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A Survey on Quantum Channel Capacities

Laszlo Gyongyosi, Sandor Imre, Hung Viet Nguyen

arXiv:1801.02019v1quant-phcs.IT

TL;DR

Quantum communication involves diverse capacity measures and transmission models, creating a need to organize their properties and differences from classical channels. This survey reviews quantum communication channels and their capacities, highlighting cases where quantum encoding improves classical transmission from 0 to 1 and entangled inputs increase transmitted information.

Abstract

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Quantum information processing exploits the quantum nature of information. It offers fundamentally new solutions in the field of computer science and extends the possibilities to a level that cannot be imagined in classical communication systems. For quantum communication channels, many new capacity definitions were developed in comparison to classical counterparts. A quantum channel can be used to realize classical information transmission or to deliver quantum information, such as quantum entanglement. Here we review the properties of the quantum communication channel, the various capacity measures and the fundamental differences between the classical and quantum channels.

I. INTRODUCTION … 2) Steps of the Communication:

Quantum channels expand communication beyond classical information transmission to private, entanglement-assisted, and quantum information delivery, while their capacities and behavior require quantum information theory. Communication proceeds through coding, noisy channel evolution, and measurement-based decoding, with performance determined by channel noise.

I. INTRODUCTION

  • Quantum communication faces difficult networking constraints because the no-cloning theorem complicates quantum repeaters and switches needed to connect distant users [267],, [551],.Source: §I · p. 1
  • Quantum channels support classical, private, entanglement-assisted classical, and quantum information, but their capacities lack a universal formula and are generally non-additive.Source: §I · p. 1

A. Applications and Gains of Quantum Communications

B. Privacy and Performance Gains of Quantum Channels

  • Quantum channels improve selected applications: QKD establishes symmetric keys because copied non-orthogonal states reveal errors,,,,,, [244],, while a bit-flip channel’s classical capacity rises from 0 to 1.Source: A. Applications and Gains of Quantum… · p. 2
  • Quantum communication can carry superpositioned states, with n qubits representing up to 2^n states corresponding to 2^n ×n bits, and quantum computing is sometimes reported as 100 millions times faster.Source: B. Privacy and Performance Gains of… · p. 3

C. Communication over a Quantum Channel

  • A quantum channel encodes classical messages by preparing quantum states, applies a Completely Positive Trace Preserving transformation, and decodes by measuring the received state.Source: C. Communication over a Quantum Channel · p. 3

1) The Quantum Channel Map:

  • Geometrically, qubit channels map the Bloch sphere to ellipsoids: unital channels preserve its center, whereas non-unital channels shift it and are more difficult to analyze.Source: 1) The Quantum Channel Map · p. 4

2) Steps of the Communication:

  • Quantum communication has three phases—channel coding, channel evolution, and measurement-based decoding—and recovery ranges from impossible for completely noisy channels to probability 1 for noiseless channels.Source: 2) Steps of the Communication · pp. 4–5

D. Formal Model … 2) Density Matrix and Trace Operator:

The paper formalizes noisy quantum-channel transmission with density matrices, unitary system–environment evolution, and partial traces, then distinguishes classical and quantum capacities and summarizes their information-theoretic foundations. Density matrices represent probabilistic mixtures of quantum states, while trace preservation and positivity characterize valid quantum states and operations.

D. Formal Model

  • Quantum-channel transmission begins with an input state ρin and pure environment state ρE=|0⟩⟨0|, forming the composite state ρin⊗ρE.The channel acts on the composite system through unitary evolution, after which tracing out the environment yields the output state and noisy channel evolution.Source: D. Formal Model · p. 5

E. Quantum Channel Capacity

  • Quantum-channel capacity measures how closely transmission approximates the ideal identity transformation and differs according to whether encoded states carry classical or quantum messages.Single-use and asymptotic capacities need not coincide, while encoding and decoding operate on blocks through quantum operators.Source: E. Quantum Channel Capacity · pp. 5–6

F. Definitions

  • Quantum information theory extends classical information theory to quantum systems, where entanglement introduces properties that classical information elements cannot describe.The paper introduces noisy channels, purification, isometric extensions, Kraus representations, von Neumann entropy, Holevo information, quantum mutual information, and quantum conditional entropy.Source: F. Definitions · p. 6, 1) Discussion · p. 6

1) Discussion:

  • Quantum channels can transmit classical information through pure orthogonal states or transmit non-orthogonal states and entanglement, motivating multiple capacity definitions.The HSW theorem establishes reliable classical transmission over noisy quantum channels [233],, while the LSD capacity characterizes asymptotic quantum transmission using quantum coherent information [134],,.Source: 1) Discussion · pp. 6–7

2) Density Matrix and Trace Operator:

  • A density matrix ρ is a classical-probability-weighted mixture of pure states, is positive-semidefinite, and reduces to a rank-one outer product for a pure state.Superpositions are quantum linear combinations, whereas mixed states are classical combinations of pure quantum states; density matrices contain all information extractable from the system.Source: 2) Density Matrix and Trace Operator · p. 8
  • The trace equals the sum of diagonal entries and eigenvalues, is linear, and satisfies cyclic product identities that support density-matrix analysis.For every density matrix, the trace is equal to one.Source: 2) Density Matrix and Trace Operator · pp. 8–9
  • Trace-based eigenvalue properties distinguish pure states from mixed reduced states, including the subsystem of an entangled pair.The section applies trace operations to density matrices and relates their eigenvalues to state characterization.Source: 2) Density Matrix and Trace Operator · p. 9
  • Unitary transformations map density matrices to transformed states, while Completely Positive Trace Preserving operations preserve the density-matrix conditions.The resulting matrix remains a valid density matrix under a CPTP operator G.Source: 2) Density Matrix and Trace Operator · p. 9

3) Quantum Measurement:

Quantum measurements use Hermitian projectors and trace-based probabilities, transforming superpositions into basis-state mixtures with diagonal density matrices. Projective measurements are a special case of general POVMs, which enable probabilistic discrimination of non-orthogonal states and support zero-error quantum-channel capacities.

3) Quantum Measurement:

  • Projective measurements use Hermitian projectors satisfying completeness, with outcome probabilities obtained from the trace rule and post-measurement states determined by the corresponding projector.For a pure state α|0⟩+β|1⟩, the computational-basis outcomes occur with probabilities |α|2 and |β|2.Source: 3) Quantum Measurement · p. 9
  • Measurement in an orthonormal basis removes off-diagonal density-matrix entries, converting a quantum superposition into a classical probabilistic mixture of basis states.For α|0⟩+β|1⟩, the resulting state is p|0⟩⟨0|+(1−p)|1⟩⟨1|, where p=|α|2.Source: 3) Quantum Measurement · p. 10

a) Orthonormal Basis Decomposition

  • In an orthonormal basis decomposition, the squared magnitudes |b_i|2 give measurement probabilities, appearing as diagonal entries of the resulting density matrix.The density matrix is diagonal with the basis-state probabilities on its diagonal.Source: 3) Quantum Measurement · p. 10

b) The Projective and POVM Measurement

  • Projective, or von Neumann, measurements associate outcomes with eigenvalues of a Hermitian operator, while POVMs generalize measurements to non-orthogonal operators and need not have orthogonal elements.POVMs may contain more elements than the Hilbert-space dimension and need not specify the post-measurement state when only outcome probabilities matter.Source: 3) Quantum Measurement · pp. 10–11
  • Because unknown non-orthogonal states cannot be distinguished with certainty, POVMs add an inconclusive outcome; their apparatus is used to distinguish quantum codewords with zero error and achieve zero-error channel capacity.The extra operator M_m+1 reports that the input cannot be identified, and it can be constructed when the states are linearly independent.Source: 3) Quantum Measurement · p. 11

G. Geometrical Interpretation of the Density Matrices

The Bloch sphere represents qubit states geometrically, distinguishing pure states on its surface from mixed states in its interior. Density-matrix geometry also describes unitary transformations, convex decompositions, and purification through environmental tracing.

Bloch-sphere representation

  • The unit-radius Bloch sphere represents pure qubit states on its surface and mixed states inside a three-dimensional real vector space.It provides a more expressive geometric description of two-level systems and can represent physically realized systems such as photon polarization, spins, and atomic energy levels.Source: G. Geometrical Interpretation of the… · p. 11

Density-matrix geometry

  • Every density matrix is a linear combination of Pauli matrices and corresponds to a point with Bloch vector r=(rX, rY, rZ).The vector components are r=(sinθcosφ, sinθsinφ, cosθ), with norm at most one.Source: G. Geometrical Interpretation of the… · p. 11

Unitary transformations

  • Under a unitary transformation U, the Bloch vector transforms as r′=UrU†, realizing a rotation.This provides the geometric interpretation of unitary evolution in the Bloch-sphere representation.Source: G. Geometrical Interpretation of the… · p. 12

Mixed-state decompositions

  • Mixed states are probabilistic mixtures of pure states, but their decompositions are not unique, whereas a pure-state statistical representation is unique.The maximally mixed state σ=1/2I lies at the Bloch-sphere center and can have multiple pure-state decompositions.Source: G. Geometrical Interpretation of the… · p. 12

Purification and partial trace

  • A mixed state can be recovered from a global pure purification by tracing out the environment with TrE.Discarding the environment decouples it from the receiver’s state and recovers the mixed density matrix.Source: G. Geometrical Interpretation of the… · p. 12

H. Channel System Description … 4) The von Neumann Entropy:

The paper models quantum-channel communication through classical registers, quantum inputs, purification, environmental interaction, and channel outputs. It then introduces purification, isometric and Kraus representations, and von Neumann entropy as tools for describing noise and quantum information.

H. Channel System Description

  • The channel model comprises Alice’s register X, input A, output B, purification P, and environment E, with input ensemble {pX(x), ρx}.The register variable x determines the prepared quantum state ρx, while the environment exposes the channel’s noise interaction.Source: H. Channel System Description · p. 12, Figure 5 · p. 13

1) Purification:

  • Purification represents the input density operator ρA using a joint system of purification P and channel input A [538].Tracing out P recovers ρA, and the resulting joint state represents a noisy state.Source: 1) Purification · pp. 12–13

2) Isometric Extension:

  • Isometric extension treats the channel and environment as a closed physical system, yielding a unitary representation of the quantum channel.This enables a one-sender, two-receiver view in which Bob and the environment receive outputs, with the environment traced out to obtain the channel output.Source: 2) Isometric Extension · p. 13

3) Kraus Representation:

  • The Kraus representation expresses a quantum channel through operators Ni satisfying Σi Ni=I and provides an equivalent form after tracing out the environment.It also represents the channel’s action on basis operators |k⟩⟨l|.Source: 3) Kraus Representation · p. 13

4) The von Neumann Entropy:

  • The von Neumann entropy extends Shannon entropy to quantum states and measures information through the uncertainty of a quantum state.It can be expressed using the Shannon entropy of the density matrix’s eigenvalue distribution, whose eigenvalues are λi.Source: 4) The von Neumann Entropy · p. 14
  • Classical Shannon entropy H(X) is defined from a variable’s probability distribution p(X), with 1≤H(X)≤log(|X|).Here |X| denotes the cardinality of the variable’s set.Source: 4) The von Neumann Entropy · p. 14
  • The Holevo bound determines how much information can be extracted from a single qubit when states ρi are sent with probabilities pi over an ideal quantum channel.At Bob’s receiver, the transmitted ensemble produces a mixed state.Source: 5) The Holevo Quantity · p. 14

5) The Holevo Quantity: … 1) Early Years of quantum information theory:

The paper develops quantum-information measures that extend classical information theory, including the Holevo bound, conditional and mutual information, relative entropy, and Rényi entropy. It also situates these quantities within the historical development of quantum information theory and its applications to communication and computation.

5) The Holevo Quantity:

  • One qubit contains at most one classical bit of information according to the Holevo bound.The Holevo quantity bounds the maximal classical mutual information obtainable from quantum states.Source: 5) The Holevo Quantity · p. 14

6) Quantum Conditional Entropy:

  • Quantum conditional entropy can be negative, reaching −1 for either subsystem of a maximally entangled two-qubit pure state.For an entangled pure state, the joint entropy is 0 while each subsystem has entropy 1.Source: 6) Quantum Conditional Entropy · p. 14

7) Quantum Mutual Information:

  • Quantum mutual information is nonnegative, can reach 2 for an entangled pure state, and is additive when maximized for a quantum channel.For an entangled pure state, I(A:B)=1+1−0=2.Source: 7) Quantum Mutual Information · p. 14

8) Quantum Relative Entropy:

  • Quantum relative entropy measures quantum-state distinguishability, is noncommutative, and reduces to classical Kullback–Leibler relative entropy for simultaneously diagonalizable matrices.It is nonnegative, vanishes exactly for equal states, and may diverge when the reference state has zero eigenvalues.Source: 8) Quantum Relative Entropy · p. 15
  • Quantum mutual information equals the relative entropy between the joint state and the tensor product of its marginals.I(A:B)=D(ρAB∥ρA⊗ρB)=S(ρA)+S(ρB)−S(ρAB).Source: 8) Quantum Relative Entropy · p. 15

9) Quantum R´enyi-Entropy:

  • Rényi entropy generalizes von Neumann entropy, depends on a parameter r≥0, and is especially relevant to describing quantum entanglement.It equals von Neumann entropy at the specified limiting condition, while r=0 relates it to the density matrix rank.Source: 9) Quantum R´enyi-Entropy · p. 15

I. Related Work

  • Quantum information theory remains an active field with fundamental open problems requiring knowledge of its foundational results.The paper presents these foundations before discussing more advanced results.Source: §I · p. 15

1) Early Years of quantum information theory:

  • Quantum information theory combines Shannon’s communication framework with quantum theory, whose entropy predates Shannon’s information-theoretic entropy and whose results support communication and computational applications.Deutsch first claimed in 1985 that quantum-information results could solve computational problems, while later work addressed quantum simulation and encoding information in physical particles.Source: 1) Early Years of quantum information… · pp. 15–16

2) Quantum Coding and Quantum Compression: … C. Transmission of Classical Information over Noisy Quantum Channels

The paper develops quantum coding, entanglement, channel models, and capacity theory, culminating in three distinct classical capacities for noisy quantum channels. It situates these results within classical coding limits and broader surveys of quantum information theory.

2) Quantum Coding and Quantum Compression:

  • Quantum compression established quantum noiseless coding, qubit encoding, and rate bounds, extending classical noiseless channel coding to quantum information.Schumacher’s framework addressed pure states, while later work extended coding to mixed states and universal compression.Source: 2) Quantum Coding and Quantum… · pp. 16–17

3) Quantum Entanglement:

  • Entanglement concentration and purification generate or improve shared entangled states, underpinning teleportation, dense coding, and quantum-channel capacity results.The discussed protocols include concentration, dilution, purification, transformations, and their communication costs.Source: 3) Quantum Entanglement · p. 17

4) Quantum Channels:

  • Quantum mutual information measures classical transmission through noisy quantum channels, while entanglement causes Holevo information to be non-additive but does not affect quantum mutual information [4].The section reviews Holevo information, channel capacities, entropy properties, and additivity-related results [231, 233, 486].Source: 4) Quantum Channels · pp. 17–18

5) Comprehensive Surveys:

  • Comprehensive surveys and textbooks consolidate quantum information theory, quantum Shannon theory, classical compression, random quantum channels, and entanglement-related communication [538].The cited works also cover quantum computation, coding theorems, entropy, and channel structure.Source: 5) Comprehensive Surveys · pp. 18–19

III. CLASSICAL CAPACITIES OF A QUANTUM CHANNEL

  • The classical-capacity section introduces noisy quantum channels, encoder-decoder settings, maximal transmittable classical information, and relevant channel maps.These theoretical results are presented as foundations for analyzing more advanced quantum-communication properties.Source: §III · p. 19

A. Extended Formal Model

  • A quantum channel is modeled as a completely positive trace-preserving map, with input states interacting unitarily with an environment and producing Bob’s mixed output state.Purification supplies a reference system that remains inaccessible during transmission.Source: A. Extended Formal Model · p. 19

B. Capacity of Classical Channels

  • Classical channel capacity is the maximum mutual information reliably transmissible per channel use, with rates R≤C(N) achievable asymptotically and exponentially small decoding error.The asymptotic regime uses N ⊗n as n→∞, whereas n=1 defines single-use capacity.Source: B. Capacity of Classical Channels · p. 19

C. Transmission of Classical Information over Noisy Quantum Channels

  • Quantum channels support classical or quantum transmission, and their classical capacities divide into unentangled HSW C(N), private P(N), and entanglement-assisted C_E(N) measures.P(N) concerns secure classical communication, while C_E(N) assumes pre-shared entanglement, as in superdense coding.Source: C. Transmission of Classical… · pp. 20–21

1) The Holevo-Schumacher-Westmoreland Capacity: · 2) Various Classical Capacities of a Quantum Channel:

The HSW theorem characterizes product-state classical capacity under joint measurements through maximized Holevo information, with reliable transmission whenever the code rate is below capacity. Quantum-channel classical capacities depend on input entanglement and measurement settings, and asymptotic capacities may require regularization.

1) The Holevo-Schumacher-Westmoreland Capacity:

  • The HSW theorem gives product-state classical capacity under joint measurement, with reliable communication when the code rate satisfies R<C(N)=max_{p_i,ρ_i}χ.Rates above capacity do not support arbitrarily large codes with arbitrarily small error.Source: 1) The Holevo-Schumacher-Westmoreland… · p. 21

2) Various Classical Capacities of a Quantum Channel:

  • Asymptotic capacity is treated as the definitive measure, while single-use capacity describes only a special case and regularization computes the asymptotic limit.The HSW theorem supplies an explicit expression for maximal classical information in the product-input, joint-measurement setting.Source: 2) Various Classical Capacities of a… · p. 21
  • At p=0.5, the classical bit-flip channel has C(N)=1−H(p)=0, whereas optimizing quantum input ensembles for the corresponding quantum bit-flip channel achieves C(N)=1.The example attributes the improvement to encoding in quantum states, whose additional degrees of freedom can reduce channel uncertainty.Source: 2) Various Classical Capacities of a… · p. 21
  • With product inputs and independent output measurements, classical capacity is expressed by maximized quantum mutual information, and single-use and asymptotic capacities coincide.This setting transmits product-state codewords through repeated channel uses with separate measurements.Source: 2) Various Classical Capacities of a… · p. 21
  • With product inputs and joint measurements, quantum mutual information is insufficient; maximized Holevo information instead determines the capacity, which equals the asymptotic capacity.The equality applies to the restricted product-input, joint-measurement setting.Source: 2) Various Classical Capacities of a… · p. 22
  • Hastings showed that entangled inputs with joint measurements can increase classical information transmission, making C(N) differ from χ(N) and requiring an asymptotic Holevo-capacity expression.This demonstrates that the Holevo capacity need not equal the asymptotic classical capacity when entangled inputs are allowed.Source: 2) Various Classical Capacities of a… · p. 22
  • For entangled inputs with single measurements, no quantum channel model is known to increase capacity because the measurement setting removes the benefits of entanglement.The discussion contrasts this setting with entangled inputs combined with joint measurements, where benefits can occur.Source: 2) Various Classical Capacities of a… · pp. 22–23

3) Brief Summary: … 1) Classical Zero-Error Capacities of Quantum Channels:

The paper distinguishes classical, private, entanglement-assisted, and zero-error capacities of quantum channels by their information resources, secrecy requirements, and error criteria. It emphasizes when capacities require asymptotic regularization and characterizes zero-error transmission through perfect distinguishability of quantum codewords.

3) Brief Summary:

  • The HSW capacity maximizes the Holevo quantity for unentangled product-state inputs, whereas entangled inputs require its asymptotic formula.With product inputs and a single measurement setting, classical capacity is expressed through maximized quantum mutual information; joint measurements instead yield HSW capacity.Source: 3) Brief Summary · p. 23

D. The Private Classical Capacity

  • Private classical capacity P(N) measures the maximum reliable classical rate that Bob can receive without information leakage to an eavesdropper.Its single-use private information is the difference between Alice–Bob and Alice–Eve mutual informations, while the true capacity requires asymptotic treatment because this difference is nonadditive.Source: D. The Private Classical Capacity · pp. 23–24

E. The Entanglement-assisted Classical Capacity

  • Entanglement-assisted classical capacity CE(N) is determined by maximized quantum mutual information and requires no asymptotic regularization, making it additive and easier to compute.Shared entanglement does not improve the channel’s asymptotic classical capacity, although it can increase single-use transmission success in the CHSH game.Source: E. The Entanglement-assisted Classical… · p. 24
  • Classical and private capacities require regularization because their asymptotic and single-use formulas are not equal.This contrasts with entanglement-assisted classical capacity, for which no regularization is needed.Source: E. The Entanglement-assisted Classical… · pp. 24–25

F. The Classical Zero-Error Capacity

  • Zero-error capacity specifies the classical or quantum information rate transmitted perfectly through a noisy quantum channel, with error probability exactly zero rather than merely vanishing asymptotically.The paper connects zero-error quantum communication with graph-theoretic characterization.Source: F. The Classical Zero-Error Capacity · p. 25

1) Classical Zero-Error Capacities of Quantum Channels

  • For product input states, classical zero-error capacity C0(N) is achievable exactly when quantum codewords are pure states measured by a receiver POVM.The construction encodes classical messages into n-length product quantum codewords and decodes them with measurement operators.Source: 1) Classical Zero-Error Capacities of… · pp. 25–26
  • Zero-error transmission requires pairwise orthogonal codewords, because only non-adjacent outputs can be perfectly distinguished by the receiver.The number of non-adjacent codewords determines the maximum transmittable classical information.Source: 1) Classical Zero-Error Capacities of… · p. 25
  • In a d-dimensional Hilbert space, at most d pairwise distinguishable states exist, limiting n-use product codes to at most d^n distinguishable codewords.The decoder must identify every input message with unit probability; otherwise the channel has no zero-error capacity.Source: 1) Classical Zero-Error Capacities of… · pp. 25–26

2) Formal Definitions of Quantum Zero-Error Communication: · 3) Achievable Zero-Error Rates in Quantum Systems:

Quantum zero-error communication is defined by input states whose channel outputs are perfectly distinguishable, enabling error-free message transmission. Its achievable classical rates are determined by the maximum number of mutually non-adjacent inputs and are asymptotically upper bounded by the HSW capacity.

2) Formal Definitions of Quantum Zero-Error Communication:

  • A quantum channel has positive classical zero-error capacity exactly when two input states produce perfectly distinguishable, orthogonal-support outputs under a suitable measurement.This non-adjacency condition is written as ρ1⊥ρ2 and is equivalent to orthogonal output supports.Source: 2) Formal Definitions of Quantum… · p. 26
  • Non-adjacent codewords can be distinguished with probability one using measurement operators that identify the respective input codewords.The construction uses orthogonal projectors, with the measurement restricted to projective or von Neumann operators.Source: 2) Formal Definitions of Quantum… · p. 26
  • For n-length tensor-product codewords, non-adjacency holds when at least one corresponding state pair is perfectly distinguishable through the channel.Thus, a single non-adjacent component pair suffices to make the full output codewords distinguishable.Source: 2) Formal Definitions of Quantum… · p. 27
  • Zero-error discrimination of general tensor-product codewords requires joint measurement, which is also sufficient when the output states have orthogonal supports, Tr(ρiρj)=0 [67].For n channel uses, joint-measurement projectors are dn×dn matrices, compared with d×d matrices for single measurements.Source: 2) Formal Definitions of Quantum… · p. 27

3) Achievable Zero-Error Rates in Quantum Systems:

  • The single-use classical zero-error capacity is determined by the maximum number of messages, equivalently the largest set of mutually non-adjacent inputs.This quantity is represented by K(N) for one use of the noisy quantum channel.Source: 3) Achievable Zero-Error Rates in… · p. 27
  • The asymptotic zero-error capacity is determined by the growth of K(N ⊗n), the maximum number of n-length classical messages transmitted with zero error.Here N ⊗n denotes n uses of the channel.Source: 3) Achievable Zero-Error Rates in… · p. 27
  • The asymptotic classical zero-error capacity C0(N) of a quantum channel is upper bounded by its HSW capacity.The section then turns to the connection between zero-error quantum codes and graph theory.Source: 3) Achievable Zero-Error Rates in… · p. 27

4) Connection with Graph Theory: · G. Entanglement-assisted Classical Zero-Error Capacity

Zero-error transmission is represented through confusability graphs, where non-adjacent codewords are distinguishable, while block length changes the achievable capacity. Shared entanglement extends this framework via hypergraphs and orthogonal measurements, increasing the number of perfectly distinguishable messages and, for special codes, reaching the HSW capacity.

4) Connection with Graph Theory:

  • Confusability graphs connect input messages that can produce a common output with nonzero probability, so independent sets represent zero-error codewords.For block length n, the graph G_n analyzes which quantum codewords are non-adjacent and therefore non-confusable.Source: 4) Connection with Graph Theory · p. 28
  • For the pentagon graph, one channel use permits at most two mutually non-adjacent codewords, giving C0(N) = 1.Other non-adjacent pairs are possible, but the maximum independent-set size remains two.Source: 4) Connection with Graph Theory · p. 28
  • With block length n = 2, the pentagon confusability graph becomes more complex and supports five two-length zero-error codewords.The resulting capacity is achieved with quantum block codes of length two.Source: 4) Connection with Graph Theory · pp. 28–29

G. Entanglement-assisted Classical Zero-Error Capacity

  • Entanglement-assisted zero-error communication replaces the confusability-graph description with a hypergraph whose vertices are inputs and hyperedges collect inputs capable of producing the same output.The hypergraph can have substantially fewer hyperedges than the corresponding confusability graph as the input set grows.Source: G. Entanglement-assisted Classical… · p. 29
  • The protocol shares a maximally entangled state, encodes each message through a basis measurement, and uses Bob’s orthogonal conditional states to recover the message with certainty.For the example, Alice and Bob share a rank-four state, and Alice’s six possible bases correspond to six messages.Source: G. Entanglement-assisted Classical… · p. 30
  • Entanglement increases the single-use message count in the example from K = 5 without shared entanglement to K_E = 6 with shared entanglement.The hypergraph contains five independent vertices, yet its entanglement-assisted protocol supports at least six maximally transmittable messages.Source: G. Entanglement-assisted Classical… · pp. 30–31
  • The Cubitt-Leung-Matthews-Winter theorem [123] establishes that entanglement can increase asymptotic classical zero-error capacity, while special Pauli-graph codes can reach the classical HSW capacity.For those special codes, entanglement-assisted zero-error capacity equals the maximal transmittable classical information and the upper bound of classical zero-error capacity.Source: G. Entanglement-assisted Classical… · pp. 31–32

H. Related Work · IV. THE QUANTUM CAPACITY OF A QUANTUM CHANNEL · A. Preserving Quantum Information

The paper surveys classical and quantum capacities of quantum channels, then focuses on preserving quantum information through noisy transmission. Quantum capacity is characterized using coherent information, while fidelity and entanglement fidelity quantify preservation relative to an inaccessible reference.

H. Related Work

  • Related work distinguishes classical, entanglement-assisted, private, and zero-error capacities, with entanglement increasing classical and zero-error transmission in the relevant settings, [54], [123], [294].Private classical capacity is closely related to quantum capacity and equals coherent information for degradable channels [134], [501].Source: H. Related Work · pp. 33–35
  • The HSW theorem determines reliable classical transmission with product inputs, while entanglement assistance yields a capacity described by maximized quantum mutual information and bounded above by the HSW capacity,,,.The BSST result requires no regularization because its single-use and asymptotic capacities coincide.Source: H. Related Work · pp. 33–34

IV. THE QUANTUM CAPACITY OF A QUANTUM CHANNEL

  • Quantum capacity Q(N) is defined as the maximal quantum-information transmission rate and is characterized by maximizing coherent information in the LSD channel-capacity framework [134],,.Coherent information represents quantum information lost to the environment during communication.Source: §IV · p. 35
  • Quantum-channel analysis covers encoding, decoding, coherent information, and the final maximal-transmission formula, paralleling classical noisy-channel capacity theory.The section presents these topics in sequence, beginning with quantum transmission over noisy channels and ending with the capacity measure.Source: §IV · p. 35

A. Preserving Quantum Information

  • Quantum messages are encoded as nonorthogonal, superposed, or entangled states; mutually orthogonal pure states reduce the quantum-information setting to classical information.The encoder maps m input states into n intermediate systems sent through independent channel instances, and the code rate is m/n.Source: A. Preserving Quantum Information · p. 35
  • Because environmental entanglement makes channel outputs mixed, fidelity is introduced to quantify how closely transmitted quantum states preserve their original properties.The output decoder state is typically mixed because of channel noise.Source: A. Preserving Quantum Information · p. 35
  • Entanglement fidelity measures preservation between the transmitted system and an inaccessible reference, whose entanglement is affected when the system passes through the noisy channel.The reference remains isolated while the input system is transmitted, and many independent channel uses can carry quantum messages.Source: A. Preserving Quantum Information · p. 36

B. Quantum Coherent Information · C. Connection between Classical and Quantum Information · 1) Quantum Coherent Information and Quantum Mutual Information:

Quantum coherent information measures entanglement transmission by comparing information received by Bob with information leaked to the environment. It connects quantum capacity to Holevo information while differing from quantum mutual information in its nonadditivity.

B. Quantum Coherent Information

  • The entropy exchange measures environmental entropy increase, or equivalently entanglement between the purification system and the environment.It is analogous to classical conditional entropy, but describes quantum information.Source: B. Quantum Coherent Information · p. 37
  • Quantum coherent information measures a channel’s capability to transmit entanglement.It is maximized when the input systems are maximally entangled and the channel is noiseless; nonmaximal entanglement or channel noise decreases it.Source: B. Quantum Coherent Information · p. 37

C. Connection between Classical and Quantum Information

  • Holevo information provides an expression for quantum coherent information and connects transmittable quantum information to a classical-information measure.The channel’s quantum capacity can be expressed using Holevo quantities for Bob’s output and information leaked to the environment.Source: C. Connection between Classical and… · pp. 37–38
  • The channel-environment transformation produces outputs for Bob and the environment by taking partial traces of the joint system.This conceptual picture treats Bob and the environment as two receivers of the channel’s transformed state.Source: B. Quantum Coherent Information · p. 37, Figure 25 · p. 38

1) Quantum Coherent Information and Quantum Mutual Information:

2) Quantum Coherent Information of an Ideal Channel: … V. QUANTUM CHANNEL MAPS AND CAPACITIES

The section surveys quantum channel capacities through coherent information, including ideal, asymptotic, assisted, zero-error, and classical–quantum capacity relations. It also summarizes foundational results and the role of channel maps in quantum communication.

2) Quantum Coherent Information of an Ideal Channel:

  • For an ideal noiseless channel NAB=I, quantum capacity is appropriately described by coherent information, maximized by maximally mixed inputs or one half of an EPR state.Environmental interaction can eliminate quantum capacity while preserving classical transmission of pure orthogonal states, and sufficiently strong noise can eliminate both capacities.Source: 2) Quantum Coherent Information of an… · p. 38

D. The Lloyd-Shor-Devetak Formula

  • The Lloyd-Shor-Devetak formula identifies asymptotic quantum capacity with coherent information, which is also expressible through Holevo information and extends the single-use capacity.Coherent information is fundamental for quantifying maximal quantum transmission through a channel and plays the central role in LSD capacity.Source: D. The Lloyd-Shor-Devetak Formula · p. 39

E. The Assisted Quantum Capacity

  • Assisted capacity combines a channel with a symmetric channel, enabling superactivation in which two zero-capacity channels can yield positive joint capacity.The symmetric channel may have unbounded dimension, so assisted capacity cannot generally be evaluated.Source: E. The Assisted Quantum Capacity · p. 39

F. The Zero-Error Quantum Capacity

  • Quantum zero-error capacities use coherent encoders and coherent POVM decoders to preserve quantum information, and their asymptotic values are bounded above by LSD capacity.K(N ⊗n) denotes the maximum number of n-length mutually non-adjacent quantum messages transmissible with zero error.Source: F. The Zero-Error Quantum Capacity · p. 39

G. Relation between Classical and Quantum Capacities of Quantum Channels

  • Quantum capacity cannot generally exceed classical capacity with entangled inputs and joint measurement, while some channels are conjectured to satisfy C (N) <Q (N) under single-setting classical decoding.The comparison depends on the encoder and measurement setting used to define classical capacity.Source: G. Relation between Classical and… · pp. 39–40

H. Related Work

1) Quantum Coherent Information:

  • Foundational work established coherent information, fidelity-based capacity bounds, and the LSD theorem, with Schumacher and Nielsen introducing coherent information in 1996 and Lloyd, Shor, and Devetak completing the capacity formulation [134],,.The LSD result states that asymptotic quantum capacity is at least the single-use capacity, so it is not generally equal to single-use coherent information.Source: 1) Quantum Coherent Information · p. 40, 2) Proofs on Quantum Capacity · pp. 40–41

2) Proofs on Quantum Capacity:

  • Proofs and extensions connected quantum capacity to error correction, private classical capacity, noisy and bosonic channels, amplitude damping, and simultaneous classical–quantum transmission [134],,,.The survey emphasizes that quantum capacity requires transmission of arbitrary quantum states and reviews related encoder, decoder, and measurement settings.Source: 2) Proofs on Quantum Capacity · p. 41, §V · p. 41

A. Channel Maps … 5) The Pancake Map:

The section surveys quantum channel maps, covering Pauli, depolarizing, damping, dephasing, and non-CP pancake transformations. It also connects decoherence modeling with environmental interactions, simulation limits, and mitigation through error correction and decoupling.

A. Channel Maps

1) The Pauli Channel:

  • The Pauli channel applies X, Y, and Z errors with probabilities whose sum is the depolarizing probability p, while time-dependent error rates depend on T1 and T2.Its input-state transformation is a probabilistic mixture of the original state and Pauli-conjugated states.Source: 1) The Pauli Channel · p. 41

2) The Depolarizing Channel:

  • The depolarizing channel is presented as a unital transformation parameterized by p, with encoding based on two orthogonal states ρ0 and ρ1.The resulting state follows application of the channel transformation N.Source: 2) The Depolarizing Channel · p. 41

3) The Damping Channel:

  • Damping channels model environmental energy exchange in a two-level atom, including spontaneous dissipation or absorption that changes the qubit between |0⟩ and |1⟩.Amplitude damping describes energy loss, while phase damping models random phase kicks; independent qubit-environment interactions yield temporally and spatially uncorrelated decoherence.Source: 3) The Damping Channel · p. 42
  • Combined amplitude-and-phase damping captures both amplitude and phase changes, but its exact classical simulation requires a 2^N-dimensional Hilbert space and may be approximated by a Pauli model.The approximation is introduced to facilitate efficient classical simulations of N-qubit channels.Source: 3) The Damping Channel · p. 42

4) The Dephasing Channel Model:

  • The dephasing map is a unitary channel that creates relative phase differences, shrinks the Bloch sphere more than a phase-flip map, and converts superpositions into mixtures.Its coupling to the environment is characterized by the positive parameter γ(t).Source: 4) The Dephasing Channel Model · p. 42

5) The Pancake Map:

  • The pancake map is a non-CP, physically forbidden transformation that preserves equatorial information while destroying information along the Bloch sphere’s z axis.Despite being nonphysical, it can theoretically transmit some information; practical decoherence reduction relies mainly on quantum error-correcting codes and decoupling methods, [492].Source: 5) The Pancake Map · pp. 42–43

B. Capacities … C. Implementation of Quantum Channel in FSO-based Quantum Key Distribution

The paper derives classical and quantum capacities for several quantum channels, then models practical channel behavior in asymmetric hardware and FSO-based QKD. It emphasizes channel-specific capacity thresholds and physically grounded transmission-loss bounds.

B. Capacities

  • Closed-form classical capacities and corresponding quantum capacities are derived and compared for erasure, phase-erasure, mixed erasure/phase-erasure, and amplitude damping channels.The study first develops classical-capacity expressions, then gives quantum capacities for the same channels.Source: B. Capacities · p. 43

1) Erasure Quantum Channel:

  • The erasure channel transmits the input unchanged with probability 1−p, loses classical capacity at p=1, and loses quantum capacity at p=1/2.It transmits some classical information for 0≤p<1 and some quantum information for 0≤p<1/2.Source: 1) Erasure Quantum Channel · p. 43

2) Phase-Erasure Quantum Channel:

  • The phase-erasure channel removes phase information with probability p without disturbing amplitudes, leaving orthogonal-state distinguishability intact while changing quantum-capacity behavior.Its map uses a phase transformation and a flag qubit; classical capacity is therefore unaffected by phase errors on |0⟩ and |1⟩.Source: 2) Phase-Erasure Quantum Channel · p. 43

3) Mixed Erasure/Phase-Erasure Quantum Channel:

  • The mixed channel erases the input with probability p, erases phase with probability q, and leaves it unchanged with probability 1−p−q≥0, with both capacities plotted against p+q.Its classical and quantum capacity expressions combine the corresponding erasure and phase-erasure results.Source: 3) Mixed Erasure/Phase-Erasure Quantum… · p. 43

4) Amplitude damping Quantum Channel:

  • For amplitude damping, classical capacity vanishes at γ=1, while quantum capacity vanishes when γ≥0.5, mirroring the erasure channel’s threshold behavior.The classical capacity depends on the population parameter τ and Shannon entropy, whereas quantum capacity is expressed through a maximization.Source: 4) Amplitude damping Quantum Channel · p. 44

VI. PRACTICAL IMPLEMENTATIONS OF QUANTUM CHANNELS

  • Practical channel modeling uses depolarizing channels for hardware decoherence and transmission-medium errors, with asymmetric versions representing phase flips that greatly exceed bit flips.The asymmetry ratio α satisfies α=1 for symmetric depolarization and commonly takes values 10^2, 10^4, or 10^6 in relevant materials.Source: §VI · p. 44, A. Realistic Material: Asymmetric… · p. 44

B. Acting Time in Asymmetric Channels

C. Implementation of Quantum Channel in FSO-based Quantum Key Distribution

  • FSO-based QKD uses polarized photons, while diffraction, atmospheric turbulence, and extinction reduce the received fraction γ; near-field and far-field bounds provide a more accurate γ range.The bounds are combined across the transition region, illustrated in Fig. 30, to characterize channel power transfer over distance L.Source: C. Implementation of Quantum Channel… · pp. 45–46, Figure 30 · p. 46

D. Quantum Channel Codes for Approaching Quantum Channel Capacity · E. Quantum Network Coding for Entanglement Distribution · VII. CONCLUSIONS

The paper reviews entanglement-assisted quantum error-correction codes for mitigating channel perturbations and approaches capacity, while examining quantum network coding for distributing entanglement across communication networks. It concludes that quantum channels support diverse information types and enable advanced communication technologies despite richer noise models.

D. Quantum Channel Codes for Approaching Quantum Channel Capacity

  • Quantum error-correction codes mitigate quantum perturbations, while entanglement assistance extends coding frameworks to symmetric and asymmetric depolarizing channels,,,,,.Asymmetric depolarization in practical devices spans α = 10^2 to α = 10^6, motivating entanglement-assisted quantum error-correction codes.Source: D. Quantum Channel Codes for… · p. 46
  • The Hashing bound provides a lower benchmark for achievable quantum depolarizing-channel capacity, while EXIT charts analyze iterative quantum-code decoding convergence [85],,,,,.The framework considers entanglement-assisted coding and capacity for both symmetric and asymmetric depolarizing channels.Source: D. Quantum Channel Codes for… · p. 46

E. Quantum Network Coding for Entanglement Distribution

  • Classical network coding improves throughput, energy use, and delay by combining multiple packets at intermediate network nodes,,.These principles have also been applied across diverse disciplines.Source: E. Quantum Network Coding for… · pp. 46–47
  • Although no-cloning initially suggested quantum network coding was impossible, preshared entanglement or abundant low-cost classical communication can make QNC feasible,.Thus, additional resources overcome a fundamental constraint of quantum communications.Source: E. Quantum Network Coding for… · p. 47
  • Entanglement enables teleportation, remote state preparation, remote measurement, and secret sharing, making its distribution central to quantum communication protocols.Entangled qubits must be delivered to distant nodes for these applications.Source: E. Quantum Network Coding for… · p. 47
  • For M = 2 source-to-target user pairs connected by N = 1 backbone hop, entanglement swapping achieves higher fidelity performance than QNC, whereas large-scale QNC can provide benefits over ES,,,.Entanglement distribution across multihop, multinode networks can use entanglement swapping or QNC, with QKD applications extending to satellite, terrestrial, and handheld communications.Source: E. Quantum Network Coding for… · p. 47

VII. CONCLUSIONS

  • Quantum channels transmit classical, entanglement-assisted classical, private classical, and quantum information, while supporting advanced communication primitives through optical-fiber and wireless-optical implementations.Their broader noise models introduce diverse errors, but quantum information processing may expand classical computation and communication capabilities.Source: §VII · p. 47

APPENDIX … C. Fidelity

The appendix defines partial trace for extracting subsystem states, distinguishes separable states from entangled states, and introduces fidelity for quantifying transmission quality under environmental noise.

A. Partial Trace

  • Partial trace extracts subsystem states from a joint density matrix: ρA=TrB(ρAB) and ρB=TrA(ρAB).For an uncorrelated joint state, tracing out subsystem B recovers ρA because ⟨ψB | ψB⟩=1.Source: A. Partial Trace · p. 58
  • For product operators, partial trace uses Tr(|ψ1⟩⟨ψ2|)=⟨ψ2 | ψ1⟩ to calculate subsystem operators from tensor-product components.The appendix extends this rule to operators A=|i⟩⟨k| and B=|j⟩⟨l|.Source: A. Partial Trace · p. 58

B. Quantum Entanglement

  • A composite quantum system is separable when its joint state factors as ρAB=ρA⊗ρB, representing independent subsystems combined by tensor product.This factorized structure parallels classical composite systems formed from independent states.Source: B. Quantum Entanglement · p. 58
  • Entanglement is a uniquely quantum phenomenon in which a composite state cannot be characterized as a tensor product of independent subsystem states.Bell, or EPR, states are given as examples of entangled states.Source: B. Quantum Entanglement · p. 58

C. Fidelity

  • Fidelity measures the distance between an intended pure quantum state and the mixed state received after environmental interactions introduce noise.Quantum channels can entangle with their environment, producing mixed outputs from probability-weighted sums of pure states.Source: C. Fidelity · p. 58
  • Fidelity is concave in its second argument: F(ρ, aσ1+(1−a)σ2) ≥ aF(ρ, σ1)+(1−a)F(ρ, σ2), for a∈[0,1].The appendix lists this among the major properties of fidelity.Source: C. Fidelity · p. 58

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