Quantum Mathematics Learning Guide 32: Quantum Shannon Theory, Holevo Bounds, Channel Capacities and Coding Theorems

Quantum Shannon theory asks how rapidly information can be transmitted or protected when the carriers themselves obey quantum mechanics. Its central objects are not single messages but asymptotic rates, coding theorems and converse bounds.

Classical Shannon theory separates source, channel, code and decoding rule. Quantum Shannon theory keeps that architecture but changes the objects: messages can be encoded into nonorthogonal quantum states, channels are completely positive maps, receivers may perform joint measurements across many outputs, and entanglement can be a resource or part of the information being transmitted.

This guide develops the Holevo bound, the Holevo–Schumacher–Westmoreland coding theorem, regularised channel capacities, entanglement-assisted classical capacity and quantum capacity. The emphasis is on what each capacity means operationally and which assumptions are hidden in a compact formula.

Message ensemble → quantum channel → block code → collective decoding → achievable rate → converse bound → channel capacity.

1. Classical information first

For a classical random variable X with probabilities p(x), Shannon entropy is

H(X)=−Σ_x p(x)log₂p(x).

For jointly distributed X and Y, mutual information is

I(X:Y)=H(X)+H(Y)−H(X,Y).

It quantifies how much observing Y reduces uncertainty about X on average.

2. A classical-quantum ensemble

Suppose a sender chooses classical symbol x with probability p_x and encodes it into quantum state ρ_x.

The ensemble is {p_x,ρ_x}. Its average state is

ρ_bar=Σ_x p_xρ_x.

A receiver performs a measurement with classical outcome Y. Different measurements produce different conditional distributions p(y|x), and therefore different classical mutual informations I(X:Y).

3. Accessible information

The accessible information of an ensemble is the maximum classical mutual information obtainable over all allowed measurements on the received quantum system:

I_acc=max_measurements I(X:Y).

If the signal states are nonorthogonal, they cannot all be perfectly distinguished. Accessible information can therefore be smaller than the Shannon entropy H(X) of the message source.

The receiver’s limitation is not lack of clever classical processing after measurement; it is the geometry of the quantum states before measurement.

4. Holevo information

For ensemble {p_x,ρ_x}, define

χ=S(ρ_bar)−Σ_x p_xS(ρ_x),

where S(ρ)=−Tr(ρlog₂ρ) is von Neumann entropy.

Holevo’s theorem gives

I_acc≤χ.

This is the Holevo bound. It limits the classical information extractable from a quantum ensemble by measurement. [1]

5. Pure-state ensembles simplify χ

If every signal state ρ_x=|ψ_x⟩⟨ψ_x| is pure, then each S(ρ_x)=0. Therefore

χ=S(ρ_bar).

The information limit is determined entirely by how mixed the average ensemble becomes.

If the pure states are mutually orthogonal and used with probabilities p_x, ρ_bar has eigenvalues p_x, so χ=H(X). Orthogonal quantum codewords can therefore carry their full classical source entropy.

6. Worked nonorthogonal binary ensemble

Send |0⟩ or |+⟩ with equal probability 1/2. Their overlap magnitude is

|⟨0|+⟩|=1/√2.

The average state has eigenvalues

λ_±=(1±1/√2)/2.

Because the signals are pure,

χ=H₂((1+1/√2)/2)≈0.600876 bits.

The source contains one classical bit of entropy per symbol, but no measurement on one copy can extract more than about 0.6009 bits of mutual information on average from this ensemble.

7. Holevo bound is not a claim that every qubit carries at most one useful bit in every resource setting

An unassisted noiseless qubit channel has classical capacity one bit per channel use. But superdense coding uses pre-shared entanglement plus one transmitted qubit to communicate two classical bits.

The resource models differ. Holevo analysis must include the full system available to the decoder and any pre-shared entanglement explicitly.

A capacity statement without a resource model is incomplete.

8. A quantum channel

A memoryless quantum channel 𝒩 maps input density matrices to output density matrices. For classical communication, the sender chooses an ensemble {p_x,ρ_x} and the receiver sees {p_x,𝒩(ρ_x)}.

The one-use Holevo information is

χ({p_x,𝒩(ρ_x)})=S(Σ_xp_x𝒩(ρ_x))−Σ_xp_xS(𝒩(ρ_x)).

Optimising over input ensembles gives a one-use Holevo quantity often denoted χ*(𝒩).

9. HSW coding theorem

The Holevo–Schumacher–Westmoreland theorem shows that rates up to the ensemble Holevo information are asymptotically achievable for classical communication through a memoryless quantum channel using long block codes and suitable collective decoding. [2,3]

The proof does not decode each output symbol independently. Codewords extend across many channel uses, and the receiver may perform one joint quantum measurement on the entire output block.

This collective measurement is one reason one-shot accessible information and asymptotic communication rate should not be conflated.

10. Coding theorem structure

An achievability theorem has four conceptual stages:

• draw codewords from an input ensemble;
• show typical codewords produce output states concentrated in typical subspaces;
• construct a collective measurement capable of distinguishing sufficiently separated codewords;
• show average decoding error tends to zero when the communication rate lies below the stated information quantity.

A converse theorem shows that rates above capacity cannot achieve vanishing error under the same resource model.

11. Why regularisation appears

For a general quantum channel, entangled input codewords across multiple channel uses can make the optimised Holevo quantity nonadditive.

The unrestricted unassisted classical capacity is therefore written

C(𝒩)=lim_{n→∞}(1/n)χ*(𝒩^⊗n).

This regularisation means one cannot universally replace the capacity by a simple one-letter optimisation over one channel use.

For many important channels, additivity or symmetry simplifies the expression, but that is additional structure.

12. Noiseless qubit classical capacity

A noiseless qubit channel transmits a two-dimensional quantum system perfectly.

Choose orthogonal code states |0⟩ and |1⟩ with equal probabilities. The average state is I/2 with entropy one bit; individual states are pure. Thus χ=1.

No unassisted ensemble can exceed log₂2=1 bit per channel use because the output system has dimension two. Therefore C=1 classical bit per noiseless qubit use without pre-shared entanglement.

13. Entanglement-assisted classical capacity

Now suppose sender and receiver share unlimited prior entanglement that is not counted as channel use.

The entanglement-assisted classical capacity is

C_E(𝒩)=max_ρ I(R:B)

where |ψ⟩_RA purifies input ρ_A, the A system passes through 𝒩 to B, and I(R:B) is the quantum mutual information of the resulting state. [4]

This formula is single-letter and additive.

14. Dense coding recovered as a capacity example

For the noiseless qubit identity channel, choose maximally mixed input ρ=I/2 with a Bell-state purification.

The output joint state RB is pure and maximally entangled, so S(R)=S(B)=1 and S(RB)=0.

Therefore

I(R:B)=1+1−0=2 bits.

Thus C_E=2 bits per noiseless qubit channel use, exactly matching superdense coding with one ebit consumed or supplied per use in the asymptotic resource model.

15. Quantum capacity is a different task

Classical capacity asks how many classical bits can be transmitted reliably. Quantum capacity asks how many unknown qubits—or equivalently how much entanglement—can be transmitted reliably per channel use.

A measurement-and-reprepare channel can have nonzero classical capacity while having zero quantum capacity because measurement destroys arbitrary quantum coherence.

No-cloning makes quantum coding fundamentally different from classical repetition.

16. Coherent information

For input purification |ψ⟩_RA and channel output state ρ_RB, coherent information can be written

I_c(R⟩B)=S(B)−S(RB)=−S(R|B).

Equivalently, using a Stinespring dilation with environment E and globally pure RBE,

I_c=S(B)−S(E).

Positive coherent information means the receiver retains more relevant quantum information than leaked to the environment in this entropy sense.

17. Quantum capacity requires regularisation

For a general memoryless channel, quantum capacity is given by a regularised coherent-information optimisation:

Q(𝒩)=lim_{n→∞}(1/n) max_ρ I_c(ρ,𝒩^⊗n).

Coherent information can be superadditive, so one channel use may not reveal the ultimate asymptotic rate. [5]

For degradable channels, the formula single-letters: one-use coherent information suffices.

18. Erasure channel as a worked capacity family

A d-dimensional quantum erasure channel transmits the input perfectly with probability 1−p and replaces it by an orthogonal erasure flag with probability p.

The unassisted classical capacity is

C=(1−p)log₂d.

The quantum capacity is

Q=max(0,1−2p)log₂d.

The entanglement-assisted classical capacity is

C_E=2(1−p)log₂d.

For p≥1/2 the channel still carries classical information when not erased, but its quantum capacity vanishes because the environment obtains too much complementary information for reliable unknown-state transmission.

19. Entanglement-breaking channels

A channel is entanglement breaking if applying it to one half of any bipartite input always produces a separable output.

Such channels have zero quantum capacity because they cannot preserve entanglement with a reference system.

They may still have nonzero classical capacity. A classical communication channel implemented by measure-and-prepare operations is the clearest example.

20. Data processing

Processing the receiver’s output through another channel cannot increase correlations with the original reference beyond the appropriate information measure.

For quantum mutual information,

I(R:B')≤I(R:B)

when B→B’ is a quantum channel.

This is the information-theoretic form of the principle that local processing cannot recover information already irreversibly discarded.

21. Joint decoding can outperform symbol-by-symbol decoding

A receiver who measures each channel output separately commits early to classical data and may destroy correlations useful for discriminating entire codewords.

Collective decoding across a long block can distinguish codewords using the geometry of the joint tensor-product states.

This is one of the deepest operational lessons of quantum Shannon theory: channel capacity can be an emergent block property rather than a simple sum of the best one-shot measurements.

22. Finite blocklength

Capacity is an asymptotic first-order rate as blocklength tends to infinity and error is driven toward a specified limit.

Real communication systems use finite blocklength. The achievable rate then depends on allowed error probability, channel dispersion-like second-order terms and code construction.

A capacity of one bit per use does not mean ten channel uses can always carry exactly ten bits with zero error under every coding constraint.

23. Strong converse questions

A weak converse says rates above capacity cannot achieve vanishing error. A strong converse says that above capacity the decoding error tends to one.

Whether a strong converse holds depends on the channel, capacity notion and assistance resources.

Do not attach “error goes to one” automatically to every capacity formula without checking the theorem.

24. Private classical capacity

A channel may be used to send classical information reliably to Bob while keeping it secret from the environment or an eavesdropper holding the complementary channel output.

Private capacity optimises a difference between information delivered to Bob and information leaked to Eve, again with regularisation in the general case.

Private classical capacity and quantum capacity are closely related through decoupling and coherent-information methods, but they are not identical operational tasks.

25. Resource inequalities connect the capacities

Teleportation and dense coding can be written schematically as resource conversions:

1 qubit channel + 1 ebit ≥ 2 classical bits

and

2 classical bits + 1 ebit ≥ 1 qubit channel

in the ideal asymptotic sense with the appropriate direction and resource accounting.

Quantum Shannon theory systematises such conversions among noiseless and noisy communication resources.

26. Common misconception: Holevo χ is always the measured information

χ is an upper bound on accessible information for a fixed ensemble and becomes an achievable asymptotic coding rate in the HSW setting. One-shot accessible information can be strictly smaller.

27. Common misconception: classical capacity is always one-letter

For general quantum channels, unrestricted classical capacity uses regularisation because entangled input codewords across several uses can affect the optimised Holevo quantity.

28. Common misconception: quantum capacity counts how many classical bits a quantum channel carries

Quantum capacity measures reliable transmission of quantum information or entanglement. Classical capacity is a different operational quantity and can be positive when quantum capacity is zero.

29. Worked synthesis problem

An equal-probability sender uses |0⟩ and |+⟩ as classical signal states.

Step 1: Average state.

ρ_bar=(|0⟩⟨0|+|+⟩⟨+|)/2.

Step 2: Spectrum. The two eigenvalues are (1±1/√2)/2.

Step 3: Individual entropy. Both signal states are pure, so their entropies vanish.

Step 4: Holevo information.

χ=H₂((1+1/√2)/2)≈0.600876 bits.

Step 5: Interpretation. Although the classical label X has entropy one bit, the nonorthogonal quantum encoding limits recoverable classical information. Block coding may approach the Holevo rate for the specified ensemble, but no measurement extracts one full bit per independently encoded signal from this ensemble.

30. Practice set

1. Define a classical-quantum ensemble.
2. What is accessible information?
3. Write Holevo χ.
4. State the Holevo bound.
5. How does χ simplify for pure signal states?
6. Why can collective measurements matter for channel coding?
7. What does HSW establish operationally?
8. Why does unrestricted classical capacity require regularisation in general?
9. What is the unassisted classical capacity of a noiseless qubit channel?
10. What is its entanglement-assisted classical capacity?
11. Define coherent information.
12. Why can a channel have positive classical capacity but zero quantum capacity?

Answers

1. A probability distribution over quantum signal states {p_x,ρ_x}.
2. The maximum classical mutual information between the label and a measurement outcome over allowed receiver measurements.
3. S(Σp_xρ_x)−Σp_xS(ρ_x).
4. I_acc≤χ.
5. χ is just the entropy of the average state.
6. Measuring outputs separately can discard quantum distinctions that remain accessible to a joint block measurement.
7. Rates up to the relevant Holevo information are asymptotically achievable with suitable coding and collective decoding.
8. The optimised Holevo quantity need not be additive across channel uses.
9. One classical bit per use.
10. Two classical bits per use with unlimited pre-shared entanglement.
11. I_c=S(B)−S(RB)=−S(R|B) for a purified input.
12. A measure-and-prepare or entanglement-breaking channel can transmit classical symbols while destroying arbitrary quantum coherence and entanglement.

Sources and further study

[1] A. S. Holevo, Bounds for the Quantity of Information Transmitted by a Quantum Communication Channel, Problems of Information Transmission 9, 177–183 (1973). The Holevo bound on accessible classical information.

[2] Benjamin Schumacher and Michael D. Westmoreland, Sending classical information via noisy quantum channels, Physical Review A 56, 131 (1997). An HSW coding theorem treatment for noisy quantum channels.

[3] A. S. Holevo, The Capacity of the Quantum Channel with General Signal States. The general mixed-state classical-quantum capacity result completing the HSW picture.

[4] Charles H. Bennett, Peter W. Shor, John A. Smolin and Ashish V. Thapliyal, Entanglement-Assisted Classical Capacity of Noisy Quantum Channels. The entanglement-assisted capacity formula in terms of quantum mutual information.

[5] I. Devetak, The private classical capacity and quantum capacity of a quantum channel. A modern route to quantum-capacity and private-capacity coding theorems using coherent information.

Batch 08 series navigation

• Guide 29: Bell Inequalities, CHSH, Tsirelson Bounds and No-Signalling
• Guide 30: Quantum Teleportation, Superdense Coding and Entanglement Swapping
• Guide 31: Entanglement Transformations, LOCC, Majorisation and Distillation
• Guide 32: Quantum Shannon Theory, Holevo Bounds, Channel Capacities and Coding Theorems
• Return to the BTT Mathematics Learning Hub

Educational note: channel-capacity formulas are resource- and model-dependent. Unassisted, entanglement-assisted, private and quantum capacities answer different operational questions and may require regularisation or additional channel assumptions.