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When Entanglement meets Classical Communications: Quantum Teleportation for the Quantum Internet (Invited Paper)

Angela Sara Cacciapuoti, Marcello Caleffi, Rodney Van Meter, Lajos Hanzo

arXiv:1907.06197v2quant-phcs.NI

TL;DR

Quantum teleportation requires a communication model that accounts for entanglement, no-cloning constraints, and quantum decoherence. The paper develops a communications-engineering treatment of teleportation and its impairments, combining theoretical modeling with experiments. It finds that composite impairments are multiplicative and asymmetric across Bloch-vector coordinates.

  • Problem

    Quantum teleportation enables qubit transfer without particle transmission, but its entanglement-based operation requires redesigning the classical communication-system model and accounting for decoherence.

  • Method

    The paper reviews teleportation fundamentals and entanglement schemes, proposes a noiseless communication model, models imperfections, and analyzes IBM Q teleportation experiments.

  • Results

    Theoretical analysis and experiments reveal that composite quantum impairments are multiplicative and affect Bloch-vector coordinates asymmetrically.

  • Takeaways & Limitations

    Quantum teleportation is the core functionality of the Quantum Internet, but its practical employment must address severe quantum impairments.

Abstract

from arXiv · show

Quantum Teleportation is the key communication functionality of the Quantum Internet, allowing the "transmission' of qubits without either the physical transfer of the particle storing the qubit or the violation of the quantum mechanical principles. Quantum teleportation is facilitated by the action of quantum entanglement, a somewhat counter-intuitive physical phenomenon with no direct counterpart in the classical word. As a consequence, the very concept of the classical communication system model has to be redesigned to account for the peculiarities of quantum teleportation. This re-design is a crucial prerequisite for constructing any effective quantum communication protocol. The aim of this manuscript is to shed light on this key concept, with the objective of allowing the reader: i) to appreciate the fundamental differences between the transmission of classical information versus the teleportation of quantum information; ii) to understand the communications functionalities underlying quantum teleportation, and to grasp the challenges in the design and practical employment of these functionalities; iii) to acknowledge that quantum information is subject to the deleterious effects of a noise process termed as quantum decoherence. This impairment has no direct counterpart in the classical world; iv) to recognize how to contribute to the design and employment of the Quantum Internet.

I. INTRODUCTION

The Quantum Internet must accommodate qubits and entanglement, whose quantum-mechanical constraints make direct transmission unreliable and require a redesigned communication model. The paper introduces teleportation, its communication links, noise mechanisms, and experimental perspective.

  • Motivation: Quantum networks connect remote nodes through qubits or distributed entangled states, enabling functionalities without direct classical counterparts.Examples include secure communications, blind computing, increased quantum-computing power, and advanced quantum sensing.
  • Motivation: Unknown qubits cannot be copied or measured, so photon corruption during direct transmission irreversibly destroys the encoded quantum information.Direct qubit transmission is therefore not practically feasible unless applications tolerate information loss or low success rates.
  • Quantum Teleportation Overview: Quantum teleportation transfers qubits without physically transferring their stored particles, using entanglement while preserving quantum-mechanical principles.Teleportation has been experimentally verified over distances up to 1200 kilometers.
  • Communication Model: Teleportation requires parallel quantum and classical links: entanglement generation and distribution, plus transmission of two classical bits.The communication model therefore differs from a conventional single-channel classical system.
  • Scope: The paper reviews quantum preliminaries, teleportation schemes, a noiseless communication model, realistic imperfections, and IBM Q experiments.Its stated objectives include clarifying classical-versus-quantum transmission, practical challenges, and quantum decoherence.
  • Quantum Noise: Quantum decoherence is environmental quantum noise with no direct classical counterpart, and quantum impairments are multiplicative and asymmetric across a qubit’s Cartesian coordinates.Bit-flip, phase-flip, and combined errors can occur, with differing probabilities.

B. Phase

Quantum phase is a property of superposed states that distinguishes physically different states and governs interference. Composite quantum systems further expand the state space through tensor products and superpositions.

  • B. Phase: The global phase does not affect measurement statistics, whereas the relative phase between amplitudes represents different quantum states.States with equal amplitude magnitudes but different relative phases are physically distinct.
  • B. Phase: Relative phase is critical to quantum computation because it determines constructive or destructive interference patterns.Interference can enhance or reduce the probability of particular measurement outcomes.
  • Composite States: Composite quantum-system state spaces combine constituent spaces through tensor products, represented compactly by concatenated ket notation.The tensor-product basis provides the representation for multi-qubit states.
  • Composite States: An n-qubit register can occupy a superposition of all 2^n basis states, unlike n classical bits, which occupy one state at a time.Quantum algorithms manipulate amplitudes and phases to create interference patterns.

D. Entanglement

Entanglement describes composite quantum states that cannot be decomposed into individual subsystem states, while quantum operations and noise determine how such states are transformed and characterized.

  • D. Entanglement: Entangled states are composite states that cannot be written as tensor products of individual qubit states.Separable states can be expressed as tensor products with respect to a specified subsystem decomposition.
  • D. Entanglement: Bell states, or EPR pairs, are examples of maximally entangled two-qubit states whose measurement outcomes are individually random but perfectly correlated.For |Φ+⟩, matching outcomes occur when the two measurement results are compared.
  • D. Entanglement: Entanglement is not an absolute property: a state may be entangled under one subsystem decomposition and unentangled under another.The relevant tensor decomposition must therefore be specified or clear from context.
  • Quantum Transformations: Closed quantum systems evolve through deterministic, reversible unitary operations, represented in the circuit model as quantum gates.Unitary operators satisfy U†U = I and have inverses given by U^-1 = U†.
  • Quantum Transformations: The no-cloning theorem prevents copying unknown quantum states, making decoherence-induced information loss irreversible rather than recoverable from a source copy.This constraint has major consequences for quantum-communication design.
  • Mixed States: Quantum fidelity measures the overlap or distinguishability between a mixed state and a desired pure state, ranging from 0 to 1.Fidelity equals 1 for a pure state and decreases as decoherence degrades state quality.

G. The Bloch Vector

The Bloch vector provides a one-to-one Cartesian representation of any single-qubit pure or mixed state, while quantum teleportation uses an unknown state and shared entanglement to reconstruct that state remotely.

  • The Bloch Vector: Mixed states lie inside the Bloch sphere, with ||r|| < 1, whereas pure states lie on its surface, with ||r|| = 1.
  • The Bloch Vector: The Bloch vector r = [rx, ry, rz] ∈ R3 maps each quantum state density matrix ρ to Cartesian coordinates.This representation applies to both pure and mixed states.
  • The Bloch Vector: Quantum gates transform Bloch-vector coordinates according to the corresponding unitary evolution ρout = UρinU†.The listed transformations include sign changes for Pauli gates and coordinate rearrangement for the Hadamard gate.
  • The Bloch Vector: The Bloch-vector representation visualizes how quantum gates alter state coordinates, including the Pauli-X gate leaving rx unchanged while changing ry and rz.
  • Quantum Teleportation: Quantum teleportation begins with an unknown state |ψ⟩ and a shared EPR pair, then reconstructs |ψ⟩ at Bob after Alice’s measurement result arrives classically.The original qubit cannot be copied or measured to reveal unknown amplitudes without altering the state.

B. Quantum Teleportation: Mathematical Details

Quantum teleportation combines a shared EPR pair, local quantum operations, Alice’s joint measurement, and a finite-delay classical message to reconstruct an unknown qubit at Bob.

  • B. Quantum Teleportation: Mathematical Details: Teleportation takes the unknown state |ψ⟩ and a pre-agreed shared Bell state, such as |Φ+⟩, as inputs.Any of the four Bell states can be used when selected in advance by Alice and Bob.
  • B. Quantum Teleportation: Mathematical Details: Alice first applies a CNOT gate to her qubits and then applies a Hadamard gate to the qubit being teleported.
  • B. Quantum Teleportation: Mathematical Details: Alice’s joint measurement yields each of 00, 01, 10, and 11 with 25% probability, regardless of α and β.
  • B. Quantum Teleportation: Mathematical Details: Bob recovers |ψ⟩ only after receiving Alice’s two classical measurement bits through a finite-delay link obeying the speed of light.
  • B. Quantum Teleportation: Mathematical Details: The teleportation of one qubit requires both EPR-pair generation and distribution and a finite-delay classical channel, tightly integrating quantum and classical resources.

C. Practical Entanglement Generation and Distribution

Practical quantum teleportation depends on generating and distributing entanglement across a quantum link, interfacing matter and flying qubits, and integrating these resources with classical processing.

  • C. Practical Entanglement Generation and Distribution: Spontaneous parametric down-conversion generates polarization-entangled photon pairs that travel to Alice and Bob, where transducers transfer entanglement to matter qubits.
  • C. Practical Entanglement Generation and Distribution: Entanglement generation may occur at a midpoint, at the source, or at both endpoints, but every scheme requires a quantum channel between Alice and Bob.
  • C. Practical Entanglement Generation and Distribution: Photons serve as flying qubits because they offer moderate environmental interaction, convenient optical control, and high-speed low-loss transmission.
  • C. Practical Entanglement Generation and Distribution: Alternative schemes use atoms in optical cavities linked by photonic channels or remote-atom entanglement created through a beam-splitter Bell-state measurement.
  • C. Practical Entanglement Generation and Distribution: Moving stationary qubits and their hardware after entanglement generation is described as far from scalable.
  • IV. A NOISELESS COMMUNICATION SYSTEM MODEL OF QUANTUM TELEPORTATION: The proposed communication model revises Shannon’s classical model by tightly integrating classical and quantum operations and communications.Its blocks include quantum information, teleportation transmission and reception, and entanglement generation and distribution.
  • IV. A NOISELESS COMMUNICATION SYSTEM MODEL OF QUANTUM TELEPORTATION: A quantum equivalent of the classical source encoder may not be feasible because quantum information cannot generally be read or copied.Classical parity- or repetition-based error correction therefore cannot be directly employed in a quantum network.

V. IMPERFECT QUANTUM TELEPORTATION

Imperfect quantum teleportation is shaped by decoherence and operation errors, requiring theoretical noise modeling and experiments to characterize cumulative impairments. The analysis identifies multiplicative, coordinate-dependent quantum impairments and highlights unresolved modeling challenges.

  • Decoherence affects realistic quantum communication systems and the entanglement process needed for teleportation.
  • Teleportation operations introduce additional imperfections that further degrade the teleported qubit.
  • Modeling quantum-domain impairments that capture the different degradations affecting teleportation remains an open problem.
  • The paper combines communications-engineering theory with IBM Q experiments because operation errors depend on the technology representing the qubit.
  • Quantum impairments are multiplicative rather than additive, resembling classical fading more than Brownian-motion noise.
  • Quantum impairments act asymmetrically on the Bloch-vector coordinates, producing spatial selectivity.

B. Phase Damping

Phase damping erodes quantum information without energy loss by exponentially damping the x- and y-coordinates of a qubit while leaving z unchanged. It transforms an initially pure state into a mixed state inside the Bloch sphere.

  • Phase damping models quantum-information erosion without energy loss through the Lindblad operator Lz = √γzσz.
  • Compensating the Hamiltonian-induced phase rotation around the z-axis isolates the spatial selectivity of phase damping.
  • The off-diagonal density-matrix elements decay exponentially at rate γz, causing the information they encode to erode over time.
  • Unlike unitary Pauli-Z evolution, phase damping damps x and y rather than flipping them.
  • The initial pure state becomes a mixed state with ||r(t)|| < 1 for t > 0.
  • The x- and y-coordinates undergo multiplicative exponential damping governed by γz, while the z-coordinate remains unchanged.

C. y-z Damping

y-z damping is a non-unitary bit-flip-related impairment that exponentially attenuates the y- and z-coordinates while preserving x. The qubit therefore approaches a mixed state along the x-axis.

  • y-z damping is modeled by the Lindblad operator Lx = √γxσx and is also known as a bit-flip error process.
  • The initial pure state becomes mixed with ||r(t)|| < 1 for t > 0.
  • Although associated with the Pauli-X gate, y-z damping represents non-unitary evolution rather than unitary bit flipping.
  • After compensating Hamiltonian-induced phase evolution, the qubit asymptotically approaches r = [rx(0), 0, 0].
  • The left plot includes orbital evolution around the z-axis, whereas the right plot compensates it to reveal coordinate-selective damping.
  • The y- and z-coordinates undergo multiplicative exponential damping governed by γx, while the x-coordinate remains unchanged.

D. Combined y-z-Phase Damping

Combined y-z-phase damping imposes multiplicative, coordinate-dependent exponential decay on all Bloch-vector coordinates. The resulting evolution can be represented as a communications-engineering model and is experimentally visualized after compensating Hamiltonian-induced phase evolution.

  • D. Combined y-z-Phase Damping: The combined process uses Lindblad operators Lx and Lz and is analyzed as a composite depolarizing process.The paper refers to this process as combined y-z-phase damping.
  • D. Combined y-z-Phase Damping: The impairments multiply Bloch-vector coordinates and damp all three exponentially at different decay rates.This asymmetric behavior represents spatial selectivity across the Bloch-vector coordinates.
  • D. Combined y-z-Phase Damping: Time-domain evolution confirms that all Bloch coordinates move toward the sphere’s center, faster than in the y-z-phase damping comparison shown previously.The right plot displays compensated Bloch-coordinate trajectories over time.
  • D. Combined y-z-Phase Damping: After compensating Hamiltonian-induced phase evolution, the x-, y-, and z-coordinates decay toward zero at rates γz, γx+γz, and γx, respectively.The y-coordinate therefore has the combined decay rate γx+γz.
  • D. Combined y-z-Phase Damping: The combined damping effects are modeled by applying multiplicative damping to every qubit coordinate with coordinate-specific exponential rates.The model is summarized in Fig. 14.
  • D. Combined y-z-Phase Damping: The broader arbitrary decoherence model likewise represents decoherence as exponential multiplication of Bloch coordinates with exponents determined by system-environment interaction.The three spatial directions share the mechanism but not necessarily the exponent.

VI. IBM Q EXPERIMENTAL RESULTS

The IBM Q experiments investigate cumulative impairments during quantum teleportation using quantum process tomography. They evaluate effects arising from entanglement generation, imperfect gates, and decoherence on a real five-qubit processor.

  • VI. IBM Q EXPERIMENTAL RESULTS: The experiments use a real IBM Q quantum computer to obtain experimental insights into composite impairments affecting teleported qubits.The campaign focuses on cumulative effects during teleportation.
  • VI. IBM Q EXPERIMENTAL RESULTS: Quantum process tomography characterizes teleportation dynamics and cumulative impairments from entanglement generation, quantum gates, and decoherence.The IBM Q platform does not account for channel effects during entanglement distribution.
  • VI. IBM Q EXPERIMENTAL RESULTS: Over 8 million experiments ran on the 5-qubit IBM Tenerife ibmqx4 processor from January 14 to January 19, 2019.The teleported state occupied q[0], while the EPR pair was created between q[1] and q[2].

A. Teleporting a basis state

Teleportation of the basis state |0⟩ shows that cumulative quantum impairments transform the intended pure state into a mixed state inside the Bloch sphere. The coordinate distributions reveal attenuation, small drifts, correlations, and imperfect normal fits.

  • A. Teleporting a basis state: The teleported |0⟩ state becomes mixed and lies inside the Bloch sphere rather than remaining at [0, 0, 1].This behavior agrees with the theoretical decoherence analysis.
  • A. Teleporting a basis state: The x-coordinate is roughly normal with µx ≃ 0 and σx ≃ 0.021, and almost all values lie in (−0.044, 0.056).Rigorous distribution fitting has low confidence because some intervals show shape deviation between theory and measurement.
  • A. Teleporting a basis state: The y-coordinate is roughly normal with variance σy ≃ 0.013 and drifts slightly toward positive values with µy around 0.02.The distribution is characterized through the marginal CDF in Fig. 16b.
  • A. Teleporting a basis state: The z-coordinate shrinks from 1 to µz ≃ 0.66, remains approximately within (0.605, 0.712), and is modeled with variance σz ≃ 0.022.The reported probability outside that interval is approximately zero.
  • A. Teleporting a basis state: The experimental teleported states form a sphere centered near [µx = 0, µy = 0.02, µz = 0.66] with radius around 0.1.This density description makes the drift from the original pure state explicit.
  • A. Teleporting a basis state: Impairment effects are correlated: positive drift in x is coupled with lower drift in y.The correlation is shown in the joint distribution analysis.

B. Teleporting superposed states

Experiments with superposed states confirm multiplicative, asymmetric impairments that drive teleported states into mixed states and attenuate Bloch coordinates unequally. The results support a y-z-phase damping interpretation, but its parameters and coordinate correlations require further investigation.

  • B. Teleporting superposed states: For |+⟩, cumulative impairments corrupt the intended pure state into a mixed state inside the Bloch sphere.The ideal input is located at Bloch coordinates [1, 0, 0].
  • B. Teleporting superposed states: The |+⟩ experiment shows the x-coordinate shrinking from 1 to an average µx = 0.61.This confirms the multiplicative nature of cumulative impairments for a superposed state.
  • B. Teleporting superposed states: The y-coordinate distribution has local peaks near −0.18, −0.09, and 0.01, which the authors associate with device calibration procedures.Restricting data between calibrations may yield an approximately normal coordinate distribution, but goodness-of-fit metrics are low.
  • B. Teleporting superposed states: The experiments find coordinate correlations, including an association between y- and z-coordinate values.The joint PDF is used to visualize these dependencies.
  • B. Teleporting superposed states: For a state with all three Bloch coordinates nonzero, average coordinates are µx = 0.47, µy = 0.16, and µz = 0.39.The experiment uses marginal CDFs to characterize cumulative effects.
  • B. Teleporting superposed states: The composite impairments agree with the y-z-phase damping model under equal rates γx = γz, although this hypothesis and the decay-rate relationships need further confirmation.Coordinate correlations also require further investigation to characterize decay rates.
  • VII. OUTLOOK: The paper frames teleportation as a core Quantum Internet function but identifies open communication-system problems needed to account for all quantum impairments.Noise affects entanglement generation/distribution and quantum pre/post-processing blocks.

A. Mitigating Impairments in Quantum Pre/Post-Processing

Mitigating quantum communication impairments requires quantum-specific processing for information and entanglement, plus quantum repeaters for long-distance distribution. Key design choices remain open, including whether jointly designing encoder/decoder blocks improves performance.

  • Quantum Pre/Post-Processing: Redundancy can be introduced at Alice and exploited by a complementary block at Bob to mitigate impairments, forming a quantum analogue of channel coding.The manuscript presents these blocks as modifications to the quantum communication system model.
  • Quantum Pre/Post-Processing: Classical encoder–decoder techniques cannot be directly transferred because the no-cloning theorem prevents copying quantum information.Quantum error-correction techniques must therefore be specifically designed for quantum systems.
  • Entanglement Resources: Entanglement distillation converts multiple imperfectly entangled pairs into one almost-maximally entangled pair when contamination remains below a threshold.This improvement requires additional processing.
  • Long-Distance Distribution: Entanglement distribution decays exponentially with Alice–Bob distance, and classical amplify-and-forward or decode-and-forward strategies cannot address this impairment.Quantum repeaters instead use entanglement swapping to establish entanglement across long links through shorter links.
  • Open Problems: Separate encoder/decoder designs are possible, but whether a more difficult joint design provides superior performance remains unclear.The paper concludes that substantial frontier research is still required for the Quantum Internet’s open challenges.
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