to fundamental tests of entanglement.
  • Applications in Entanglement Swapping: The first high-dimensional entanglement-swapping experiment projected onto many two-dimensional subspaces and conditioned the result on two-fold detections.The output was an incoherent superposition of multiple two-dimensionally entangled states.
  • Applications in Entanglement Swapping: Average fidelities reached F = 80% after background subtraction and F = 57% raw across six two-dimensionally entangled systems.The authors explicitly distinguish these swapped subspaces from genuine high-dimensional entanglement.
  • Applications in Entanglement Swapping: The swapping method was applied to ghost imaging, where the imaging photon and the photon interacting with the object did not share a common past.This setup raises whether quantum or classical correlations are required for the task.
  • Future directions: Future progress depends on more efficient photon sources, near-unity detectors, multi-outcome detection, and scalable integrated optics.These technologies are needed to increase count rates and exploit high-dimensional information in multi-photon systems.
Loading 1911.10006v2…