A Survey on Quantum Channel Capacities

Laszlo Gyongyosi; Sandor Imre; Hung Viet Nguyen

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A Survey on Quantum Channel Capacities

Laszlo Gyongyosi, Sandor Imre, Hung Viet Nguyen

arXiv:1801.02019v1Jan 6, 2018quant-phcs.IT

TL;DR

Quantum communication channels raise unresolved questions about how information and entanglement can be transmitted across noisy links and networks. This survey reviews channel properties, capacity measures, communication phases, and classical–quantum differences, highlighting that quantum encoding can increase a bit-flip channel’s classical capacity from 0 to 1.

Problem
Quantum communication requires distinct capacity measures and difficult network components because quantum channels differ fundamentally from classical channels and cannot be freely copied.
Method
The paper synthesizes quantum-channel properties, capacity definitions, coding and evolution phases, measurements, and entanglement-assisted protocols.
Results
Quantum encoding increases the bit-flip channel’s classical information transmission capability from 0 to the maximum 1, while entangled inputs can increase classical information over noisy quantum channels.
Takeaways & Limitations
Quantum encoding and entanglement can improve classical transmission and support zero-error communication across quantum channels.

Abstract

from arXiv · show

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 extend classical communication by supporting multiple forms of information transmission, but their capacities and network implementation raise distinct theoretical and practical challenges. Communication proceeds through encoding, noisy channel evolution, and measurement-based decoding, governed by physically allowed quantum transformations.

I. INTRODUCTION: Quantum networks require repeaters and switches or routers to connect distant links and many users, but the no-cloning theorem makes their construction difficult.The introduction identifies the quantum Internet as requiring these network entities.
I. INTRODUCTION: Quantum communication research focuses on the capacity of quantum channels, whose distinct capacity measures remain an active area with open questions.A channel’s capacity describes faithful, recoverable information delivery, while quantum capacities emerged in the 1990s.
A. Applications and Gains of Quantum Communications: Quantum channels support classical, entanglement-assisted classical, private classical, and quantum information, enabling applications including secret transmission, superdense coding, and teleportation.Entanglement-based transmission has been demonstrated over 144 km terrestrial and 1200 km earth-station-to-satellite distances.
B. Privacy and Performance Gains of Quantum Channels: Quantum communication can provide randomness and security while improving information transmission and computing performance relative to binary-based systems.The paper also describes quantum encoding improving transmission for certain channels and entanglement-assisted communication transferring 2 classical bits using 1 quantum bit.
B. Privacy and Performance Gains of Quantum Channels: Quantum mechanics supports secure communication through no-cloning, measurement randomness, and entanglement, while n qubits can carry up to 2^n states corresponding to 2^n × n bits.The paper reports that quantum computing has in some cases been proved 100 millions times faster than classical computing.
C. Communication over a Quantum Channel: Communication over a quantum channel encodes messages by preparing quantum states and decodes them by measuring the received states after physically allowed channel transformations.Quantum channels transform input quantum states into output quantum systems through Completely Positive Trace Preserving transformations.
1) The Quantum Channel Map:: Algebraically, a quantum channel is a linear CPTP map on density matrices, and geometrically a qubit channel can be represented as an affine transformation of Bloch-sphere states.For unital channels, the channel ellipsoid is centered on the Bloch sphere and preserves the average of system states; non-unital channels shift the center.
2) Steps of the Communication:: Quantum communication has three phases: encoding to compensate channel noise, channel evolution that disturbs the state, and measurement for decoding and information extraction.A completely noisy channel yields a maximally mixed state whose measurement is fully undetermined, whereas a noiseless channel permits recovery of the encoded information.

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

Quantum channel transmission is modeled by unitary evolution of an input state together with a pure environment, followed by tracing out subsystems to obtain output states. The paper distinguishes classical and quantum capacities and introduces density matrices as positive-semidefinite, unit-trace representations of quantum states.

D. Formal Model: The input state ρin and pure environment state ρE=|0⟩⟨0| form the composite state ρin⊗ρE before transmission.The channel acts on this composite system during the initial transmission phase.
D. Formal Model: Unitary evolution through channel N produces the joint output state, whose system and environment states are obtained by tracing out the complementary subsystem.TrE traces out the environment, while TrB traces out the output system B.
E. Quantum Channel Capacity: Quantum-channel capacity measures how closely transmission approaches the ideal identity transformation and must distinguish classical-message from quantum-message transmission.Classical messages encode outputs from classical information sources, whereas quantum messages originate from quantum information sources.
1) Discussion:: Quantum channels support multiple capacity definitions because they can transmit orthogonal states, non-orthogonal states, quantum information, or quantum entanglement.The text contrasts these possibilities with classical communication channels, which have one capacity type.
1) Discussion:: Quantum entanglement cannot be handled by classical information theory, although transmission of product states can be described similarly to classical information.Classical information theory is presented as a subset of the larger quantum-information framework.
2) Density Matrix and Trace Operator:: Every density matrix is positive-semidefinite, and its trace equals one.The trace is the sum of diagonal entries and also the sum of eigenvalues.
2) Density Matrix and Trace Operator:: A density matrix ρ represents a probabilistic mixture of pure states, with pure states written as ρ=|ψ⟩⟨ψ| and having rank one.Mixed states are classical probability-weighted combinations of pure quantum states, while superpositions remain pure states.
2) Density Matrix and Trace Operator:: Completely Positive Trace Preserving operations map density matrices to matrices that remain density matrices.The operator G describes physically admissible operations when the transformation is not unitary.

3) Quantum Measurement:

Quantum measurements use Hermitian projectors and trace-based probabilities to map quantum states into measurement outcomes and post-measurement states. Projective measurements are special cases of POVMs, which can handle non-orthogonal states and support zero-error quantum-channel capacities.

Quantum Measurement: Projective measurement operators are Hermitian projectors, and the trace operator gives the probability of outcome j associated with projector Pj.The projectors satisfy Pj=Pj†.
Quantum Measurement: For |ψ⟩=α|0⟩+β|1⟩, measuring the computational basis yields probabilities |α|^2 and |β|^2 for |0⟩ and |1⟩, respectively.These probabilities follow from trace expressions involving the corresponding projectors.
The Projective and POVM Measurement: In projective measurement, orthogonal projectors Pm correspond to eigenspaces of a Hermitian operator Z, with outcome m associated with eigenvalue λm.The projectors are pairwise orthogonal, and measurement probabilities are assigned to the corresponding outcomes.
The Projective and POVM Measurement: POVMs generalize measurement to non-orthogonal states without requiring exact post-measurement operators, using an additional inconclusive outcome when perfect discrimination is impossible.POVMs are the most general measurement formula, include projective measurements as a special case, and are key to zero-error quantum-channel capacity.

G. Geometrical Interpretation of the Density Matrices

The Bloch sphere represents qubit states geometrically, placing pure states on its surface and mixed states in its interior. This representation also describes unitary transformations as rotations and supports the analysis of quantum channel capacities.

G. Geometrical Interpretation of the Density Matrices: The Bloch sphere is a unit-radius representation of two-level quantum systems in a three-dimensional real vector space.Pure states lie on the sphere’s surface, while mixed states occupy its interior.
G. Geometrical Interpretation of the Density Matrices: The Bloch-sphere geometry can represent physically realized one-qubit systems and the geometrical expression of quantum channel capacities.Examples of physical systems include photon polarization, spins, and atomic energy levels.
G. Geometrical Interpretation of the Density Matrices: Every density matrix corresponds to a point in three-dimensional real space through linear combinations of the Pauli matrices.The Bloch vector is r=(rX, rY, rZ)=(sinθcosφ, sinθsinφ, cosθ), with norm at most 1.
G. Geometrical Interpretation of the Density Matrices: A unitary transformation maps the Bloch vector by r′=UrU†, realizing a rotation.Applying a unitary transformation U to the density matrix produces this geometric rotation.
G. Geometrical Interpretation of the Density Matrices: Mixed states are probabilistic mixtures of pure states, but their decompositions into pure states are not unique.A pure state has a unique statistical representation, whereas a mixed state can have different equivalent decompositions.

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

The paper models quantum-channel communication using classical registers, quantum inputs, purification, environmental interactions, and output systems. It then introduces Kraus representations and von Neumann entropy to characterize channel behavior and quantum information.

H. Channel System Description: The refined channel model includes Alice’s register X, purification state P, input A, output B, and environment E.Input states ρx occur with probabilities pX(x), forming the ensemble {pX(x), ρx}x∈X.
H. Channel System Description: Mixed or non-orthonormal density operators ρx prevent measurement operators M={|x⟩⟨x|}x∈X from identifying pX(x) and x.Alice’s register X and quantum system A are represented as a tensor product, with x correlated to ρx in an orthonormal basis.
H. Channel System Description: Alice’s input system A is described by the density matrix A=|ψx⟩⟨ψx|A, representing the input state |ψx⟩A.The purification offers a viewpoint on the origin of quantum-channel noise through the spectral decomposition of ρA.
1) Purification:: Purification introduces an ensemble {pX(x), |x⟩} and uses orthonormal basis vectors {|x⟩P}x∈X for the purification system P.The original state ρA is recovered from the purified state |ϕ⟩PA by tracing out P with TrP(·).
2) Isometric Extension:: The isometric extension represents the quantum channel as a unitary evolution of the joint channel-and-environment system.This yields a one-sender-and-two-receiver view in which Bob and the environment receive outputs, while the channel output is obtained after tracing out the environment.
3) Kraus Representation:: The Kraus representation expresses a channel N acting on input ρA through Kraus operators Ni satisfying ∑i Ni=I.Tracing out the environment establishes equivalence between the Kraus and isometric representations.
4) The von Neumann Entropy:: Von Neumann entropy S(ρ) extends Shannon entropy to quantum systems and measures the information contained in quantum state ρ.It can be expressed through the Shannon entropy of ρ’s eigenvalue distribution, where d is the system level and λi are density-matrix eigenvalues.

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

The paper develops quantum information measures from the Holevo bound through conditional and mutual information, relative entropy, and Rényi entropy, then situates these results within the field’s historical development. It emphasizes that quantum channels support distinct classical, quantum, and entanglement-based communication tasks.

5) The Holevo Quantity:: The Holevo quantity bounds the maximal classical mutual information obtainable from quantum states, and one qubit can contain at most one classical bit.This bound is contrasted with classical entropy bounds and can exceed mutual information for mixed or pure non-orthogonal states.
6) Quantum Conditional Entropy:: Quantum conditional entropy can be negative, unlike classical conditional entropy, because entanglement can make joint entropy smaller than the sum of component entropies.For a pure maximally entangled state, S (ρAB) = 0 while S (ρA) = S (ρB) = 1, yielding S (B| A) = −S (ρA) ≤0.
7) Quantum Mutual Information:: Quantum mutual information is nonnegative, can exceed classical mutual information, and equals 2 for a pure maximally entangled two-qubit system.The entangled-state calculation is I (A:B) = S (ρA) +S (ρB) −S (ρAB) = 1 + 1 −0 = 2.
7) Quantum Mutual Information:: Quantum mutual information is additive for a quantum channel when maximized, while entanglement enables quantum information storage in correlations between quantum states.Quantum channels may transmit classical information using pure orthogonal states or transmit non-orthogonal states and quantum entanglement, motivating multiple capacity definitions.
8) Quantum Relative Entropy:: Quantum relative entropy measures informational distance and state distinguishability, is generally noncommutative, and reduces to classical Kullback-Leibler relative entropy for simultaneously diagonalizable matrices.It is nonnegative when states differ, vanishes when they are equal, and may diverge when the reference state has zero eigenvalues.
8) Quantum Relative Entropy:: Quantum mutual information can be expressed as the relative entropy between the joint state and the tensor product of its individual subsystem states.The relation is I (A:B) =D (ρAB∥ρA⊗ρB) = S (ρA) +S (ρB) −S (ρAB).
9) Quantum R´enyi-Entropy:: Rényi entropy is another quantum entropy function, parameterized by r≥0, that is particularly relevant to describing quantum entanglement and connects to von Neumann entropy.Its limiting cases include r converging to infinity and r= 0, where it can be expressed using the density matrix rank.
I. Related Work: Quantum information theory extends classical information theory while sometimes producing fundamentally different answers, and its foundations draw on von Neumann’s quantum mechanics and Shannon’s communication framework.Quantum entropy was discovered in the 1930s, quantum information methods were proposed for computational problems in 1985, and major fundamentals were established during the 1990s.

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

The paper surveys quantum coding, entanglement, channel models, and capacity measures, emphasizing that quantum channels support distinct classical, private, entanglement-assisted, and quantum information transmission tasks. It develops the classical-capacity framework from channel modeling through reliable coding and quantum-channel capacity definitions.

2) Quantum Coding and Quantum Compression:: Schumacher’s 1995 results established foundational quantum information compression and introduced the term “qubit,” with upper and lower bounds on quantum compression rates.These results are presented as analogous in importance to Shannon’s noiseless channel coding theorem and as mathematically similar to Shannon’s proof.
2) Quantum Coding and Quantum Compression:: Quantum information theory developed noiseless coding, mixed-state and universal compression, quantum error correction, quantum Huffman coding, and links between compression and entanglement concentration.The section also distinguishes Schumacher’s encoding method from entanglement concentration and surveys later coding-theorem developments.
3) Quantum Entanglement:: Entanglement distinguishes quantum from classical information and supports communication protocols, concentration, purification, distribution, broadcasting, and quantum-capacity-related resource transformations.Devetak and Winter connected entanglement distillation with quantum coherent information, while purification was identified as important for quantum-channel capacity.
4) Quantum Channels:: Quantum-channel analysis relates Holevo information, quantum mutual information, additivity, relative entropy, and entanglement to the characterization of channel capacities.The survey notes that quantum mutual information measures classical transmission through noisy quantum channels but not maximal transmittable quantum information, whereas entanglement affects Holevo-information additivity.
5) Comprehensive Surveys:: The survey consolidates broader resources on quantum information, including textbooks, quantum Shannon theory, depolarizing-channel communication, random quantum channels, and entanglement of random subspaces.It also points to perspectives on quantum computing and additional channel research topics.
III. CLASSICAL CAPACITIES OF A QUANTUM CHANNEL: Communication over quantum channels is bounded by corresponding capacities, and the classical-capacity section introduces noisy-channel descriptions, encoder-decoder settings, maximal transmittable classical information, and channel maps.These results are presented as prerequisites for analyzing advanced quantum-communication properties.
A. Extended Formal Model: A quantum channel is modeled as a CPTP map describing quantum operations, with input states interacting unitarily with an initially pure environment and producing output, environment, and purification states.For mixed inputs, purification allows restoration of the original state by taking a partial trace.
B. Capacity of Classical Channels: Classical channel capacity is the maximum reliable information rate per channel use, achieved when R≤C(N), while rates above capacity have exponentially vanishing successful-decoding probability as n increases.The asymptotic capacity concerns repeated use of the channel, denoted by N ⊗n, with n→∞.

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

The HSW theorem characterizes classical information transmission through noisy quantum channels with product-state inputs and joint measurements via maximized Holevo information. Capacity relations depend on input entanglement and measurement settings, with entangled inputs and joint measurements potentially exceeding the Holevo capacity.

1) The Holevo-Schumacher-Westmoreland Capacity:: The HSW theorem defines the product-state classical capacity as the maximum classical information transmissible through a noisy quantum channel with joint output measurement.The optimization ranges over ensembles of input quantum states, and the capacity is the channel’s Holevo capacity χ(N).
1) The Holevo-Schumacher-Westmoreland Capacity:: Codewords transmit with arbitrarily small error when the code rate satisfies R<C(N)=max_all_pi,rho_i χ.Rates above C(N) cannot support arbitrarily large error-free quantum codes.
2) Various Classical Capacities of a Quantum Channel:: The asymptotic channel capacity is treated as the true measure of channel capacities, whereas single-use capacity applies only to special cases.Regularization computes capacity as a limit and connects single-use lower bounds with asymptotic results.
2) Various Classical Capacities of a Quantum Channel:: At p=0.5, the classical binary symmetric channel has C(N)=1−H(p)=0, while quantum-state encoding can achieve C(N)=1 through suitable Pauli X eigenstates.The quantum improvement arises from optimizing over input ensembles and exploiting additional state degrees of freedom.
2) Various Classical Capacities of a Quantum Channel:: For product-state inputs and single measurements, the single-use and asymptotic classical capacities are equal.The channel can transmit codewords across repeated uses, represented by N ⊗n.
2) Various Classical Capacities of a Quantum Channel:: For product-state inputs and joint measurements, the classical capacity is expressed by maximized Holevo information rather than maximized quantum mutual information.The Holevo capacity χ(N) equals the asymptotic channel capacity in this restricted setting.
2) Various Classical Capacities of a Quantum Channel:: With entangled inputs and joint measurements, the equality C(N)=χ(N) fails, and Hastings showed that entanglement can increase transmitted classical information.The asymptotic capacity must therefore use a regularized Holevo-capacity formula.
2) Various Classical Capacities of a Quantum Channel:: No known quantum channel increases capacity with entangled inputs and single measurements because changing the joint measurement to a single measurement removes entanglement’s benefits.Single measurement was conjectured to support the relevant formula, but joint measurement is required to realize entanglement benefits.

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

The paper reviews classical, private, entanglement-assisted, and zero-error capacities of quantum channels, emphasizing their operational conditions and differences between single-use and asymptotic formulations. Zero-error communication requires perfect transmission, while entanglement assistance can yield an additive capacity computable without regularization.

3) Brief Summary:: The HSW capacity maximizes the Holevo quantity over non-entangled product input states to quantify reliably transmittable classical information through a noisy quantum channel.The Holevo quantity measures classical information remaining after transmission, while the HSW capacity is also called product-state channel capacity.
D. The Private Classical Capacity: Private classical capacity P (N) is the maximum reliable classical communication rate that leaks no information about the original message to an eavesdropper.Its single-use expression maximizes the difference between information received by Bob and information obtained by Eve.
D. The Private Classical Capacity: The difference of two quantum mutual information functions is not additive, so the true private classical capacity requires an asymptotic formulation.Although quantum mutual information itself is additive, its difference is not.
E. The Entanglement-assisted Classical Capacity: Entanglement-assisted classical capacity CE (N) measures classical information transmitted when Alice and Bob share entanglement before transmission, rather than entangling input states.It is expressed using maximized quantum mutual information and can be derived from its single-use version.
E. The Entanglement-assisted Classical Capacity: Shared entanglement does not improve the classical capacity in the cited formulation but can increase single-use transmission success, including through the CHSH game.The paper states that transmitting a single quantum bit can achieve higher success probability with shared entanglement.
E. The Entanglement-assisted Classical Capacity: Entanglement-assisted classical capacity CE (N) requires no regularization and is therefore always additive, making it easier to compute than capacities requiring regularization.Shared entanglement does not change the additivity of maximized quantum mutual information.
F. The Classical Zero-Error Capacity: Zero-error capacity describes the amount of classical or quantum information transmitted perfectly through a noisy quantum channel with zero probability of error.Unlike standard capacity, which permits asymptotically small non-vanishing error, zero-error communication permits no errors.
1) Classical Zero-Error Capacities of Quantum Channels:: For product input states, classical zero-error capacity C0 (N) requires pure inputs whose output codewords are pairwise orthogonal and therefore perfectly distinguishable.In a d-dimensional Hilbert space, at most d pairwise distinguishable states exist, so n systems provide at most d^n distinguishable n-length codewords.

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

Quantum zero-error communication is characterized by non-adjacent inputs whose channel outputs are perfectly distinguishable, including for arbitrary-length codewords. Its single-use and asymptotic classical zero-error capacities are determined by non-adjacent message sets, with the asymptotic capacity upper bounded by the HSW capacity.

2) Formal Definitions of Quantum Zero-Error Communication:: Non-adjacent inputs are those whose channel outputs are perfectly distinguishable, allowing perfect identification under a suitable POVM.The outputs lie in orthogonal subspaces, and the corresponding codewords can be distinguished with probability one.
2) Formal Definitions of Quantum Zero-Error Communication:: A quantum channel has positive classical zero-error capacity if at least two input states are non-adjacent with respect to a POVM.Equivalently, the channel maps the states into orthogonal subspaces.
2) Formal Definitions of Quantum Zero-Error Communication:: The measurement construction uses orthogonal projectors whose sum, together with the remaining projector, equals the identity.The measurement can be restricted to projective measurements and replaced by von Neumann operators.
2) Formal Definitions of Quantum Zero-Error Communication:: For n-length tensor-product codewords, non-adjacency requires at least one corresponding state pair to be perfectly distinguishable through the channel.Thus, one distinguishable pair can make the complete codewords non-adjacent.
2) Formal Definitions of Quantum Zero-Error Communication:: At least two non-adjacent input states are necessary and sufficient for non-zero zero-error capacity, with joint measurement required to distinguish the output codewords.Single measurement measures each state individually, whereas joint measurement waits for all n states and measures them together.
3) Achievable Zero-Error Rates in Quantum Systems:: The single-use zero-error capacity is determined by K(N), the maximum number of mutually non-adjacent inputs or messages transmitted in one channel use.This defines the largest zero-error message set for a single use of N.
3) Achievable Zero-Error Rates in Quantum Systems:: The asymptotic zero-error capacity is based on K(N ⊗n), the maximum number of n-length classical messages transmitted with zero error over n channel uses.The quantity K(N ⊗n) describes the largest zero-error message set for the n-use channel.
3) Achievable Zero-Error Rates in Quantum Systems:: The asymptotic classical zero-error capacity C0(N) of a quantum channel is upper bounded by the HSW capacity.This establishes an upper bound relating zero-error classical communication to the HSW capacity.

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

The paper recasts zero-error transmission through confusability graphs, then extends the framework with hypergraphs and shared entanglement. Entanglement can increase the number of perfectly transmissible messages and, for special codes, reach the classical HSW capacity.

4) Connection with Graph Theory:: Zero-error codeword selection is represented by a confusability graph, where vertices are input messages and edges connect codewords that can produce a common output.Two inputs are adjacent when some output has positive probability conditioned on either input.
4) Connection with Graph Theory:: For the pentagon graph with one channel use, only two non-adjacent vertices are available, so the classical zero-error capacity remains C0 (N) = 1.Other non-adjacent codeword pairs exist, but the maximum independent set still contains only two vertices.
4) Connection with Graph Theory:: Increasing the pentagon block-code length from n=1 to n=2 changes the graph structure and enables five two-length zero-error codewords.The construction uses the computational basis {|0⟩, |1⟩, |2⟩, |3⟩, |4⟩}.
4) Connection with Graph Theory:: For the pentagon graph, the maximum zero-error transmission rate over the noisy quantum channel is achieved with quantum block-code length two.This conclusion is stated from an engineering perspective for the pentagon graph.
G. Entanglement-assisted Classical Zero-Error Capacity: The entanglement-assisted protocol begins with shared maximally entangled states, uses message-dependent measurements by Alice, and lets Bob identify the transmitted message after channel decoding.The example allows Alice to choose among K possible bases, with her measurement outcome sent through the classical channel.
G. Entanglement-assisted Classical Zero-Error Capacity: With shared entanglement, zero-error communication is modeled by a hypergraph whose vertices are channel inputs and hyperedges group inputs capable of producing the same output.Hypergraphs can represent the same non-adjacency structure with fewer hyperedges than the corresponding confusability graph as the input set grows.
G. Entanglement-assisted Classical Zero-Error Capacity: In the example, entanglement permits K=6 zero-error messages, whereas communication without shared entanglement permits K=5.The six-message construction uses a rank-four maximally entangled qudit state and partitions the hypergraph into six cliques of size d=4.
G. Entanglement-assisted Classical Zero-Error Capacity: For special Pauli-graph codes, entanglement-assisted classical zero-error capacity can reach the maximal classical HSW capacity, the upper bound for classical zero-error capacity.The CLMW theorem and related results establish that entanglement can increase the asymptotic zero-error capacity and the number of possible input messages.

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

The paper surveys classical and entanglement-assisted capacities of quantum channels before turning to quantum-information transfer, where fidelity and entanglement preservation characterize transmission through noisy channels.

H. Related Work: Quantum channels encompass classical communication because classical information can be encoded into qubits or quantum states.The section reviews major results concerning classical capacity.
H. Related Work: The HSW theorem determines reliable classical information transmission through a noisy quantum channel using product input states and repeated channel uses.It uses Holevo information and extends Shannon’s classical channel-coding framework.
H. Related Work: Entanglement-assisted classical capacity CE(N) quantifies classical communication through a quantum channel with shared entanglement.Its formulation was completed by Bennett et al. in 1999 and is based on superdense-coding-like encoding and decoding.
H. Related Work: The private classical capacity P(N) measures private classical information and is at least as large as the single-use quantum capacity of any quantum channel.The concept was introduced by Devetak in 2003 and Cai et al. in 2004.
IV. THE QUANTUM CAPACITY OF A QUANTUM CHANNEL: Quantum-capacity analysis focuses on transmitting quantum information through noisy channels and introduces fidelity and quantum coherent information as key quantities.Fidelity compares input and output states, while coherent information represents quantum-information loss to the environment.
A. Preserving Quantum Information: Quantum messages are encoded as non-orthogonal, superposed, or entangled states, mapped by encoder E into n intermediate systems, independently transmitted through N, and decoded by D.The input and output each contain m quantum states.
A. Preserving Quantum Information: Because interaction with the environment produces mixed output states, successful quantum transmission requires preserving the original superposition and entanglement with an inaccessible reference system P.Entanglement fidelity FE measures preservation of the initial entangled state, and repeated uses of N transmit quantum messages over channel copies.

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

Quantum coherent information quantifies entanglement transmission through a quantum channel and decreases when the input is not maximally entangled or the channel is nonideal. It also connects quantum transmission to Holevo information and differs from quantum mutual information because its maximized form is not always additive.

B. Quantum Coherent Information: Entropy exchange measures environmental entropy increase, equivalently the entanglement between the purifying system and environment after channel evolution.It is analogous to classical conditional entropy, but concerns quantum information.
B. Quantum Coherent Information: Quantum coherent information is maximized when the input systems are maximally entangled and the channel is completely noiseless.Under an ideal channel, the joint input state remains pure after transmission.
B. Quantum Coherent Information: When the input is not maximally entangled or the channel is not ideal, quantum coherent information decreases.The reduction reflects nonideal input or channel conditions.
B. Quantum Coherent Information: Quantum coherent information measures a channel’s capability to transmit entanglement.It is associated with the difference between information received by Bob and information received by the environment.
C. Connection between Classical and Quantum Information: The amount of transmittable quantum information can be derived from Holevo information, which measures classical information.The connection is expressed through Holevo quantities involving Alice, Bob, and the environment.
C. Connection between Classical and Quantum Information: Quantum capacity can be expressed using Bob’s Holevo quantity and the information leaked to the environment during transmission.The relevant quantities are denoted XAB for Bob’s output and XAE for the environment.
1) Quantum Coherent Information and Quantum Mutual Information:: Icoh (ρA:N (ρA)) = S (ρB) −S (ρAB) = −S (A| B), so coherent information corresponds to negative conditional entropy.Unlike quantum mutual information, its expression omits the term S (ρA) after the channel transforms Alice’s state.
1) Quantum Coherent Information and Quantum Mutual Information:: The maximized quantum mutual information is always additive, whereas the maximized quantum coherent information is not always additive.This marks a fundamental difference between the two channel-information measures.

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

The section surveys quantum channel capacities and maps, emphasizing coherent information, the LSD capacity, assisted and zero-error capacities, and their relation to classical capacities. It also summarizes foundational results and related work on quantifying quantum information transmission.

2) Quantum Coherent Information of an Ideal Channel:: For a completely noiseless ideal channel NAB=I, quantum coherent information appropriately measures the channel’s quantum capacity.The capacity can be calculated without maximization for NAB=I.
2) Quantum Coherent Information of an Ideal Channel:: Maximal coherent information for NAB=I requires maximally mixed input states or one half of an EPR state, where von Neumann entropies are maximal.Environmental interaction can make quantum capacity vanish while classical capacity remains nonzero, although stronger interaction may also eliminate classical capacity.
D. The Lloyd-Shor-Devetak Formula: Quantum coherent information determines the asymptotic Lloyd-Shor-Devetak capacity and is fundamental for describing the maximal transmittable quantum information through N.The asymptotic capacity can also be expressed through Holevo information, with XAB denoting information sent to Bob and XAE information sent to the environment.
E. The Assisted Quantum Capacity: Assisted quantum capacity combines a quantum channel N with a symmetric channel A and can enable superactivation of channels having zero LSD capacity.A zero-capacity Horodecki channel combined with a zero-capacity symmetric channel can have positive joint capacity.
F. The Zero-Error Quantum Capacity: Quantum zero-error capacities use coherent encoders and coherent POVM decoders, and the zero-error capacity is upper bounded by the LSD capacity Q (N).K (N ⊗n) is the maximum number of n-length mutually non-adjacent quantum messages transmissible with zero error.
G. Relation between Classical and Quantum Capacities of Quantum Channels: The quantum capacity cannot generally exceed classical capacity achieved with entangled inputs and joint measurement, while some channels are conjectured to satisfy C (N) <Q (N).The inequality is stated for classical capacity measured with a classical encoder and a single measurement setting.
H. Related Work; 1) Quantum Coherent Information:; 2) Proofs on Quantum Capacity:: The paper reviews quantum coherent information as a measure of state preservation, the HSW theorem’s classical role, and historical proofs establishing quantum capacities for noisy channels.Schumacher and Nielsen introduced the exact coherent-information measure in 1996, while Lloyd, Shor, Devetak, and Hayden contributed proofs for noisy-channel capacity.
V. QUANTUM CHANNEL MAPS AND CAPACITIES: The survey introduces important quantum channel maps and studies capacity formulas, directing readers to a separate reference for state-vector definitions.The section’s stated purpose is to survey channel maps and capacity formulas.

A. Channel Maps … 5) The Pancake Map:

The section surveys Pauli, depolarizing, damping, dephasing, and pancake channel maps, including their transformations, environmental effects, assumptions, and physical status. It also notes that the nonphysical pancake map can theoretically transmit information despite practical decoherence challenges.

1) The Pauli Channel:: The Pauli channel maps input state ρ through weighted identity and Pauli X, Y, and Z error operations.Its depolarizing probability is p = px + py + pz, while error probabilities depend on relaxation time T1 and dephasing time T2.
2) The Depolarizing Channel:: The depolarizing channel is presented as a unital channel transformation whose encoding may use two orthogonal states, ρ0 and ρ1.The mixed input state follows when Alice uses those two orthogonal encoding states.
3) The Damping Channel:: Damping channels model environmental energy exchange and decoherence in two-level atoms, including amplitude damping, phase damping, and their combination.Independent qubit–environment interactions may be treated as temporally and spatially uncorrelated, while combined amplitude-and-phase damping can be approximated by a Pauli model for classical simulation.
4) The Dephasing Channel Model:: The dephasing channel is a unitary decoherence map that creates relative phase differences and shrinks the Bloch sphere more than the phase-flip map.Its environmental coupling is characterized by the positive real parameter γ(t).
5) The Pancake Map:: The pancake map is a physically not allowed non-CP transformation that preserves equatorial Bloch-sphere information while demolishing information along the z axis.The passage describes this as having no Completely Positive Trace Preserving map with those properties.
5) The Pancake Map:: Despite being nonphysical, the pancake map can theoretically transfer some information, while decoherence remains difficult to eliminate in practical quantum systems.The section connects decoherence reduction to the engineering requirement of designing quantum systems appropriately.

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

The paper derives classical and quantum capacities for erasure-related and amplitude-damping quantum channels, then models practical quantum-channel behavior in hardware and free-space optical QKD. It characterizes channel transmission using material-dependent decoherence parameters and near-field/far-field bounds on photon-transfer fraction γ.

B. Capacities: The study derives closed-form classical capacities and quantum capacities for erasure, phase-erasure, mixed erasure/phase-erasure, and amplitude-damping channels.These capacities are compared across the four channel types.
1) Erasure Quantum Channel:: At p=1, the erasure channel’s classical capacity vanishes, while 0≤p<1 permits classical transmission; its quantum capacity vanishes at p=1/2 and permits quantum transmission for 0≤p<1/2.The erasure channel either erases the input with probability p or transmits it unchanged with probability 1−p.
2) Phase-Erasure Quantum Channel:: The phase-erasure channel erases phase with probability p without disturbing amplitude, leaving orthogonal |0⟩ and |1⟩ states distinguishable for classical communication.Its quantum-capacity behavior differs because phase errors affect quantum transmission.
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 expressed from the component-channel results.Its classical and quantum capacities are plotted against total erasure probability p+q.
4) Amplitude damping Quantum Channel:: For amplitude damping, classical capacity vanishes at γ=1 but remains nonzero for 0≤γ<1, whereas quantum capacity vanishes when γ≥0.5.The paper presents quantum capacity as a maximization and compares both capacities as functions of damping parameter γ.
A. Realistic Material: Asymmetric Depolarising Channel: Quantum depolarizing channels model hardware imperfections, including decoherence- and gate-induced qubit flips, as well as state flips imposed by transmission media.The channel is characterized by bit-flip, simultaneous bit-and-phase-flip, and phase-flip probabilities px, py, and pz.
B. Acting Time in Asymmetric Channels: For short coherent-operation duration t << T1, α ≈2T2/T1−1, enabling pz to be determined from α and px; absolute t, T1, and T2 improve characterization across materials.The paper notes that using only precalculated α values may not closely characterize systems with different absolute parameters.
C. Implementation of Quantum Channel in FSO-based Quantum Key Distribution: In FSO-based QKD, diffraction, atmospheric turbulence, and extinction reduce the transmitted photon stream to a fraction γ over distance L, whose accurate range is bounded across near-field, transition, and far-field regimes.The near-field and far-field parameters depend on aperture diameters, optical wavelength, and the spherical-wave structure function; the transition region is illustrated in Fig. 30.

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

Quantum channel codes mitigate perturbations and use entanglement assistance to improve quantum error-correction performance, while quantum network coding distributes entanglement for protocols underpinning practical quantum communications and the quantum Internet. Quantum channels extend classical communication by transmitting several forms of classical and quantum information through advanced communication primitives.

D. Quantum Channel Codes for Approaching Quantum Channel Capacity: Quantum perturbations obstruct practical quantum communications, while Quantum Error Correction Codes mitigate their deleterious effects.The section motivates quantum-channel coding through quantum-computing parallelism and related communications applications.
D. Quantum Channel Codes for Approaching Quantum Channel Capacity: Entanglement assistance was suggested to further improve Quantum Error Correction Codes in symmetric quantum-channel settings.The passage specifically identifies entanglement assistance as a performance-enhancing resource for QECCs.
D. Quantum Channel Codes for Approaching Quantum Channel Capacity: The Hashing bound provides a lower limit for achievable quantum depolarizing-channel capacity and benchmarks Quantum Error Correction Code schemes.Entanglement Assisted Quantum Channel capacity was investigated for benchmarking EAQECC designs, alongside EXIT-chart methods.
E. Quantum Network Coding for Entanglement Distribution: Classical network coding increases throughput while reducing energy per packet and packet-travel delay by allowing intermediate nodes to combine incoming data packets.This describes the classical network-coding mechanism that motivates its quantum counterpart.
E. Quantum Network Coding for Entanglement Distribution: Although impossible cloning prompted negative answers, Quantum Network Coding studies confirm feasibility when extra resources such as preshared entanglement or low-cost classical communications are available.The passage contrasts initial impossibility concerns with later resource-assisted QNC results.
E. Quantum Network Coding for Entanglement Distribution: Entanglement enables quantum teleportation, remote state preparation, quantum remote measuring, and secret sharing.It is described as a valuable enabler of quantum-communication protocols.
E. Quantum Network Coding for Entanglement Distribution: Entangled qubits must be distributed to distant nodes, supporting QKD across satellite, terrestrial, and handheld communications and laying foundations for the quantum Internet.The passage presents entanglement distribution as a practical networking requirement and highlights QKD as a popular application.
VII. CONCLUSIONS: Quantum channels transmit classical information, entanglement assisted classical information, private classical information, and quantum information through communication primitives unavailable to classical channels.Quantum entanglement and superposed states carry quantum information that cannot be described classically.

APPENDIX … C. Fidelity

The appendix explains partial traces for subsystem states, distinguishes separable systems from entangled states, and introduces fidelity for comparing pure inputs with noisy mixed outputs.

A. Partial Trace: A subsystem’s reduced density matrix is obtained from the joint state by taking the partial trace, ρA=TrB(ρAB).For uncorrelated subsystems, the larger system can be represented as a tensor product of reduced density matrices.
A. Partial Trace: For a product state, tracing subsystem B gives TrB(ρAB)=|ψA⟩⟨ψA|=ρA because ⟨ψB|ψB⟩=1.The calculation uses Tr(|ψ1⟩⟨ψ2|)=⟨ψ2|ψ1⟩.
A. Partial Trace: For operators on two systems, the partial trace is calculated using tensor-product basis expressions such as |i⟩⟨k|⊗|j⟩⟨l|.The tensor-product operator is written as |i⟩|j⟩(|k⟩|l⟩)T.
B. Quantum Entanglement: A quantum system is separable when its joint density matrix factors as ρAB=ρA⊗ρB, representing independent states combined by tensor product.Quantum mechanics also permits entanglement, a phenomenon absent from the description of independent composite systems.
B. Quantum Entanglement: Bell states, also called EPR states after Einstein, Podolsky, and Rosen, are examples of entangled quantum states.They contrast with product states formed from independent subsystems.
C. Fidelity: Quantum channels can transform pure input states into mixed output states because interaction with the environment introduces noise.Mixed states are classical probability-weighted sums of pure states.
C. Fidelity: Fidelity describes the relation between Alice’s pure input |ψ⟩ and the received mixed quantum system σ.It is introduced to characterize transmission when the original superposition is disturbed by environmental interaction.
C. Fidelity: Fidelity is concave in its second argument: F(ρ,aσ1+(1−a)σ2)≥aF(ρ,σ1)+(1−a)F(ρ,σ2), with a∈[0,1].This is listed among the major properties of fidelity.