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Quantum hyperentanglement and its applications in quantum information processing

Fu-Guo Deng, Bao-Cang Ren, Xi-Han Li

arXiv:1610.09896v5quant-ph

TL;DR

Hyperentanglement offers high-capacity quantum information processing by entangling multiple degrees of freedom, but distributed states undergo decoherence. This review surveys generation and applications, including Bell-state analysis, concentration, purification, and hyperparallel gates, reporting protocols and states such as a fully entangled fraction reaching 0.99.

  • Problem

    Hyperentangled systems lose entanglement through channel noise and storage, motivating methods for high-capacity long-distance quantum communication and computation.

  • Method

    The review synthesizes hyperentanglement preparation and applications across photon degrees of freedom, including communication protocols, concentration, purification, and hyperparallel computation.

  • Results

    The reviewed approaches include a generated state with fully entangled fraction up to 0.99 and a hyper-ECP with success probability P = 4|βγ|2.

  • Takeaways & Limitations

    Hyperentanglement can increase communication channel capacity and support quantum communication protocols and hyperparallel photonic computation.

Abstract

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Hyperentanglement is a promising resource in quantum information processing with its high capacity character, defined as the entanglement in multiple degrees of freedom (DOFs) of a quantum system, such as polarization, spatial-mode, orbit-angular-momentum, time-bin and frequency DOFs of photons. Recently, hyperentanglement attracts much attention as all the multiple DOFs can be used to carry information in quantum information processing fully. In this review, we present an overview of the progress achieved so far in the field of hyperentanglement in photon systems and some of its important applications in quantum information processing, including hyperentanglement generation, complete hyperentangled-Bell-state analysis, hyperentanglement concentration, and hyperentanglement purification for high-capacity long-distance quantum communication. Also, a scheme for hyper-controlled-not gate is introduced for hyperparallel photonic quantum computation, which can perform two controlled-not gate operations on both the polarization and spatial-mode DOFs and depress the resources consumed and the photonic dissipation.

I. INTRODUCTION

Hyperentanglement entangles multiple degrees of freedom, offering high-capacity resources for quantum communication and computation. This review surveys its preparation, communication applications, entanglement recovery, and hyperparallel photonic gates.

  • Hyperentanglement is simultaneous entanglement across multiple degrees of freedom, including polarization, spatial mode, time, energy, and orbital angular momentum.
  • Combining single-degree-of-freedom entanglement techniques enables preparation of diverse hyperentangled photon states.Examples include polarization-spatial and polarization-spatial-time-energy states.
  • Hyperentanglement supports dense coding, complete Bell-state analysis, hyper-teleportation, swapping, concentration, purification, and hyperparallel quantum gates.Complete hyperentangled-Bell-state analysis is described as prerequisite for hyperentanglement-based communication protocols.
  • Channel noise and storage cause decoherence in distributed hyperentangled photons, making concentration and purification necessary for long-distance high-capacity communication.
  • Hyperparallel computation encodes information in multiple photon degrees of freedom and can reduce resource consumption and photonic dissipation.
  • Reported states include a 144-dimensional hyperentangled system with CHSH violation in each degree of freedom and maximum fidelity F = 0.974.
  • A six-qubit cluster state achieved fidelity F = 0.6350±0.0008, 7 % better than the best previous six-qubit graph-state result.
  • A generated state’s fully entangled fraction reached a maximum of 0.99, while another hyperentanglement protocol produced the stated hyperentangled state through repeated preparation.

A. Status of Bell-state analysis for photonic quantum systems

Bell-state analysis is foundational for entanglement-based communication and quantum repeaters. Hyperentanglement enables complete discrimination of the 16 polarization-spatial hyperentangled Bell states through separate spatial-mode and polarization analyses.

  • Bell-state analysis distinguishes the four orthogonal Bell states and is a prerequisite for entanglement-based communication protocols and quantum repeaters.
  • Hyperentanglement-assisted linear-optical protocols can completely distinguish all four Bell states in one photonic degree of freedom, improving on 50% success probability.
  • Earlier linear-optical hyperentangled analysis distinguished only 7 of 16 hyperentangled Bell states, motivating nonlinear schemes for complete discrimination.
  • The reviewed protocol analyzes spatial-mode Bell states with parity-check QNDs and cross-Kerr nonlinearities, followed by a beam-splitter transformation and coherent-beam measurement.
  • A polarization parity-check QND, half-wave plate, and single-photon detectors distinguish the polarization Bell states from their detection patterns.
  • Combining spatial-mode and polarization parity-check QNDs with cross-Kerr nonlinearity completely distinguishes all 16 orthogonal hyperentangled Bell states.

C. Teleportation with a hyperentangled channel

Hyperentanglement supports teleportation of an unknown single-photon state encoded in two degrees of freedom. Alice performs hyperentangled Bell-state analysis, and Bob applies outcome-dependent corrections to recover the state.

  • Hyperentanglement allows two-qubit unknown information to be transferred by teleporting a single photon.
  • The protocol encodes the input photon's unknown state in polarization and spatial-mode degrees of freedom and supplies Alice and Bob with a hyperentangled Bell pair.
  • Alice performs hyperentangled Bell-state analysis on the input photon and her entangled partner, projecting Bob's photon into a corresponding two-degree-of-freedom state.
  • Bob uses outcome-dependent polarization and spatial-mode unitary corrections to obtain the original unknown single-photon state.

D. Hyperentanglement swapping

Hyperentanglement swapping creates polarization-spatial entanglement between photons that initially do not interact. The protocol performs hyperentangled Bell-state analysis on Alice's photons and applies outcome-dependent corrections.

  • Entanglement swapping creates entanglement between particles that initially have no interaction and is used in quantum repeaters and communication protocols.
  • Two initially hyperentangled photon pairs are distributed so Alice holds the measured photons while Bob and Charlie hold the remote photons.
  • Alice performs hyperentangled Bell-state analysis on photons B and C to create correlations between photons A and D.
  • The Bell-state-analysis outcome projects photons A and D into the corresponding polarization-spatial Bell state, with signs and degrees of freedom determined by the outcome.
  • After Alice publishes the result, Bob applies outcome-dependent unitary operations so the remote photons can share the desired hyperentangled Bell state.

A. Development of entanglement concentration

Hyperentanglement concentration developed from single-DOF entanglement concentration toward protocols for multiple DOFs, including known- and unknown-parameter cases. The parameter-splitting method concentrates known partially hyperentangled states with maximal success probability using linear optics.

  • A. Development of entanglement concentration: Entanglement concentration distills nonlocal partially entangled pure states into maximally entangled states, with protocols classified by whether state parameters are known or unknown.Known parameters can enable efficient concentration using one nonlocal photon system, whereas unknown parameters motivate different protocols.
  • A. Development of entanglement concentration: 2013 marked the introduction of hyperentanglement concentration for polarization and spatial-mode DOFs with known coefficients using parameter splitting.The method extracts maximally entangled photons with linear-optical elements and requires no optical nonlinearity.
  • A. Development of entanglement concentration: The parameter-splitting method applies local linear-optical operations to one remote photon and can achieve the maximal concentration success probability.The method is designed for nonlocal partially entangled pure states whose parameters are accurately known to the remote users.
  • A. Development of entanglement concentration: For a partially hyperentangled Bell state, the parameter-splitting protocol first processes the spatial-mode state, then splits the polarization parameter through polarization operations.The setup uses local unitary operations on both DOFs of photon A, while photon B is not operated on.
  • A. Development of entanglement concentration: If photon A is not detected in the designated failure mode, the spatial-mode state becomes maximally entangled and the polarization state can yield a maximally hyperentangled Bell state.Detection in other spatial modes projects one DOF to a product state, causing protocol failure.
  • A. Development of entanglement concentration: P = 4|βγ|2 is the maximal success probability for obtaining a maximally hyperentangled Bell state from the partially hyperentangled Bell state.The parameter-splitting method is also described as applicable to known-parameter concentration involving one or multiple DOFs.

1. Hyper-ECP with linear optical elements

The Schmidt projection hyper-ECP addresses unknown partially hyperentangled Bell-state parameters by processing two identical photon pairs with parity checks. Successful detector outcomes project the remaining pair into a maximally hyperentangled Bell state, with success probability P = 4|αβγδ|2.

  • 1. Hyper-ECP with linear optical elements: The Schmidt projection method requires two identical partially hyperentangled photon pairs, AB and CD, shared between Alice and Bob.The pairs use the same unknown polarization and spatial-mode parameters.
  • 1. Hyper-ECP with linear optical elements: After polarization bit flips on photons C and D, the protocol performs polarization parity checks on AC and spatial-mode parity checks on BD.PBSs implement the polarization check, while a beam splitter uses the Hong–Ou–Mandel effect for the spatial-mode check.
  • 1. Hyper-ECP with linear optical elements: Selecting even-parity polarization states of AC and odd-parity spatial-mode states of BD yields the successful detector condition for the four-photon system.Alice and Bob each need only one detector clicked in the corresponding parity outcomes.
  • 1. Hyper-ECP with linear optical elements: Hadamard operations on the spatial-mode and polarization DOFs of photons C and D transform the postselected four-photon state before the final projection.These operations are applied after the parity-check stage.
  • 1. Hyper-ECP with linear optical elements: An even-parity polarization and even-parity spatial detector outcome projects photon pair AB into the maximally hyperentangled Bell state.Odd-parity outcomes require a phase-flip operation on photon B to obtain the same target state.
  • 1. Hyper-ECP with linear optical elements: P = 4|αβγδ|2 is the success probability when Alice and Bob each register one detector click, producing a maximally hyperentangled Bell state.Other detector outcomes cause the hyper-ECP to fail; successful events can be identified by postselection with imperfect detectors.

2. Hyper-ECP with nonlinear optical elements

The nonlinear-optical hyper-ECP replaces polarization and spatial-mode parity checks with cross-Kerr-based QNDs, iteratively distilling partially hyperentangled states into maximally hyperentangled Bell states. Parity measurements identify the relevant modes, while repeated rounds improve the total success probability.

  • Cross-Kerr-based P-QND and S-QND replace polarization and spatial-mode parity checks in the hyper-ECP.The QNDs distinguish parity without affecting the corresponding spatial-mode or polarization states.
  • X-quadrature measurements distinguish even- from odd-parity polarization and spatial-mode Bell states through coherent-state phase shifts.A phase shift θ or −θ indicates even parity, whereas no phase shift indicates odd parity.
  • 4|αβγδ|2 is the first-round success probability when both measured photon-pair DOFs have odd parity.The protocol projects the four-photon system into |Φ1⟩ under this outcome.
  • The protocol produces partially hyperentangled Bell states in other parity branches, each distillable to a maximally hyperentangled Bell state after another round.These branches include states with either polarization or spatial-mode maximally entangled while the other DOF remains partially entangled.
  • Iterative application improves the total hyper-ECP success probability substantially, while swap gates can further improve each round’s success probability.The protocol concentrates polarization and spatial-mode DOFs independently; swap gates transfer useful information between nonlocal partially hyperentangled states.

D. Hyper-ECP for polarization-time-bin hyperentangled Bell state

The polarization-time-bin hyper-ECP uses Schmidt projection, parity checks, time-bin-selective polarization flips, and single-photon measurement to distill unknown partially hyperentangled states. Its baseline success probability is |αβγδ|2, increasing to 4|αβγδ|2 with an improved measurement.

  • The protocol addresses unknown polarization-time-bin hyperentangled Bell states using the Schmidt projection method.The input consists of two nonlocal partially hyperentangled photon pairs with unknown normalized parameters.
  • Polarization and time-bin bit flips prepare the four-photon state for parity-based selection.Half-wave plates and active switches implement the polarization and time-bin bit-flip operations.
  • The protocol selects even-parity polarization outcomes, then performs time-bin parity checking and retains the even-parity polarization branch.The selected four-photon state is projected with probability 2|αβ|2 before subsequent time-bin processing.
  • Single-photon measurement on photons C and D converts the selected state into a maximally hyperentangled Bell state of photons A and B.An unbalanced interferometer separates early, middle, and late arrival times; the middle-time events are retained and local corrections are applied.
  • |αβγδ|2 is the baseline success probability, while an improved single-photon measurement increases it to 4|αβγδ|2.The baseline is stated as one quarter of the polarization-spatial hyper-ECP success probability.

A. History of entanglement purification

Entanglement purification developed from early CNOT- and linear-optics protocols toward deterministic and hyperentanglement-specific methods for noisy nonlocal systems. Hyper-EPPs address the added complexity of purifying multiple entangled degrees of freedom.

  • Entanglement purification distills high-fidelity entangled states from mixed nonlocal systems with less entanglement.It is described as a passive way to reduce noise effects and as indispensable in quantum repeaters.
  • Early purification protocols progressed from Bennett’s CNOT-based Werner-state EPP to linear-optics schemes and cross-Kerr-based polarization purification.Subsequent proposals and experiments expanded the available optical implementations.
  • Deterministic purification was introduced through two-step and one-step protocols using hyperentanglement in additional DOFs.These developments included spatial-mode and frequency resources for polarization-entanglement purification.
  • Hyperentanglement purification is more complex than purification in a single DOF, motivating dedicated hyper-EPPs for polarization-spatial photon states.Later proposals combined conventional purification with photon-loss amplification and local entanglement resources.

B. Two-step hyper-EPP

The two-step hyper-EPP purifies mixed polarization-spatial hyperentangled Bell states using a phase-check QND and QSJM. QSJM joins states across photons, allowing cases retained for a second purification step and improving efficiency.

  • Quantum-state joining: QSJM transfers photon A’s polarization state to photon B’s polarization degree of freedom through a double-sided QD-cavity system.Conditional bit-flip and phase-flip operations recover photon B’s final polarization-spatial state.
  • Two-step hyper-EPP: The protocol targets mixed hyperentangled Bell states with polarization bit-flip and spatial-mode phase-flip errors using a P-S-QND and QSJM.It can also address nonlocal mixed hyperentangled Bell states with arbitrary errors in the two DOFs.
  • Two-step hyper-EPP: The two-step process retains high-fidelity outcomes from first-step cases that would otherwise require discarding or further processing.The second step combines additional photon-pair states through QSJM.
  • Fidelity and efficiency: Iterative application of the two-step hyper-EPP can greatly improve the fidelity of the resulting nonlocal hyperentangled photon pair.The review presents this behavior in Fig. 23.
  • Fidelity and efficiency: Introducing the second QSJM-assisted step greatly improves hyper-EPP efficiency compared with the first-step process.The comparison is shown for initial states with F1 = F2; the efficiency definition includes outcomes recoverable with QSJM.

VI. HYPERPARALLEL PHOTONIC QUANTUM COMPUTATION

Hyperparallel photonic quantum computation encodes information in multiple photon degrees of freedom to operate on two or more channels simultaneously. The reviewed hyper-CNOT scheme acts on polarization and spatial-mode DOFs using double-sided QD-cavity nonlinearities without auxiliary modes.

  • Concept and motivation: Hyperparallel photonic computation encodes each photon’s quantum states in multiple DOFs and performs universal gates on two-photon or multiphoton systems.This differs from conventional parallel computation based on one DOF or an equivalent encoding.
  • Hyper-CNOT gate: The reviewed hyper-CNOT operates simultaneously on the polarization and spatial-mode DOFs of a two-photon system using double-sided QD-cavity nonlinearities.The scheme is presented as a route to scalable hyperparallel computation.
  • Advantages: Hyperparallel quantum logic gates can reduce resource consumption and photonic dissipation in quantum circuits.The reviewed hyper-CNOT scheme is described as avoiding auxiliary spatial or polarization modes.
  • Hyper-CNOT gate: The gate sequence prepares two QD electron spins, applies Hadamard operations, and sends photon spatial modes through QD-cavity circuits and polarization components.The two photons begin with independent polarization and spatial-mode superpositions.
  • Hyper-CNOT gate: The resulting four-qubit hybrid CNOT gate uses photon A’s polarization and spatial modes as controls and photon B’s corresponding DOFs as targets.Conditional sign corrections follow measurements of the two electron spins.

VII. DISCUSSION AND SUMMARY

The review defines hyperentanglement as entanglement across multiple photon DOFs and surveys its preparation and QIP applications. It emphasizes increased communication capacity and concentration or purification against distribution and storage noise.

  • Discussion and summary: Hyperentanglement is entanglement in multiple DOFs of a photon system, prepared by combining techniques that create entanglement in individual DOFs.The review discusses its use in quantum communication and computation.
  • Discussion and summary: Hyperentanglement can greatly increase quantum-channel capacity while assisting quantum communication protocols.The review places this use alongside applications in quantum information processing.
  • Discussion and summary: Locally produced maximally entangled photons can decohere during distribution and storage because of environmental noise.The review identifies this as a practical issue for quantum information processing.
  • Discussion and summary: Entanglement concentration and purification are presented as passive methods for nonlocal quantum systems to counter noise in long-distance communication assisted by quantum repeaters.These methods address the adverse influence of noise on distributed photon systems.
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