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Continuous Variable Quantum Information Processing

Ulrik L. Andersen, Gerd Leuchs, Christine Silberhorn

arXiv:1008.3468v1quant-ph

TL;DR

Continuous-variable quantum information addresses how continuous observables can be used to encode, process, and detect quantum information alongside discrete descriptions. The review surveys the preparation, processing, and detection stages, including Gaussian and non-Gaussian operations, protocols, memories, and optical implementations. It identifies controlled non-Gaussian operations as a central unresolved requirement for universal CV quantum computation while documenting progress across the field.

  • Problem

    CV quantum computing and related protocols require operations and memories that are difficult to realize, especially controlled non-Gaussian transformations for universal computation.

  • Method

    The paper reviews CV quantum information processing across state preparation, optical transformations, detection, protocols, quantum memories, and experimental developments.

  • Results

    The review covers progress in CV quantum key distribution, quantum memories, Gaussian and non-Gaussian operations, optical squeezing, computation, and information distribution.

  • Takeaways & Limitations

    Continuous-variable descriptions can be more efficient than discrete descriptions depending on the system and experiment, but universal CV computation still depends on mastering a controlled non-Gaussian operation.

Abstract

from arXiv · show

Observables of quantum systems can posses either a discrete or a continuous spectrum. For example, upon measurements of the photon number of a light state, discrete outcomes will result whereas measurements of the light's quadrature amplitudes result in continuous outcomes. If one uses the continuous degree of freedom of a quantum system either for encoding, processing or detecting information, one enters the field of continuous variable (CV) quantum information processing. In this paper we review the basic principles of CV quantum information processing with main focus on recent developments in the field. We will be addressing the three main stages of a quantum informational system; the preparation stage where quantum information is encoded into CVs of coherent states and single photon states, the processing stage where CV information is manipulated to carry out a specified protocol and a detection stage where CV information is measured using homodyne detection or photon counting.

I. INTRODUCTION

The introduction frames continuous-variable quantum information as an alternative representation for quantum systems, emphasizing that discrete and continuous descriptions can coexist and that experimental context determines the useful choice. It presents CV processing through preparation, controlled operations or decoherence, and measurement, with homodyne detection providing access to field quadratures.

  • Discrete and continuous variables: Quantum systems can be described using discrete or continuous variables, with light showing discrete photon-number spectra but continuous amplitude and phase spectra.The distinction reflects how quantization appears in the measured observables.
  • Discrete and continuous variables: The same light field can be represented in photon-number or continuous-variable language, and the preferable representation depends on the experiment and detector.Photon counting favors the photon-number basis, whereas homodyne detection favors continuous variables.
  • Protocol stages: Quantum information protocols comprise preparation of an optical quantum state, controlled processing or environmental decoherence, and measurement for characterization or decoding.The review addresses states such as coherent states and single photons within this three-stage framework.
  • Continuous-variable encoding: CV encoding uses laser light and homodyne measurements to obtain information about field quadratures related to amplitude and phase.Quadratures form an infinite-dimensional Hilbert space and can encode information as continuous superpositions of eigenstates.
  • Continuous-variable encoding: The Wigner function represents quantum states in phase space, while integrating over one quadrature yields the probability distribution for measuring the conjugate quadrature.This provides a direct link between phase-space descriptions and homodyne detection.
  • Quantum states: Gaussian states are characterized by Gaussian marginal distributions and their first and second moments, while squeezing reduces one quadrature’s uncertainty at the expense of the conjugate quadrature.Extreme squeezing would require infinite energy and is therefore not physically attainable.
  • Beyond Gaussian states: Recent CV research explores non-Gaussian states and hybrid systems because Gaussian states with Gaussian operations are insufficient for entanglement distillation or CV quantum computation.These approaches combine homodyne detection and photon counting within one system.
  • Hybrid degrees of freedom: Discrete and continuous degrees of freedom can coexist in one quantum system, so a single photon may support both discrete-variable and continuous-variable processing.Coupled degrees of freedom can require a multimode description.

III. QUANTUM DETECTION

Quantum detection extracts photon or field statistics using methods suited to bright or dark states, including direct detection, homodyne detection, photon counting, and time multiplexing. Each method has distinct information and resolution limits.

  • Detector choice determines whether measurements access field statistics, photon number, or quadrature information, while experimental imperfections limit resolution and noise performance.Direct detection, homodyne detection, and photon counting provide complementary measurement capabilities.
  • Direct detection is effective for intense light and provides a fixed phase reference, but it lacks direct information about the phase quadrature.The bright beam supplies its own local oscillator, avoiding measurement-basis synchronization between communicating parties.
  • Homodyne detection amplifies dark quantum states through interference with a strong local oscillator, with the relative phase selecting the measured quadrature.This makes homodyne detection suitable when ordinary direct detection lacks sufficient sensitivity.
  • A time-multiplexed detector divides one input pulse into eight output pulses measured by two APDs, whose click statistics reveal the incident photon statistics.The fiber network uses three 50/50 couplers and variable-length fiber loops; detector tomography can specify the resulting statistics.
  • Photon statistics alone do not fully characterize a quantum light state because they determine probabilities but not phase relationships between photon-number components.The Wigner function at the origin is nevertheless directly related to photon-number parity.
  • Parity measurements combined with phase-space displacements can probe the Wigner function point by point, although this had not been achieved for traveling light fields.The principle was demonstrated for a field stored in a high-Q cavity.

IV. QUANTUM OPERATIONS

Quantum operations are classified by whether they preserve Gaussianity: Gaussian operations map Gaussian states to Gaussian states, whereas non-Gaussian operations do not. The section introduces these two transformation classes.

  • Gaussian operations map Gaussian states onto Gaussian states, while non-Gaussian operations map Gaussian states onto non-Gaussian states.

A. Gaussian operations

Gaussian transformations can be decomposed into elementary optical operations and implemented across multiple modes using squeezers, interferometers, measurements, and feed-forward. Offline squeezed-vacuum resources enable linear-optical realizations of arbitrary multimode Gaussian operations.

  • Gaussian operations: Elementary Gaussian transformations include beam splitting, phase shifting, displacement, squeezing, and homodyne detection.Combining these operations in optical circuits implements Gaussian transformations.
  • Gaussian operations: Bloch-Messiah decomposition expresses any multimode Gaussian transformation as beam-splitter arrays surrounding single-mode squeezers.This reduces arbitrary Gaussian-operation implementation to realizing pure, efficient single-mode squeezing.
  • Gaussian operations: Feed-forward squeezing combines an input state with offline squeezed vacuum, measures an output by homodyne detection, and applies a displacement based on the continuous result.The setup uses a beam splitter and an electro-optic phase modulator to largely circumvent decoherence in squeezing fragile quantum information.
  • Gaussian operations: Arbitrary multimode Gaussian operations can be realized by beam-splitter networks, offline squeezed vacua, homodyne measurements, and feed-forward displacements.The construction uses only linear optical transformations on the quantum state when the squeezed-vacuum resources and final measurement-induced displacements are supplied.
  • Gaussian operations: A two-mode Gaussian operation uses two squeezers in a Mach-Zehnder interferometer and has been experimentally implemented as a quantum nondemolition interaction.The circuit combines four linear beam-splitting interactions, two squeezed-vacuum resources, homodyne detection, and feed-forward.
  • Gaussian operations: Optical amplification can operate at the shot-noise limit without squeezed-state resources, and its coupling strength and phase are controlled by beam-splitter parameters.The same nonlinearity-free amplification transformation has been used to make coherent-state clones at the quantum limit.
  • Gaussian operations: The feed-forward approach is physically simpler than the cluster-state approach for universal Gaussian transformations.Single-mode squeezing requires one squeezed-vacuum state and one homodyne detector with feed-forward, versus a five-mode entangled state and four homodyne detectors for the cluster-state method.

B. Non-Gaussian operations

Non-Gaussian transformations complement Gaussian operations to implement arbitrary pure operations and can be realized through measurement-induced processes using photon-sensitive detectors.

  • Non-Gaussian operations: Arbitrary pure operations require complementing Gaussian transformations with a non-Gaussian operation.Pure inputs must map to arbitrary pure outputs, but the required non-Gaussian operation is harder to realize because it involves very large third-order nonlinearities.
  • Non-Gaussian operations: Measurement-induced non-Gaussian transformations use nonlinear detectors such as avalanche photodiodes to project Gaussian states into non-Gaussian states.This approach avoids relying solely on large optical nonlinearities and can prepare single-photon states from weakly entangled states.
  • Non-Gaussian operations: Conditional state preparation requires precise control of photonic modes because avalanche-photodiode and homodyne detection can have different spectral-spatial responses.The mismatch can introduce unwanted multimode structure and mixedness.
  • Non-Gaussian operations: Photon-sensitive measurement-induced operations have been used for single-photon and cat-state preparation and proposed for arbitrary quantum operations in cluster-state schemes.The cluster-state proposal complements homodyne detectors with photon-number-resolving detectors.

V. QUANTUM RESOURCES

Continuous-variable quantum information relies on non-classical resources, including squeezed, entangled, and single-photon states, whose generation has recently become more efficient and stable.

  • V. QUANTUM RESOURCES: Classical and non-classical optical states are distinguished through the behavior of the Glauber-Sudarshan P-function.Thermal states have well-behaved P-functions, whereas squeezed states have singular or negative P-functions and are classified as non-classical.
  • V. QUANTUM RESOURCES: Non-classical resources reviewed for CV quantum information include squeezed states, entangled states, and single-photon states.These resources are presented as central components of CV quantum information processing.
  • V. QUANTUM RESOURCES: Recent progress has produced purer, more efficient, and more stable squeezing using second-order and third-order optical nonlinearities.Reported approaches include optical parametric amplification, optical fibers, and rubidium vapor.

A. Optical parametric amplification

Optical parametric amplification generates correlated signal and idler photons whose quadrature correlations can produce squeezed light, with cavity design used to improve measured squeezing.

  • A. Optical parametric amplification: Optical parametric amplification converts a pump photon into signal and idler photons satisfying ωp = ωs + ωi.The generated photons become quantum-correlated in time, frequency, and amplitude and phase quadratures.
  • A. Optical parametric amplification: Indistinguishable signal and idler photons produce quadrature squeezing through their quadrature correlations.The photons must occupy the same spatial-temporal and polarization mode.
  • A. Optical parametric amplification: A bow-tie cavity with a PPKTP crystal uses copper-oven temperature control to maintain phase matching between the waves.The temperature-controlled crystal is used for squeezing the light field.
  • A. Optical parametric amplification: More than 10dB squeezing was directly measured using a LiNbO3 monolithic cavity designed to minimize optical losses and improve phase stability.The nonlinear crystal formed the cavity, with mirrors coated onto its curved end facets.

B. Fiber system

Fiber-based Kerr squeezing uses four-photon mixing and long, strongly driven fibers, while Raman and Brillouin scattering limit performance in different energy regimes.

  • B. Fiber system: The Kerr effect generates squeezing through a four-photon process in which two degenerate photons convert into signal and idler photons.Quantum correlations between the converted photons produce quadrature squeezing, analogous to optical parametric amplification.
  • B. Fiber system: Long fibers and short pump pulses compensate for silica’s small nonlinear susceptibility and increase the effective Kerr nonlinearity.The fiber length and pulse duration are engineering choices for producing appreciable squeezing.
  • B. Fiber system: 6.8 dB quadrature squeezing was produced using a 13.2 m polarization-maintaining fiber pumped by a 140fs pulse.The model included nonlinear and stochastic Raman effects.
  • B. Fiber system: Raman scattering markedly deteriorates squeezing at higher energies, while guided acoustic-wave Brillouin scattering affects squeezing at lower energies.The two scattering mechanisms therefore constrain different operating regimes.

D. Single photons

CV quantum key distribution uses coherent states and quadrature measurements to generate shared secret keys, with security analyzed against progressively stronger eavesdropping attacks. Experiments demonstrated both discrete-state and Gaussian-modulated implementations, while reverse reconciliation improves practical transmission security.

  • Security principle: CV QKD generates shared random keys by measuring non-orthogonal states, such as conjugate quadratures, whose simultaneous values cannot be known exactly.The security mechanism follows from the uncertainty principle.
  • Practical security: Reverse reconciliation provides Alice and Bob an informational advantage at all transmission ratios, addressing the presumed secure-distance limit below 50% loss.The protocol lets Bob send error-correction and privacy-amplification messages to Alice.
  • Attack models: Three attack classes are considered: individual, collective, and coherent attacks, ordered by increasing sophistication.Individual attacks measure stored probes separately, whereas collective attacks jointly measure stored probes after basis disclosure.
  • Security results: Collective attacks are sufficient for unconditional security in qubit protocols, and this result has been partially extended to continuous-variable systems.For Gaussian coherent-state alphabets, Gaussian collective attacks are optimal.
  • Experiments: CV QKD was first experimentally demonstrated using either four coherent states or a continuous Gaussian distribution of coherent-state amplitudes and phases.In the Gaussian-modulated experiment, Bob used random homodyne measurements and reverse reconciliation to form binary secret keys.

B. Distribution of quantum information

Quantum information must be transmitted between prepared, stored, and processed network nodes through fragile quantum channels. For continuous variables, teleportation with entanglement distillation and quantum error-correction coding are the principal approaches to reducing transmission noise.

  • Network transmission: Quantum networks link nodes that prepare, store, and process quantum states through channels transmitting fragile continuous-variable information.Fault-tolerant transmission between nodes is a central task in quantum information science.
  • Error mitigation: Noise mitigation uses two main strategies: quantum teleportation combined with entanglement distillation, or quantum error-correction coding.These protocols have been extensively investigated for discrete variables, while progress for continuous variables has been slower.

1. Teleportation and distillation

CV teleportation transmits quantum information using shared entanglement, while distillation and error correction address noise accumulated during transmission. Teleportation has reached experimentally demonstrated fidelities up to 83%, but Gaussian-state distillation remains experimentally incomplete.

  • Teleportation: Teleportation transmits quantum information through a classical channel and a quantum channel sharing entanglement between sender and receiver.CV teleportation was experimentally realized in 1998 and later extended to squeezed states, entangled states, and networks.
  • Teleportation: Teleportation fidelity increased from 58% in the first experiment to as high as 83% in later experiments.These results include demonstrations involving single-mode squeezed and entangled states.
  • Teleportation-based operations: Teleportation-based operations can implement transformations by modifying the entangled resource, leaving the difficult transformation offline and the information transfer deterministic and clean.The entangled state encodes the desired operation applied to the input.
  • Distillation: Entanglement distillation extracts a smaller ensemble of highly entangled states from a larger ensemble degraded by noisy transmission.Faithful entanglement distribution is required for efficient teleportation between network nodes.
  • Distillation limits: Gaussian-state distillation has no full experimental demonstration because it requires difficult non-Gaussian transformations, whereas certain non-Gaussian noise channels can be distilled with linear optics, homodyne detection, and feed-forward.Experiments addressed time-varying transmission and phase noise.
  • Error correction: Quantum error correction cannot directly use classical encoding and decoding because quantum measurements disturb states and the no-cloning theorem forbids copying them.Proposed CV codes include schemes for non-Gaussian noise, distributed entanglement, and complete erasure noise.

C. Quantum memory

Quantum memories are essential for quantum networking, secure eavesdropping analyses, repeaters, and scalable computing. CV memories couple light to atomic ensembles using QND feedback, EIT Raman interactions, or photon-echo techniques, with experiments demonstrating storage and retrieval of quantum states.

  • Role of quantum memories: Quantum memories are crucial for optimal eavesdropping attacks, quantum repeaters, and scalable quantum computing.They must store quantum information faithfully for these applications.
  • Memory approaches: CV memories use efficient light–atomic-ensemble coupling through QND interaction with feedback, EIT via Raman interaction, or photon-echo techniques.The three approaches differ in their interaction mechanisms.
  • QND memories: A QND memory stored and retrieved a light pulse using two cesium ensembles, producing an output state with better quality than any classical memory.The realization combined the QND operation with electro-optical feedback.
  • EIT memories: EIT memories slow and stop signal pulses as dark-state polaritons and retrieve them coherently when the control pulse is restored.CV experiments stored squeezed vacuum in rubidium clouds with total efficiencies of about 10–15%.
  • Photon-echo memories: Photon-echo memories store light by absorption in an inhomogeneously broadened medium, compensate dephasing, and control release through the echo process.The approach has been experimentally realized in solid-state media.

D. Quantum computation

Continuous-variable quantum computation remains less advanced than qubit-based computing, partly because experimental realization is difficult. The review contrasts direct-gate and off-line resource approaches, including cluster-state computation, while noting that universal CV operations still require controllable non-Gaussian resources.

  • CV quantum computing has progressed less far than qubit-based computing, partly because its experimental realization faces significant difficulties.
  • The conventional circuit model applies difficult computational gates directly to quantum information through nonlinear coupling.
  • The off-line approach moves difficult non-Gaussian operations away from the computational line before gate execution.
  • Cluster-state computation uses off-line Gaussian multimode entangled resources, followed by gate operations using linear optics, detection, and feed-forward.
  • The review places CV processing within a broader alternative to the historically dominant discrete-variable description of quantum information.
  • Universal CV quantum operations remain limited by the difficulty of enabling an efficient and controllable non-Gaussian operation.
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