Quantum Communication and Channel Capacities

Introduction

Quantum information theory studies how information is represented, transmitted, protected, and processed when the physical carriers obey the laws of quantum mechanics. A communication channel is not merely a wire: it is a physical transformation that maps an input system to an output system. Noise, loss, measurement disturbance, and environmental interaction determine how much useful information can be transmitted.

The central question of this chapter is:

How many reliable bits, qubits, private bits, or entanglement bits can a noisy quantum channel transmit per channel use?

The answer depends on the resource being transmitted and on the assistance available to the communicating parties. The principal capacities are:

• classical capacity: reliable transmission of classical messages;
• quantum capacity: reliable transmission of unknown quantum states;
• private capacity: reliable transmission of classical information hidden from an eavesdropper;
• entanglement-assisted classical capacity: classical communication when sender and receiver share unlimited prior entanglement.

These capacities are generally different. A channel may transmit classical information well while being unable to preserve quantum coherence. Conversely, entanglement assistance can substantially increase classical communication without changing the physical channel itself.

Learning objectives

By the end of this chapter, a learner should be able to:

1. Explain a quantum channel using density operators and completely positive trace-preserving maps.
2. Distinguish classical, quantum, private, and entanglement-assisted capacities.
3. Calculate von Neumann entropy, mutual information, and coherent information for simple states.
4. Interpret the Holevo bound and explain why accessible information is limited.
5. Describe the regularization appearing in classical and private coding theorems.
6. Explain the role of coherent information in quantum capacity.
7. Apply capacity concepts to depolarizing, erasure, and amplitude-damping channels.
8. Identify practical, security, and governance limitations in quantum communication systems.

1. Quantum channels and communication resources

A quantum state is represented by a density operator ρ satisfying ρ ≥ 0 and Tr(ρ) = 1. A quantum channel is a map 𝒩 from input states to output states. Physically valid channels are completely positive and trace preserving (CPTP):

Here, the operators K_j are Kraus operators, I is the identity operator on the input space, and † denotes conjugate transpose. The second condition guarantees preservation of total probability. Complete positivity means that 𝒩 remains positive even when the input is entangled with an external reference system.

A communication protocol uses a channel many times. For n independent uses, the memoryless channel is represented by 𝒩^⊗n. A coding scheme consists of an encoder, transmission through the channel, and decoder. A rate is achievable if the probability of decoding error approaches zero as n becomes large. The capacity is the supremum of all achievable rates.

Rates are normally measured in units per channel use. Classical capacity is measured in bits per use, quantum capacity in qubits per use, and entanglement generation in ebits per use. A capacity is an asymptotic ideal; finite devices achieve rates below the theoretical capacity because of block-length, hardware, and error-correction constraints.

Important information measures

The von Neumann entropy of a state ρ is

It is measured in bits because the logarithm is base 2. If the eigenvalues of ρ are λ_i, then S(ρ) = −∑_i λ_i log₂ λ_i. A pure state has entropy zero, while a maximally mixed state on a d-dimensional system has entropy log₂ d.

For a bipartite state ρ_AB, the quantum mutual information is

where S(A) = S(ρ_A), S(B) = S(ρ_B), and ρ_A and ρ_B are reduced states. Mutual information measures total correlations, including both classical and quantum correlations.

The conditional entropy is

Unlike classical conditional entropy, quantum conditional entropy can be negative. Negative conditional entropy is a signature of entanglement and is central to quantum capacity.

The coherent information from A to B is

For a channel 𝒩 acting on A and a purification ψ_RA of the input, it is often written as

The reference R purifies the input and is not sent through the channel. Coherent information can be negative for some inputs, but quantum coding uses positive values.

2. Classical communication and the Holevo bound

Suppose a sender chooses a classical message x with probability p_x and encodes it as a quantum state ρ_x. The average state at the receiver is

The ensemble’s Holevo information is

The Holevo bound states that the accessible classical information obtained by any measurement is no greater than χ:

X is the sender’s classical variable and Y is the measurement outcome. The bound is not necessarily attained by a single measurement, especially when the signal states do not commute.

The Holevo bound explains why a d-dimensional quantum system cannot communicate more than log₂ d classical bits through a single use when no additional resources are available. It does not say that quantum systems are always inferior to classical systems. Rather, it limits the classical information extractable from quantum states and distinguishes information stored in a state from information obtained by measurement.

Example: two nonorthogonal states

Consider two pure states, |0⟩ and |+⟩ = (|0⟩ + |1⟩)/√2, chosen with equal probability. Since each signal state is pure, S(ρ_x) = 0. The average state is

Its matrix representation is ρ̄ = 1/4 [[3,1],[1,1]]. Its eigenvalues are (2 ± √2)/4, approximately 0.8536 and 0.1464. Therefore

Although the sender selected one of two messages, the receiver cannot reliably distinguish the nonorthogonal states perfectly. The accessible information is at most 0.600 bits, not 1 bit.

3. Classical capacity of a quantum channel

For a quantum channel 𝒩, the one-use Holevo capacity is

where ρ̄ = ∑_x p_xρ_x. For a memoryless channel, the classical capacity is generally

The limit is required because collective encodings across several channel uses can outperform product-state encodings. This phenomenon is called superadditivity. In special channels, including many highly symmetric channels, the one-use expression is additive and the limit reduces to χ(𝒩).

A classical code uses a set of codewords and a measurement at the receiver. The quantum analogue of Shannon’s coding theorem states, informally, that rates below the appropriate regularized Holevo quantity can be achieved with vanishing error, while rates above it cannot be achieved reliably.

For a noiseless d-dimensional quantum channel, orthogonal states can encode log₂ d classical bits, so C = log₂ d bits per use. For a noiseless qubit channel, C = 1 bit per use without entanglement assistance. With shared entanglement, the same qubit channel can transmit 2 classical bits per use, as shown by superdense coding.

4. Quantum capacity

Quantum capacity measures the rate at which unknown quantum states can be transmitted with arbitrarily high fidelity. It is expressed in qubits per channel use. The task is stronger than sending classical labels: the protocol must preserve superposition and entanglement with an external reference.

For a channel 𝒩, the quantum capacity is characterized by the Lloyd–Shor–Devetak theorem:

Here ρ_n is an input state over n channel uses. The regularization is important: coherent information may be superadditive, and a single-use maximization does not always give the exact capacity.

The coding theorem has two parts. The direct part states that rates below the regularized coherent information are achievable using quantum error-correcting codes. The converse states that rates above the quantum capacity cannot be achieved with vanishing error.

For a noiseless qubit channel, the receiver obtains the input without disturbance, and Q = 1 qubit per use. For a completely depolarizing channel that replaces every input with the maximally mixed state, Q = 0 because the output contains no information about the input.

Erasure channel

A qubit erasure channel transmits the input unchanged with probability 1 − ε and replaces it with an orthogonal erasure flag with probability ε. Its quantum capacity is

for 0 ≤ ε ≤ 1. Thus quantum communication is possible only when ε < 1/2. The threshold is stricter than the classical threshold because preserving quantum information requires dealing with both the erased and unerased branches coherently.

The classical unassisted capacity of the d-dimensional erasure channel is

because the receiver knows which transmissions were erased and can use the unerased fraction. For a qubit, C = 1 − ε bits per use, while Q = max{0,1 − 2ε} qubits per use.

5. Private classical capacity

A private communication protocol sends classical information to the legitimate receiver while keeping it hidden from an eavesdropper. Let B denote the receiver’s system and E the environment controlled, in principle, by an adversary. The private information of an ensemble is

where I(X:B) and I(X:E) are quantum-classical mutual informations. The first term measures information available to the receiver; the second measures information leaked to the environment.

The private capacity is generally

The regularization reflects possible advantages from correlated, multi-use encodings. Private capacity is not identical to classical capacity: a channel may transmit many classical bits while leaking almost all of them to the environment, resulting in low private capacity.

Private communication is closely related to quantum communication. Coherent versions of private classical protocols can generate entanglement, and privacy amplification can convert partially secret correlations into secure keys. However, private capacity, quantum capacity, and secret-key capacity are distinct operational quantities and should not be substituted for one another.

6. Entanglement-assisted classical capacity

Suppose sender and receiver share unlimited maximally entangled states before communication begins. The entanglement is a resource, not transmitted through the noisy channel. The entanglement-assisted classical capacity is

where ψ_RA is a purification of ρ_A, R is the reference system, and B is the channel output. Equivalently,

The quantum mutual information here measures total correlations between the retained reference and the receiver’s output.

Unlike ordinary classical capacity, C_E is single-letter: no regularization is required. Entanglement-assisted coding allows dense coding and related protocols. For a noiseless qubit channel, C_E = 2 bits per use. This does not violate relativity or create free information: one shared ebit must be available for each use, and establishing that entanglement has a physical cost.

Entanglement assistance can also improve robustness against noise. For a qubit erasure channel,

bits per use. The factor of 2 comes from the two classical bits that can be conveyed by a noiseless qubit when entanglement is shared.

7. Relationships among capacities

The capacities answer different operational questions. A useful comparison is:

General resource inequalities include

where these inequalities express broad operational relationships, although equality or strict separation depends on the channel and resource definitions. In particular, private capacity can exceed quantum capacity, and positive private capacity does not automatically imply a simple positive one-use coherent information.

The quantum capacity of a channel is also bounded by the entanglement-assisted capacity in an appropriate resource sense. Entanglement assistance cannot reduce the optimal classical rate, but it changes the communication model and therefore must be stated explicitly in any capacity claim.

8. Important channel models

Depolarizing channel

A qubit depolarizing channel applies the identity with probability 1 − p and replaces the state with a uniformly random Pauli error with total probability p. One convention is

where X, Y, and Z are Pauli operators and 0 ≤ p ≤ 1. Different texts use different parameterizations, so the definition of p must always be specified. Increasing p reduces distinguishability and generally reduces all unassisted capacities.

Exact quantum capacity is difficult for general depolarizing noise. Numerical estimates, symmetry reductions, degradability bounds, and quantum error-correction thresholds are therefore important in practice. A fault-tolerance threshold is not the same as a channel capacity: a threshold concerns scalable error correction under a specified architecture, while capacity is an asymptotic communication rate for a channel model.

Amplitude-damping channel

Amplitude damping models energy relaxation, such as spontaneous emission. With damping probability γ, Kraus operators may be chosen as

and the channel is 𝒩(ρ) = K₀ρK₀† + K₁ρK₁†. Here 0 ≤ γ ≤ 1. Unlike depolarizing noise, amplitude damping is nonunital: it does not preserve the maximally mixed state. Its asymmetry makes code design and capacity analysis more dependent on the input ensemble.

Degradable channels

A channel is degradable if its environment output can be simulated by applying another channel to the receiver’s output. For degradable channels, coherent information is additive and

This removes the difficult regularization. The qubit erasure channel is degradable for ε ≤ 1/2 and antidegradable for ε ≥ 1/2. An antidegradable channel has Q = 0 because the environment can reproduce the receiver’s output well enough to prevent reliable quantum transmission.

9. Worked example: comparing an erasure channel’s capacities

Consider a qubit erasure channel with ε = 0.20.

The unassisted classical capacity is

The quantum capacity is

The entanglement-assisted classical capacity is

The results are not contradictory. They concern different tasks:

• 0.80 bits/use describes ordinary classical messages without prior entanglement.
• 0.60 qubits/use describes preservation of arbitrary quantum states.
• 1.60 bits/use assumes unlimited pre-shared entanglement.

If a protocol uses 10,000 channel transmissions, the asymptotic ideal rates suggest approximately 8,000 ordinary classical bits, 6,000 qubits, or 16,000 entanglement-assisted classical bits, subject to finite-block and implementation losses.

10. Applied case: a metropolitan quantum link

Background

This illustrative case concerns a research network connecting two laboratories through a fibre-optic link. The link is intended to support both conventional authenticated data and quantum-key-distribution experiments. Photon loss is modelled approximately as an erasure process, although real devices also exhibit detector dark counts, mode mismatch, and memory effects.

Problem

The link has an estimated erasure probability of ε = 0.15 per transmitted photonic qubit. Engineers must decide whether to prioritize ordinary classical telemetry, entanglement-assisted communication, or secret-key generation. They must also avoid claiming that a classical capacity estimate guarantees secure key production.

Analysis

For a simplified qubit erasure model,

These are ideal asymptotic values. The 0.70-qubit rate is relevant only if the system can implement suitable quantum error correction and preserve the required coherence. The 1.70-bit rate requires pre-shared entanglement, which itself must be generated, stored, verified, and protected.

For secure key generation, the team must characterize the environment and adversarial information, not merely count detected photons. A high receiver rate with substantial leakage can produce low or zero secret-key rate. Privacy amplification and authentication consume resources and reduce the final operational rate.

Approach and outcome

The engineers select a layered architecture. Classical authenticated channels carry synchronization, calibration, and control traffic. Quantum transmissions are reserved for key-establishment experiments. Loss statistics are monitored over time, and finite-key analysis is used rather than relying solely on asymptotic formulas. Entanglement-assisted protocols are tested only after an entanglement-distribution budget has been included.

The outcome is a realistic performance report containing channel estimates, confidence intervals, detector assumptions, finite-size penalties, and separate rates for raw detections, corrected bits, and final secret bits.

Learning points

1. A capacity is meaningful only after the communication task and assistance assumptions are stated.
2. Loss alone does not determine a practical secret-key rate.
3. Ideal capacity formulas guide architecture but do not replace device characterization.
4. Classical control communication remains essential in quantum networks.

11. Coding, implementation, and engineering practice

Capacity theorems are asymptotic, whereas engineering systems use finite blocks. Practical design therefore requires:

• selecting a code family appropriate to the channel and hardware;
• estimating channel parameters from calibration data;
• balancing block length against latency and memory;
• measuring logical error rates rather than only physical error rates;
• using entanglement purification or quantum error correction when appropriate;
• accounting for decoder complexity and energy consumption;
• validating assumptions such as memorylessness and independent channel uses.

For classical information carried by quantum states, receiver design may involve minimum-error discrimination, maximum-likelihood decoding, or collective measurements. For quantum information, error correction must protect both populations and phases. A code that corrects bit flips but not phase errors may have poor quantum performance even if its classical performance appears strong.

Capacity-achieving sequences need not be practical at moderate block lengths. There can be a large gap between the asymptotic capacity and the rate achieved with available photons, memories, detectors, and computational resources. Performance should therefore be reported using block length, fidelity, error probability, latency, and resource overhead.

12. Security, ethics, and governance

Quantum communication is often associated with security, but quantum mechanics does not automatically secure an entire system. Security depends on protocol assumptions, authentication, source characterization, detector behaviour, side-channel resistance, and correct implementation.

Important governance practices include:

• defining the adversary model explicitly;
• distinguishing information-theoretic security from computational security;
• documenting trusted and untrusted devices;
• protecting classical authentication keys;
• auditing randomness generation and parameter estimation;
• avoiding unsupported claims such as “quantum means unhackable”;
• evaluating whether deployment increases surveillance or creates unequal access to secure infrastructure.

Quantum key distribution can detect certain forms of interception under its assumptions, but it does not prevent endpoint compromise or denial-of-service attacks. Capacity analysis should therefore be integrated with risk assessment rather than treated as a complete security evaluation.

13. Contemporary developments

Current research seeks tighter capacity bounds and more usable communication systems. Active areas include quantum repeaters, satellite links, bosonic channels, continuous-variable communication, network capacities, finite-blocklength theory, and fault-tolerant transduction between matter and light.

For optical channels, energy constraints matter. A bosonic channel has an infinite-dimensional input space, so a capacity statement must specify an average-energy or photon-number constraint. Without such a constraint, formal capacities can become physically meaningless. Continuous-variable systems also require careful treatment of detector noise, modulation, and security assumptions.

Quantum network theory extends point-to-point capacities to multiple senders, receivers, repeaters, and network coding tasks. In a network, bottleneck links, memory coherence time, routing, and entanglement-generation probability can dominate the end-to-end rate. The capacity of each link alone is insufficient to predict network performance.

Artificial intelligence can assist channel estimation, adaptive measurement selection, decoder design, and device calibration. However, machine-learning performance does not replace a proof of security or a capacity theorem. Training data, distribution shift, interpretability, and adversarial manipulation must be considered when AI is used in a communication control loop.

Chapter Summary

A quantum channel is a CPTP transformation that maps input density operators to output states. Communication capacities quantify asymptotic reliable rates, but each capacity corresponds to a different resource and task. The Holevo information bounds classical information extractable from a quantum ensemble and leads to the regularized classical capacity. Quantum capacity concerns faithful transmission of unknown states and is characterized by regularized coherent information. Private capacity measures receiver information minus environmental leakage, while entanglement-assisted classical capacity is determined by quantum mutual information and can be substantially larger than unassisted classical capacity.

Erasure channels provide a clear comparison: for a qubit erasure probability ε, the unassisted classical capacity is 1 − ε, the quantum capacity is max{0,1 − 2ε}, and the entanglement-assisted classical capacity is 2(1 − ε). These formulas apply to the stated ideal memoryless model and do not automatically describe finite experimental systems.

Practical communication requires coding, parameter estimation, finite-block analysis, hardware characterization, and explicit security assumptions. Capacity is a powerful benchmark, but it is not itself a complete engineering specification, security proof, or guarantee of deployment performance.

Key Terms

Quantum channel: A physically valid CPTP map from input quantum states to output quantum states.

CPTP map: A completely positive, trace-preserving quantum operation.

Density operator: A positive semidefinite, unit-trace operator representing a quantum state.

Von Neumann entropy: The quantity S(ρ) = −Tr(ρ log₂ρ) measuring quantum uncertainty.

Holevo information: The difference between the entropy of an average state and the average entropy of signal states.

Holevo bound: The upper bound on classical information obtainable from a quantum ensemble.

Accessible information: The maximum mutual information obtainable by measurements on a quantum ensemble.

Classical capacity: The maximum reliable classical-message rate of a channel.

Quantum capacity: The maximum reliable rate for transmitting unknown quantum states.

Private capacity: The maximum rate for transmitting classical information while limiting environmental leakage.

Entanglement-assisted capacity: Classical capacity when unlimited prior shared entanglement is available.

Coherent information: I_c(A⟩B) = S(B) − S(AB), an important quantity for quantum coding.

Quantum mutual information: I(A:B) = S(A) + S(B) − S(AB).

Erasure channel: A channel that either transmits a state or produces a detectable erasure flag.

Depolarizing channel: A channel that applies random Pauli noise or replaces information with a highly mixed state.

Amplitude damping: Noise associated with energy relaxation and spontaneous emission.

Degradable channel: A channel whose environment output can be simulated from the receiver output.

Regularization: Optimization over many channel uses followed by division by the number of uses.

Ebit: One shared maximally entangled pair used as an entanglement resource.

Finite-blocklength effect: The difference between asymptotic capacity and performance at practical block sizes.

Review Questions

Short Answer Questions

1. What conditions must a quantum channel satisfy to be physically valid?
2. State the Holevo bound and explain its operational meaning.
3. Define coherent information in terms of von Neumann entropies.
4. Why can quantum conditional entropy be negative?
5. What resource is assumed in entanglement-assisted classical capacity?

Long Answer Questions

1. Explain the difference between classical capacity, quantum capacity, private capacity, and entanglement-assisted classical capacity.
2. Describe why regularization appears in the classical and quantum capacity formulas.
3. Compare the communication properties of depolarizing, amplitude-damping, and erasure channels.
4. Explain how the environment affects private communication and quantum communication.
5. Discuss why an asymptotic capacity is not a complete specification of a practical quantum communication system.

Analytical / Application Questions

1. A qubit erasure channel has ε = 0.30. Calculate its ideal unassisted classical capacity, quantum capacity, and entanglement-assisted classical capacity, and interpret the differences.
2. A proposed quantum link reports a high receiver detection rate but provides no model of environmental leakage. Explain why this evidence is insufficient to establish a high private capacity.
3. Design a measurement and reporting plan for a metropolitan quantum link that distinguishes theoretical capacity, finite-block performance, security parameters, and hardware limitations.

Multiple Choice Questions with Answers

Question 1

Which condition ensures that a quantum operation preserves total probability?

A. The Kraus operators satisfy ∑_j K_j†K_j = I

B. The density operator has zero trace

C. All input states are pure

D. The channel output has maximum entropy

Correct Answer: A. The Kraus operators satisfy ∑_j K_j†K_j = I

Question 2

For an ensemble of quantum states, what does the Holevo bound limit?

A. The energy used by the transmitter

B. The accessible classical information obtained by measurement

C. The number of qubits stored in the environment

D. The fidelity of every individual signal state

Correct Answer: B. The accessible classical information obtained by measurement

Question 3

What is the coherent information I_c(A⟩B) for a bipartite state ρ_AB?

A. S(A) + S(B)

B. S(AB) − S(B)

C. S(B) − S(AB)

D. S(A) − S(B) − S(AB)

Correct Answer: C. S(B) − S(AB)

Question 4

For a qubit erasure channel with erasure probability ε, what is the ideal unassisted classical capacity?

A. ε bits/use

B. 1 − ε bits/use

C. 1 − 2ε bits/use

D. 2(1 − ε) qubits/use

Correct Answer: B. 1 − ε bits/use

Question 5

For a qubit erasure channel, at what condition does the quantum capacity become zero?

A. ε = 0

B. ε < 1/4

C. ε ≥ 1/2

D. ε = 1/10

Correct Answer: C. ε ≥ 1/2

Question 6

What additional resource is required for entanglement-assisted classical capacity?

A. A noiseless classical feedback channel only

B. Unlimited prior shared entanglement

C. A perfectly pure environment

D. A depolarizing encoder

Correct Answer: B. Unlimited prior shared entanglement

Question 7

Why is regularization used in general capacity formulas?

A. To convert bits into qubits

B. Because multiple channel uses can provide advantages over independent single-use optimization

C. Because entropy is always negative

D. To remove the need for decoding

Correct Answer: B. Because multiple channel uses can provide advantages over independent single-use optimization

Question 8

For ε = 0.20, what is the quantum capacity of an ideal qubit erasure channel?

A. 0.20 qubits/use

B. 0.40 qubits/use

C. 0.60 qubits/use

D. 0.80 qubits/use

Correct Answer: C. 0.60 qubits/use

Question 9

Private information is generally expressed as which difference?

A. Environment information minus receiver information

B. Receiver information minus environment information

C. Classical capacity minus Hilbert-space dimension

D. Quantum capacity plus environmental entropy

Correct Answer: B. Receiver information minus environment information

Question 10

What is a key practical limitation of asymptotic capacity results?

A. They apply only to classical computers

B. They ignore the distinction between bits and qubits

C. Finite block lengths and hardware imperfections reduce achievable rates

D. They require every channel to be noiseless

Correct Answer: C. Finite block lengths and hardware imperfections reduce achievable rates