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Quantum channels and their entropic characteristics

A. S. Holevo, V. Giovannetti

arXiv:1202.6480v1quant-phmath-ph

TL;DR

Quantum information theory needs channel models and information measures that capture quantum noise, irreversibility, and correlations in physical communication. This survey develops channel representations, finite-dimensional and Gaussian examples, and entropic capacity formulas. It also identifies important scope boundaries, including unresolved additivity and capacity-computation problems for some channels.

  • Problem

    The paper addresses how quantum communication channels should be structurally described and how their information-processing efficiency should be characterized by entropic quantities.

  • Method

    The paper surveys mathematical channel representations, finite-dimensional channel classes and examples, continuous-variable Gaussian systems, and entropic capacity formulas.

  • Results

    The survey presents channel capacities and related entropic quantities as characterizations of information-processing performance, while emphasizing entanglement as a communication resource.

  • Takeaways & Limitations

    Quantum channels provide a framework for analyzing reliable classical and quantum information transfer, including communication through noisy and Gaussian systems.

  • Takeaways & Limitations

    For Gaussian channels, additivity of Cχ(Φ, E) is not known in general, preventing identification with the full constrained capacity C(Φ, E).

Abstract

from arXiv · show

One of the major achievements of the recently emerged quantum information theory is the introduction and thorough investigation of the notion of quantum channel which is a basic building block of any data-transmitting or data-processing system. This development resulted in an elaborated structural theory and was accompanied by the discovery of a whole spectrum of entropic quantities, notably the channel capacities, characterizing information-processing performance of the channels. This paper gives a survey of the main properties of quantum channels and of their entropic characterization, with a variety of examples for finite dimensional quantum systems. We also touch upon the "continuous-variables" case, which provides an arena for quantum Gaussian systems. Most of the practical realizations of quantum information processing were implemented in such systems, in particular based on principles of quantum optics. Several important entropic quantities are introduced and used to describe the basic channel capacity formulas. The remarkable role of the specific quantum correlations - entanglement - as a novel communication resource, is stressed.

1. Introduction

Quantum channels provide the operational framework for modeling physical communication lines when quantum noise cannot be treated as negligible. The paper surveys their mathematical structure and characterizes transfer efficiency through entropic quantities and capacities.

  • Quantum channels are needed because every physical communication line is ultimately quantum, and quantum noise matters in optical communication and quantum system engineering.
  • Dynamical maps transform quantum states into states through positivity, extending classical Markov maps and supporting models of quantum Markovian dynamics.
  • Complete positivity strengthens positivity for reduced open-system dynamics and became central to the physical interpretation of quantum channels.
  • Coding theorems establish reliable asymptotic transmission rates and express channel capacities using entropic functionals.
  • The survey covers channel representations, finite-dimensional examples, continuous-variable quantum Gaussian systems, entropic characterizations, and open questions.Its organization includes structural theory, Gaussian systems, and summary and outlook sections.

2. Channels and open systems

Quantum information carriers require Hilbert-space descriptions rather than classical alphabets, while channels model reversible and noisy transformations under complete positivity and trace preservation. Their representations expose environmental information flow, irreversibility, and entanglement-breaking behavior.

  • Quantum carriers are represented by Hilbert spaces, unlike classical carriers described by finite phase spaces or alphabets.
  • Qubit channels act linearly on the Bloch ball, mapping it into an ellipsoid up to input and output rotations.
  • Unitary conjugations describe reversible quantum dynamics, whereas anti-unitary transformations include reflection or transposition and are excluded as proper dynamical processes.
  • A minimal environment has a channel-invariant size reflecting noisiness and irreversibility, while Kraus decompositions themselves are generally nonunique.
  • Complete positivity together with trace preservation is necessary and sufficient for Kraus and unitary representations of open-system quantum channels.
  • Channel composition is generally noncommutative and noninvertible; only unitary mappings admit CPTP inverses.
  • Complementary channels track information transferred to the environment, making perfect transmission equivalent to no quantum information entering that environment.Approximate complementarity yields an information-disturbance trade-off between a channel and its complement.
  • Entanglement-breaking channels destroy entanglement with a reference system and include depolarizing channels when p ≥ d/(d+1).Their Choi-Jamiolkowski states are separable, and every entanglement-breaking channel has the stated measure-and-prepare form.

3. Bosonic Gaussian channels

Bosonic Gaussian channels model continuous-variable quantum systems, especially optical implementations, by preserving Gaussian-state structure under constrained completely positive maps. Their action is encoded through linear transformations of canonical observables and covariance data.

  • Bosonic systems: Continuous-variable Bosonic systems use canonical observables satisfying Heisenberg commutation relations and describe many experimental quantum-information platforms.Examples include optical fibers, free-space communication, vibrational modes, atomic ensembles, and Bose-Einstein condensates.
  • Gaussian states and channels: Gaussian states include vacuum, coherent, squeezed, and thermal states, while Gaussian channels describe attenuation, amplification, and thermalization in optical implementations.Displacements induced by fixed sources are also included among Gaussian states.
  • Signal encoding: A classical signal µ can be encoded into displaced quantum states ρµ, yielding a c-q channel with continuous alphabet C and additive quantum Gaussian noise.The model generalizes to broadband channels and settings involving squeezing.
  • Channel characterization: Gaussian channels are completely characterized by real quantities (K, l, β), with β constrained by an uncertainty relation that is necessary and sufficient for complete positivity.The displacement vector l is unrestricted.
  • Channel action: The channel transforms means and correlation matrices as m → m′ = Kᵀm + l and α → α′ = KᵀαK + β, preserving Gaussianity of Gaussian inputs.When β = 0, K is symplectic and the channel represents canonical unitary evolution composed of a shift and a symplectic transformation.

4. Entropy, information, channel capacities

Classical information theory characterizes reliable communication through entropy, mutual information, and coding theorems. For memoryless noisy channels, capacity is the asymptotic errorless transmission rate obtained by optimizing mutual information over input distributions.

  • Source entropy: Shannon’s first coding theorem identifies nH(X) bits as the asymptotic number of bits needed to encode typical n-symbol source messages.Typical messages have approximately equal probabilities and number about 2^(nH(X)).
  • Channel model: A memoryless noisy channel maps independent input symbols through a single-letter conditional distribution p(y|x), whereas memory channels require joint input-output probabilities.The output distribution is induced by the input distribution and p(y|x).
  • Information measure: Mutual information I(X;Y) measures recoverable information by subtracting conditional entropy, or information loss, from the output entropy.It can equivalently be written as H(X) + H(Y) − H(X,Y).
  • Channel capacity: Shannon capacity CShan is the ultimate asymptotically errorless transmission rate, achieved through block encoding and decoding with vanishing error.Approximately 2^(nCShan) messages can be transmitted as n approaches infinity, while safely transmitting more is impossible.
  • Memorylessness: For memoryless channels, capacity is additive across independent uses, although block formulations optimize over correlated input distributions on the n-symbol alphabet.The n-block channel treats length-n strings as super-symbols.

4.2. Quantifying information in a quantum world

Quantum source coding replaces classical message-counting with the dimension of a typical subspace. The von Neumann entropy S(ρ) quantifies the asymptotic quantum resources needed to represent sequences emitted by a quantum source.

  • Quantum source coding: For n-long sequences of pure quantum symbols, the average state is ρ = Σj pj|ψj⟩⟨ψj|, and typical sequences span a subspace of dimension approximately 2^(nS(ρ)).This is the quantum analogue of classical typical-message compression.
  • Operational entropy: The quantity nS(ρ) is the logarithmic size of a quantum register that can optimally store ρ^⊗n with negligible information loss.The result holds with asymptotically unit probability in the large-n limit.
  • Operational entropy: The von Neumann entropy is therefore an operational measure of the quantum information stored in a density matrix.The same result quantifies the qubit resources needed to carry the corresponding encoding.

4.3. The classical capacity of quantum channel: part I

Quantum channel capacity depends on how nonorthogonal output states are measured and jointly decoded. Block and entangled measurements can outperform single-use strategies, while energy-constrained Gaussian channels yield finite entropic capacity formulas.

  • c-q channel capacity: For memoryless c-q channels, independent inputs produce product output states, and decoding requires a collective quantum measurement on the tensor-product output space.Optimizing over measurements on H^⊗n defines the n-use capacity C(n).
  • Collective decoding: Entangled output measurements can make C(n) exceed nC(1), so collective decoding may transmit more classical information than independent single-use optimization.This strict superadditivity is a consequence of the available entangled measurements.
  • Holevo capacity: The Holevo coding theorem shows that block coding saturates the Holevo upper bound, giving a compact one-letter expression for classical capacity.The maximization is over input probabilities while the output states remain fixed.
  • Examples: For a binary coherent-state channel, Cχ/C(1) exceeds 1 and tends to infinity in the weak-signal limit as the state overlap approaches 1.The overlap is ε = exp(−2|z|²), with ε → 1 as z → 0.
  • Examples: Overcomplete nonorthogonal pure-state ensembles can attain Cχ = log2 d, while C(1) remains strictly smaller unless the states form an orthonormal basis.For the three-state qubit example, Cχ = 1 and C(1) ≈ 0.645.
  • Gaussian channels: Continuous-alphabet Gaussian channels require an input-energy constraint; maximizing the average output entropy gives a Gaussian input distribution and a finite capacity.The displaced states share entropy S(ρµ) = S(ρ0) = g(N), while the maximizing average entropy is g(N + E).
  • Gaussian channels: At large energies, the Gaussian quantum capacity becomes Cχ = log2(1 + E/N), compared with the classical additive-noise formula containing a factor of 1/2.The quantum expression lacks the factor because one quantum optical degree of freedom corresponds to two real amplitudes.

4.4. The classical capacity of quantum channel: part II

The classical capacity of a quantum channel may require regularization because entangled input block-letters can make the χ-capacity superadditive. When additivity holds, the capacity reduces to a single-letter maximization, but nonadditive channels exist in sufficiently high dimensions.

  • Capacity regularization: The classical capacity is obtained from regularized χ-capacities of n-fold channel uses, optimizing over ensembles of possibly entangled input codewords.The resulting rate is Cχ(Φ^⊗n)/n, with the asymptotic limit giving the ultimate achievable rate.
  • Capacity regularization: The regularization in the classical-capacity formula reflects possible superadditivity from entangled quantum block-letters at the encoding stage.This differs from the regularization associated with entanglement at the decoding stage.
  • Additivity: When χ-capacity is additive, C(Φ)=Cχ(Φ), so entangled input states do not increase transmitted classical information.Additivity has been established for unital qubit, depolarizing, erasure, purely lossy Bosonic, and entanglement-breaking channels.
  • Examples: For the depolarizing channel, additivity and symmetry identify an optimum ensemble of d equiprobable orthogonal pure states.This ensemble attains the channel’s classical capacity.
  • Nonadditivity: Hastings proved that channels violating χ-capacity additivity, and therefore requiring regularization, exist among mixtures of unitary channels in very high dimensions.Earlier violations involving related entropic functionals were not strong enough to imply superadditivity of Cχ.
  • Joint-channel superadditivity: The tensor-product capacity question asks whether C(Φ1⊗Φ2) can exceed C(Φ1)+C(Φ2), beyond the guaranteed rate obtained by operating the channels independently.Strict superadditivity could arise from quantum correlations introduced across the two channel inputs.

4.5. Entropy exchange and quantum mutual information

Quantum mutual information extends classical input-output correlation measures by incorporating quantum reference and environment systems. Purification and strong subadditivity yield entropy relations, subadditivity across channels, and data-processing inequalities.

  • Quantum correlations: The χ-information does not capture all correlations between a quantum channel’s input and output, motivating additional quantum generalizations of mutual information.Multiple quantum generalizations arise because joint distributions of quantum observables generally do not exist unless the observables commute.
  • Purification: Purifying the input with a reference system R lets the channel output and preserved reference define quantum mutual information as an input-output correlation measure.The reference state has the same spectrum and entropy as the input state.
  • Entropy exchange: An isometric channel representation produces a pure BER state, making the entropy of the output-reference subsystem equal to the environment’s entropy exchange.The environment state is also the output of the complementary channel.
  • Entropy quantities: Input, output, and exchange entropies generate loss and noise quantities through mutual-information combinations involving the environment.The loss couples input and environment, whereas the noise couples output and environment.
  • Entropy inequalities: Nonnegativity of quantum mutual information, loss, and noise implies triangle inequalities among the input, output, and entropy-exchange quantities.These inequalities constrain the three basic entropies of a channel use.
  • Conditional entropy: Quantum conditional entropy can be negative, unlike classical conditional entropy, because quantum entropy need not increase when systems are enlarged.This behavior is exemplified by a pure entangled BR system with S(BR)=0 while S(R)>0.
  • Operational properties: Strong subadditivity supports quantum mutual-information properties including channel subadditivity and data processing under channel composition.Specifically, mutual information does not increase through successive processing and is bounded across product channels.

4.6. Entanglement as an information resource

Shared entanglement can increase classical communication through a quantum channel, with entanglement-assisted capacity characterized by a single-use quantum mutual-information formula. The gain is at least nonnegative and can become large for noisy channels.

  • Protocol: Entanglement-assisted communication distributes an entangled state beforehand, encodes messages through operations on the sender’s share, and sends the result through the channel.The receiver measures the transmitted channel output together with its retained entangled share.
  • Capacity formula: The entanglement-assisted capacity is Cea(Φ)=maxρ I(ρ,Φ), with the maximum taken over input states for one channel use and no regularization limit.Its additivity follows from subadditivity of quantum mutual information.
  • Capacity gain: Entanglement-assisted capacity is always at least the unassisted classical capacity, Cea(Φ)≥C(Φ).Thus shared entanglement cannot reduce the supported classical communication rate.
  • Examples: For a noiseless d-dimensional channel, superdense coding gives Cea(Id)=2 log2 d=2C(Id), doubling the unassisted capacity.The same doubling occurs for the quantum erasure channel.
  • Noisy channels: For depolarizing channels, the capacity gain Cea(Φ)−C(Φ) approaches d+1 as noise p approaches 1.The entanglement-assisted optimum is attained on the chaotic state.
  • Entanglement-breaking channels: Even entanglement-breaking channels can satisfy Cea(Φ)>C(Φ), including depolarizing channels with p≥d/(d+1).The cited explanation compares the communication cost of unassisted transmission with the capacity difference.

4.7. Coherent information and perfect error correction

Coherent information links quantum mutual information to quantum capacity and perfect error correction. Perfect reversibility is equivalent to privacy because no information leaks to the environment.

  • Coherent information can be negative and lacks subadditivity and the second data processing inequality, while satisfying the first data processing inequality.
  • Perfect reversibility on a state is equivalent to the existence of a recovery channel acting after the channel output.
  • Privacy means information does not leak into the environment, and under private transmission coherent information reaches its maximal value S(ρ).
  • Coherent information is closely related to quantum capacity, which characterizes asymptotically reliable quantum-information transmission.
  • Perfect reversibility on a subspace is equivalent to the complementary channel being completely depolarizing there.

4.8. The quantum capacity

Quantum capacity is the asymptotic rate of reliable quantum-information transmission and is connected to regularized coherent information. Evaluation is difficult because coherent information is generally nonadditive, but degradable channels admit one-letter formulas.

  • Quantum capacity is the maximum quantum information per channel use transmissible with asymptotically vanishing error.
  • The capacity–coherent-information relation requires regularization over n successive channel uses, with the maximum taken over n-use input states.
  • Nonadditivity of coherent information makes quantum-capacity evaluation difficult; the depolarizing channel lacks a general analytical expression.
  • Degradable channels permit a one-letter capacity formula because their complementary channel is obtainable from the channel by adding noise.
  • The erasure channel has quantum capacity (1 − 2p) log2 d for p ∈ [0, 1/2] and 0 for p ∈ [1/2, 1].
  • PPT or binding channels have zero quantum capacity, although non-entanglement-breaking examples can transfer non-distillable bound entanglement.

4.9. Quantum wiretap channel

The quantum wiretap model separates receiver and eavesdropper outputs through complementary channels. Secret classical capacity is characterized by information shared with the receiver without informing the eavesdropper and connects to quantum-capacity proofs.

  • An isometric wiretap map sends the sender’s input into joint receiver–eavesdropper states, whose marginals define the receiver and complementary channels.
  • The wiretap description is formally the complementary-channel construction, with the environment interpreted as being controlled by the eavesdropper.
  • Secrecy measures information shared between sender and receiver without informing the eavesdropper, using the corresponding Holevo informations.
  • The identity connecting receiver and eavesdropper informations underlies the relation between secret classical and quantum capacities.
  • The resulting inequality can be strict in general, but equality holds for degradable channels.
  • Quantum cryptography is presented as a broader field connected with wiretap-channel analysis.

4.10. Capacities for Gaussian channels

Gaussian-channel capacities require energy constraints for finite classical rates, while entanglement-assisted capacity is more tractable through concave optimization. For pure-loss channels, degradability gives quantum-capacity formulas, whereas several additivity and noisy-channel cases remain unresolved.

  • Continuous-variable classical capacities require an input-energy constraint, while quantum capacity remains finite without bounded input energy.
  • Additivity of constrained Gaussian Holevo capacity is unknown in general, preventing identification with full constrained capacity; Gaussian-ensemble optimality is also generally conjectural.
  • Entanglement-assisted capacity requires no multi-use regularization and is optimized by a Gaussian state through Kuhn–Tucker equations.
  • For N0 = 0 and k2 > 1/2, attenuation/amplification channels are degradable, so quantum capacity equals maximized single-letter Gaussian coherent information.
  • For N0 = 0, the classical capacity in the plotted attenuation-channel case is exact, and entanglement-assisted quantum capacity equals half the corresponding classical capacity.
  • The attenuation-channel figure varies capacities with k2 at E = 10; for k2 ≤ 1/2, anti-degradability makes quantum capacity vanish.

4.11. The variety of quantum channel capacities

Quantum channel capacities form a hierarchy that expands when communication resources such as feedback, two-way communication, correlations, or entanglement are added. These capacities provide benchmarks for reliable information transmission in memoryless channels.

  • Q(Φ) ≤ C(Φ) ≤ Cea(Φ) relates the quantum, classical, and entanglement-assisted capacities of a quantum channel.These capacities underpin the study of capacity variants obtained by adding communication, correlation, or entanglement resources.
  • Figure 5 schematically compares capacities of a memoryless quantum channel, with vertical inequalities reflecting that each transferred qubit can carry a classical bit.The figure compares capacities under different resource assumptions.
  • Classical feedback can increase quantum capacity and enable reliable quantum transmission even through channels with zero quantum capacity.PPT channels are an exception because they retain zero quantum capacity with feedback.
  • Two channels with zero quantum capacity can jointly have positive capacity, a phenomenon called superactivation.The cited construction combines an anti-degradable channel with a PPT channel.
  • Feedback capacities Q← and C←, and the two-way-assisted quantum capacity Q↔, distinguish performance under different classical side-communication resources.The corresponding classical two-way capacity is defined under a restriction that side communication is independent of the transmitted message.
  • Quantum capacity bounds serve as benchmarks because coding theorems provide converse limits and establish asymptotic achievability.The direct theorem establishes achievability for very long transmissions but does not necessarily provide a practical implementation recipe.

4.12. Relative entropy

Relative entropy supplies a central framework for quantifying distinguishability and deriving information-processing inequalities. Its monotonicity supports results on mutual information, χ-information, coherent information, entropy gain, and channel additivity.

  • Quantum relative entropy is nonnegative and vanishes exactly when the two states are identical, while measuring their asymmetric distinguishability.
  • Relative entropy is monotone under quantum channels, meaning irreversible evolution makes states less distinguishable.This monotonicity was derived from strong subadditivity of quantum entropy.
  • Quantum mutual information and the data processing inequality arise as applications of relative-entropy monotonicity.Mutual information is positive unless the relevant joint state factorizes into a product state.
  • χ-information also has a relative-entropy representation, yielding a data processing inequality and the classical Shannon bound for quantum-to-classical channels.For q-c channels, the χ-function becomes the Shannon information of the associated measurement process.
  • Anti-degradable channels have zero quantum capacity because their coherent information is nonpositive and the coding theorem bounds the capacity accordingly.
  • The minimal entropy gain G(Φ) is additive under tensor products, unlike the minimal output entropy.The additivity follows from strong subadditivity.
  • For infinite-dimensional unital evolutions, the generalized H-theorem applies to finite-entropy inputs, while the finite-dimensional bound G(Φ) ≤ 0 no longer holds generally.For suitable Bosonic Gaussian channels, the minimal entropy gain is attained on Gaussian states and depends on |det K|.

5. Summary and outlook

The review presents quantum channels as completely positive descriptions of noisy open-system communication and surveys their capacities and entropic characterizations. It emphasizes continuous-variable Gaussian channels and shows that entanglement changes achievable information-transmission rates, including through superactivation.

  • Quantum channels describe transformations from senders’ input states to receivers’ output states when physical information carriers interact with quantum noise.
  • For memoryless communication models, the review relates channel quality to multiple classical and quantum capacities expressed through entropic formulas and coding theorems.
  • Entanglement-assisted protocols can increase information-transmission rates, with examples including superadditivity, entanglement assistance, and superactivation of zero quantum capacity.
  • Entanglement is presented as a crucial feature of the entropic characterization of noisy quantum channels.
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