Quantum Channel Capacities

When Shannon defined the capacity of a classical channel in 1948, there was only one number to define. The channel p(y | x) takes classical inputs to classical outputs, and the capacity C = \max_{p(x)} I(X; Y) is the single rate at which information can be transmitted reliably. One channel, one capacity, one clean formula.

A quantum channel \mathcal{E} breaks this unity. It takes quantum states to quantum states, and what is being transmitted through it can be classical bits, or qubits, or a classical message with entanglement assistance, or a secret key, or any of several other things. Each task has its own notion of "rate." Each rate has its own capacity. The quantum channel therefore has a zoo of capacities, each measuring a different resource-to-resource conversion the channel can support.

This chapter surveys the three most important ones — classical C, quantum Q, and entanglement-assisted classical C_E — and shows how they relate. You will see the hierarchy Q \leq C \leq C_E, the three major theorems that identify these capacities with single-letter or regularized formulas, and the depolarizing channel worked out end-to-end as a concrete example.

The three capacities — definitions

Fix a quantum channel \mathcal{E}: \mathcal{D}(\mathcal{H}_A) \to \mathcal{D}(\mathcal{H}_B), memoryless, used n times.

Classical capacity C(\mathcal{E})

The classical capacity is the largest rate R (classical bits per channel use) for which there exists a sequence of codes with M = 2^{nR} messages, block length n, and maximum decoding error \to 0 as n \to \infty. The input codewords are quantum states (products or entangled across uses); the decoder is a quantum measurement.

By the HSW theorem (hsw-theorem),

the regularized one-shot Holevo capacity. For additive channels (depolarizing, erasure, entanglement-breaking) the single-letter \chi(\mathcal{E}) equals C(\mathcal{E}); for generic channels (Hastings 2009) the regularization is strict.

Quantum capacity Q(\mathcal{E})

The quantum capacity is the largest rate R (qubits per channel use) for which there exists a sequence of encodings of k = nR qubits into n-use codewords such that Bob can decode the k qubits with fidelity \to 1. The input codewords are entangled-in-general; the decoder is a quantum operation (not just a measurement).

By the LSD theorem (Lloyd 1997, Shor 2002, Devetak 2003),

the regularized coherent information. The coherent information is

where |\psi^\rho\rangle is any purification of \rho. The second term is the entropy of the environment after the channel action — equivalently, the entropy of the complementary channel's output.

Coherent information is notoriously non-additive: Smith and Yard (2008) found channels where I_c(\mathcal{E}_1 \otimes \mathcal{E}_2) > I_c(\mathcal{E}_1) + I_c(\mathcal{E}_2) in a dramatic way — two individually zero-quantum-capacity channels can, when used together, transmit qubits. Regularization is essential; Q is typically not single-letter.

Entanglement-assisted classical capacity C_E(\mathcal{E})

The entanglement-assisted classical capacity is the classical-bit rate when Alice and Bob share an unlimited supply of pre-existing entanglement (an infinite reservoir of Bell pairs, for free). Alice uses the channel n times and consumes some of the ebits. The best rate achievable is C_E(\mathcal{E}).

By the BSST theorem (Bennett-Shor-Smolin-Thapliyal 1999, 2002),

where \Phi^\rho_{AA'} is a purification of \rho_A (shared between Alice's reference A and the channel's input A'), and I(A:B) is the quantum mutual information of the state after the channel acts on A' alone.

This formula is single-letter — no regularization needed. Quantum mutual information is additive across tensor-product states, which forces C_E(\mathcal{E}^{\otimes n}) = n\, C_E(\mathcal{E}). BSST is thus the cleanest of the three capacity theorems: one ensemble, one closed-form expression, no asymptotic limits.

The hierarchy

For any quantum channel \mathcal{E},

where d_A is the input dimension. Additionally Q(\mathcal{E}) \leq \log_2 d_A (no more qubits out than in, up to fidelity). The left inequality Q \leq C is easy: any qubit-transmission protocol can be used to send classical bits (encode a classical bit by sending |0\rangle or |1\rangle). The middle inequality C \leq C_E is obvious (entanglement is a free extra resource, can only help). The upper bound C_E \leq 2\log_2 d_A is the entanglement-assisted Holevo bound — twice the plain Holevo ceiling, reflecting the fact that superdense coding achieves 2 classical bits per qubit with pre-shared entanglement.

HSW for C — recap

The HSW theorem identifies C(\mathcal{E}) with the regularized Holevo quantity. The hsw-theorem chapter spells out the proof in full; here is the one-paragraph summary needed for this survey.

Achievability: for any input ensemble \{p_x, \rho_x\}, product-state block codewords \rho_{x_1} \otimes \cdots \otimes \rho_{x_n} with a joint pretty-good measurement decoder achieve rate \chi(\{p_x, \mathcal{E}(\rho_x)\}) with vanishing error. Maximising over ensembles gives C \geq \chi(\mathcal{E}).

Converse: Holevo bound on the block channel gives nR \leq \chi(\mathcal{E}^{\otimes n}) for any code, so C \leq \chi^*(\mathcal{E}).

The two halves squeeze to C(\mathcal{E}) = \chi^*(\mathcal{E}). For channels where \chi is additive, C = \chi(\mathcal{E}) is single-letter.

LSD theorem for Q — the coherent-information capacity

The quantum-capacity story is more subtle than the classical one. The quantum analogue of "Alice's message" is a quantum state — a qubit, or a register of qubits — that Alice wants to transmit intact. Success is measured by fidelity between the input state and Bob's reconstructed output. Noise in the channel entangles the input with the environment; recovering the state requires disentangling from the environment.

Coherent information

The key quantity is the coherent information [coherent-information]:

where S(\rho, \mathcal{E}) = S((\mathcal{E} \otimes I_R)(|\psi^\rho\rangle\langle\psi^\rho|)) is the entropy exchange — the entropy of the channel's environment after acting on a purification of \rho. Equivalently, S(\rho, \mathcal{E}) = S(\mathcal{E}^c(\rho)), where \mathcal{E}^c is the complementary channel that produces the environment's state.

Why coherent information is the right thing to maximise: Schumacher's 1996 insight was that I_c measures the decoupling between the environment and the transmitted information. High I_c means the environment knows little about the encoded state, so Bob can in principle disentangle and decode. Low I_c (in particular I_c = 0) means the environment has captured enough information that Bob cannot reconstruct the state faithfully. The rate at which qubits can be transmitted is exactly the rate at which the environment's information about them stays small — which is I_c.

LSD theorem statement

The theorem has a long history: Lloyd (1997) gave a heuristic achievability argument; Shor (2002) tightened it; Devetak (2003) gave the definitive proof of both achievability and converse. The three-author attribution ("LSD") honours the cumulative contribution.

Superadditivity of coherent information — Smith-Yard 2008

Coherent information is spectacularly non-additive. Smith and Yard (2008) [2] gave an explicit construction where

Two individually useless channels combined into a useful quantum channel. This is the superactivation phenomenon: neither channel alone can transmit any qubits (zero quantum capacity each), yet combined they can transmit a positive rate. Nothing like this exists in classical information theory — classically, if two channels have zero capacity, so does their product.

Superactivation forces regularization in the LSD formula and makes computing Q(\mathcal{E}) exactly an open problem for most channels. Even the qubit depolarizing channel's quantum capacity is known only for specific parameter ranges and is believed to have non-trivial behaviour near the "boundary" where Q = 0.

BSST theorem for C_E — the single-letter capacity

Entanglement-assisted classical capacity is the exception in quantum Shannon theory: its formula is single-letter, additive, and computationally tractable. Bennett-Shor-Smolin-Thapliyal (BSST) proved this in two papers (1999, 2002) [3].

The setup with shared entanglement

Alice and Bob share an infinite supply of maximally-entangled pairs (Bell pairs) before the protocol starts. For each channel use, Alice takes a Bell pair, applies some encoding (depending on her classical message), and sends her half through \mathcal{E} to Bob. Bob performs a joint measurement on his end of the pair and the channel output. Repeat n times; consume \sim n ebits in the process.

The question: what is the maximum classical rate R? BSST's answer:

where |\phi^\rho\rangle_{RA'} is any purification of \rho_{A'} on reference R and channel input A'. The mutual information is computed on the post-channel state \rho_{RB} = (I_R \otimes \mathcal{E})(|\phi\rangle\langle\phi|_{RA'}).

Why C_E is single-letter

Quantum mutual information is additive on tensor-product channels:

Additivity follows from the subadditivity of von Neumann entropy (S(R_1 R_2) \leq S(R_1) + S(R_2) with equality on product states). So C_E(\mathcal{E}^{\otimes n}) = n\, C_E(\mathcal{E}), and the regularization is trivial.

Further, BSST and the quantum reverse Shannon theorem (Bennett-Devetak-Harrow-Shor-Winter 2009) show C_E has a beautiful simulation interpretation: any quantum channel of capacity C_E can be simulated by a perfect classical channel of rate C_E plus unlimited shared entanglement, and vice versa. Entanglement-assisted rates thus quantify channel-to-channel interconversion in the presence of free entanglement — an elegant closure.

The doubling phenomenon

For the noiseless qubit channel \mathcal{E} = I, the classical capacity is C = \log_2 2 = 1 bit per use; the entanglement-assisted classical capacity is C_E = 2\log_2 2 = 2 bits per use — twice as much. The protocol achieving this is superdense coding: Alice manipulates her half of a Bell pair via one of four Pauli operations and sends her qubit; Bob measures both qubits in the Bell basis and recovers 2 classical bits. The single channel use carries 1 qubit; combined with the pre-shared ebit, it delivers 2 classical bits. This is the origin of the "2\log d" upper bound in the capacity hierarchy.

For noisy channels the doubling is partial — C_E / C ranges from 1 (for classical channels embedded in quantum language, where entanglement assistance gives nothing) to 2 (for the noiseless channel). Real physical channels sit somewhere between.

Depolarizing channel — all three capacities

The qubit depolarizing channel \mathcal{D}_q(\rho) = (1-q)\rho + q\, I/2 is the most important worked example in capacity theory. All three capacities have closed forms.

C(\mathcal{D}_q) = 1 - H(q/2)

By HSW (King 2003 for additivity), C(\mathcal{D}_q) = \chi(\mathcal{D}_q) = 1 - H(q/2), where H(p) = -p \log p - (1-p)\log(1-p). At q = 0: C = 1 bit/use. At q = 1: C = 0. At q = 1/2: C \approx 0.189 bits/use. (This is worked out in detail in the hsw-theorem chapter.)

Q(\mathcal{D}_q) — complicated

For the depolarizing channel, the one-shot coherent information has the closed form

This is positive only for q < q_0 \approx 0.1893 — below this threshold, Q^{(1)} > 0 and the channel can transmit qubits at rate Q^{(1)}. Above q_0, Q^{(1)} = 0 (and in fact the true Q is also zero for q > 1/4, where the channel becomes anti-degradable and cannot transmit any quantum information). The "gap region" q_0 < q < 1/4 is mysterious: Q^{(1)} = 0 but Q > 0 due to superadditivity effects — entangled codes across channel uses activate positive quantum rate. The exact Q is not known in closed form for this region; rigorous bounds and numerics exist.

C_E(\mathcal{D}_q) = 2 - H(\{1 - 3q/4, q/4, q/4, q/4\})

By BSST, C_E is computed from the maximally-entangled purification. The optimal \rho_A = I/2, and the post-channel state is a mixture of four Bell states:

Quantum mutual information I(R:B) = 2 S(B) - S(RB) with S(B) = 1 and S(RB) = H(\{1 - 3q/4, q/4, q/4, q/4\}):

At q = 0: C_E = 2 - H(1, 0, 0, 0) = 2 bits/use (superdense coding). At q = 1: C_E = 2 - H(1/4, 1/4, 1/4, 1/4) = 2 - 2 = 0. At q = 1/2: C_E = 2 - H(\{5/8, 1/8, 1/8, 1/8\}) \approx 2 - 1.549 \approx 0.451 bits/use.

Worked examples

Common confusions

• "A quantum channel has one capacity, like a classical channel." No. It has several — C, Q, C_E, P, plus more exotic ones (quantum-assisted, quantum feedback, adaptive, etc.). Each measures a different resource-to-resource conversion. The classical case is the degenerate one-variable limit.

• "Q = C always." No. Q \leq C, with strict inequality for almost every noisy channel. Depolarizing: Q \approx C/2 at q = 0.1. Amplitude-damping channel: Q = 0 for damping parameter above 1/2, while C stays positive until full damping. Quantum information is strictly harder to transmit than classical.

• "C_E = 2C always." Only for the noiseless channel. For noisy channels C_E / C \in [1, 2], depending on how much the channel damages quantum correlations. Fully-depolarizing channels have C = C_E = 0; fully-noiseless have C_E = 2C = 2\log d. Intermediate cases sit between.

• "Quantum capacity is additive because entanglement-assisted capacity is." These are different things. C_E is additive because quantum mutual information is additive across tensor products. Q is not additive — coherent information is superadditive (Smith-Yard 2008), forcing the regularization Q = \lim I_c(\mathcal{E}^{\otimes n})/n. Even worse, two zero-capacity channels can combine to a non-zero-capacity channel (superactivation), which is a strictly quantum phenomenon with no classical analogue.

• "Superdense coding proves C_E > C always." Superdense coding proves C_E = 2 \cdot C for the noiseless qubit channel, and by BSST the ratio is generally in [1, 2]. Some noisy channels have C_E = C (entanglement-breaking channels), meaning entanglement assistance provides no boost. The ratio is a channel property.

• "Private capacity P equals quantum capacity Q." Only sometimes. In general Q \leq P \leq C with both inequalities strict for typical channels. Secret-key distillation (private capacity) is a strictly weaker requirement than full quantum-state transfer, so P \geq Q. And it is a stronger requirement than classical transmission (no eavesdropper), so P \leq C.

Going deeper

Private capacity and the Q \leq P \leq C hierarchy

The private capacity P(\mathcal{E}) is the rate of classical bits transmittable securely against an eavesdropper (with complete access to the channel's environment). Devetak (2005) proved P(\mathcal{E}) = \lim \frac{1}{n} \max [\chi(\mathcal{E}) - \chi(\mathcal{E}^c)], the regularized difference between Bob's and Eve's Holevo information. In general Q \leq P \leq C, with both inequalities often strict. The left inequality comes from "every qubit is a secret bit"; the right from "Eve's information is non-negative." Private capacity is the relevant quantity for QKD security proofs.

The capacity zoo

Beyond C, Q, C_E, P there are:

• Classical capacity with quantum feedback C_{\leftarrow}(\mathcal{E}): Alice uses classical feedback from Bob. Devetak-Shor showed this can exceed C for some channels.
• Quantum capacity with classical feedback Q_{\leftarrow}(\mathcal{E}): classical feedback can boost the quantum rate.
• Trade-off capacities: simultaneous transmission of classical and quantum information at rate pair (R_c, R_q); the achievable region is a 2D convex set with a characterisation in terms of coherent information and Holevo quantity jointly (Hsieh-Wilde 2010).
• Adaptive / non-additive capacities: exotic channels where adaptive strategies (choosing the next input based on past outputs) exceed i.i.d. strategies.

Each of these is a distinct operational notion with its own capacity theorem. The Wilde textbook catalogues the full zoo.

Quantum reverse Shannon theorem

Bennett-Devetak-Harrow-Shor-Winter (2009) [4] proved the quantum reverse Shannon theorem: any quantum channel of entanglement-assisted capacity C_E can be simulated by a noiseless classical channel of rate C_E combined with unlimited pre-shared entanglement. This is a "reverse" direction — using a noiseless channel plus entanglement to simulate a noisy one — and it closes the BSST formula into an exact resource inequality. The reverse Shannon theorem is the reason C_E has such a clean operational interpretation: it is literally the classical-bit cost of channel simulation, once entanglement is free.

Superactivation and zero-capacity paradoxes

Smith-Yard (2008) [2] showed: there exist channels \mathcal{E}_1, \mathcal{E}_2 with Q(\mathcal{E}_1) = Q(\mathcal{E}_2) = 0 but Q(\mathcal{E}_1 \otimes \mathcal{E}_2) > 0. Example: a 50%-erasure channel and a particular PPT-entanglement-binding channel, neither of which alone transmits any qubits, but together transmit a positive rate. This is superactivation of quantum capacity and has no classical analogue. It also means: "no quantum channel has zero capacity in an absolute sense" — the zero-capacity property depends on which other channels it is combined with.

Related pathology: the 2009 Cubitt-Smith paper [5] and subsequent work showed quantum capacity is uncomputable in a precise sense — there is no algorithm that, given a channel's description, outputs Q(\mathcal{E}) to any specified precision. This is a striking failure of the analogy with Shannon's classically-tractable capacity.

Computational tractability summary

This table is the real "one picture is worth a thousand words" of modern quantum Shannon theory: the classical world has one capacity that is always computable; the quantum world has a zoo of capacities of varying tractability, and even the best-behaved one (C_E) required a decade of research to pin down.

Indian quantum communication — NQM and QKD infrastructure

India's National Quantum Mission (2023, ₹6000 crore) allocates a substantial pillar to quantum communication, targeting a national QKD backbone connecting government and financial institutions. ISRO's 2017 demonstration of a free-space quantum-key-distribution over 300 m laid early groundwork; the 2020 ISRO-RRI air-to-ground QKD experiment extended this to kilometres. C-DoT (Centre for Development of Telematics) and CDAC are developing QKD-compatible networking hardware, and the Raman Research Institute Bangalore runs world-class quantum-optics experiments directly relevant to optical-channel capacities. The capacity formulas in this chapter — especially C_E and P — are the theoretical foundation for rate projections in these engineering programmes.

Open problems

• Computing Q(\mathcal{D}_q) exactly for the depolarizing channel in the gap region q_0 < q < 1/4.
• A sharp characterisation of when C(\mathcal{E}) is additive vs non-additive.
• Deciding whether BSST's C_E formula has a multi-letter analogue for combined classical-quantum tasks.
• Closing the gap between achievability and converse for the private capacity P on structured channels (lossy bosonic Gaussian).
• Understanding the operational meaning of superactivation and its physical realisability.

Where this leads next

• HSW theorem — the classical capacity C = \chi^* in full detail, with the Hastings superadditivity counterexample.
• Coherent information — the quantity I_c that parameterises the quantum capacity via LSD.
• Channel superadditivity — Smith-Yard's superactivation and the broader non-additivity phenomena.
• BB84 protocol — the standard QKD protocol whose achievability rate lower-bounds private capacity.
• Superdense coding — the protocol saturating C_E = 2 for the noiseless qubit channel.
• Quantum mutual information — the single-letter quantity behind BSST's C_E formula.

References

1. Igor Devetak, The private classical capacity and quantum capacity of a quantum channel (2005) — arXiv:quant-ph/0304127. The LSD converse and private capacity.
2. Graeme Smith and Jon Yard, Quantum communication with zero-capacity channels (2008) — arXiv:0807.4935. Superactivation of quantum capacity.
3. Charles H. Bennett, Peter W. Shor, John A. Smolin, Ashish V. Thapliyal, Entanglement-assisted classical capacity of noisy quantum channels (1999) — arXiv:quant-ph/9904023. BSST theorem.
4. Charles H. Bennett, Igor Devetak, Aram W. Harrow, Peter W. Shor, Andreas Winter, The quantum reverse Shannon theorem and resource tradeoffs for simulating quantum channels (2009) — arXiv:0912.5537. Channel simulation and C_E operational meaning.
5. Toby S. Cubitt, David Elkouss, William Matthews, Maris Ozols, Diego Pérez-García, Sergii Strelchuk, Unbounded number of channel uses may be required to detect quantum capacity (2015) — arXiv:1408.5115. Uncomputability of quantum capacity.
6. Mark M. Wilde, Quantum Information Theory (2nd ed., 2017), Ch. 20–24 — arXiv:1106.1445. Textbook reference for all capacity theorems and the zoo.