The private classical capacity and quantum capacity of a quantum channel

1 Introduction

• Secret classical information and quantum information are related through coherent operations, although measuring entanglement into a key does not allow literal recovery of the entanglement.A Bell state can yield a secret key by local measurements, while the reverse conversion is not available after measurement.Source: §1 · p. 1
• The motivation is Schumacher and Westmoreland’s information-theoretic approach to secret-key generation and its connection to coherent information.The paper presents an exact secret-key capacity formula where earlier work had only suggested a role for coherent information.Source: §1 · pp. 1–2
• The paper represents classical-quantum ensembles as quantum systems in an enlarged Hilbert space, enabling quantum entropy and mutual-information tools for classical-quantum resources.This framework identifies classical-quantum systems as special cases of quantum systems and defines corresponding information quantities.Source: §1 · pp. 2–3
• Classical-quantum Markov chains model auxiliary classical variables that correlate with an input while leaving the conditional quantum state unchanged.The construction augments XQ with a classical variable T satisfying T → X → Q.Source: §1 · p. 3
• The paper targets private-information and key-generation capacities for classical-quantum and noisy quantum channels, then connects them to entanglement and quantum capacities.The sections define capacities for classical-quantum channels, apply the results to noisy quantum channels, and derive entanglement and quantum-capacity expressions.Source: §1 · p. 3

2 Private information transmission and key generation over classical-quantum channels

• Private-channel codes require Bob to identify Alice’s message reliably while Eve obtains almost no information, whereas key-generation codes only require secret correlations with Bob.The two protocols differ in whether Alice’s key is a function of the randomization variable rather than a separately transmitted message.Source: §2 · pp. 3–4
• The capacity formula uses a regularized optimization over auxiliary variables T satisfying T → X^l → Q^lE^l.The limit exists, but its l → ∞ form is not generally effectively computable.Source: §2 · pp. 4–5
• The converse reduces secret-key capacity to the same private-information expression, using Fano’s inequality, Holevo bounds, and the Markov structure of the protocol.Since any private channel can generate a secret key, K(W) ≥ Cp(W), so achievability for private transmission and a converse for key generation suffice.Source: §2 · pp. 4–5
• The direct coding theorem achieves private rate I(X;Q)−I(X;E) without using the public channel, so forward public communication cannot increase Cp(W) or K(W).The construction uses random codes, typicality, operator concentration, and expurgation to obtain reliable decoding and exponentially small information leakage.Source: §2 · pp. 5, 8
• The paper also gives a new classical direct-coding proof using large-deviation techniques rather than hashing or extractors.This remark identifies the classical analogue of the theorem and its proof-method distinction.Source: §2 · p. 8

3 Private information transmission and key generation over quantum channels

• A noisy quantum channel is represented by an isometric extension whose output Q goes to Bob and environment E is controlled by Eve.For input ρP, the channel outputs ωQ and σE, and coherent information is defined from these states.Source: §3 · pp. 8–9
• Private transmission over N optimizes I(X;Q^l)−I(X;E^l) over input state preparations and regularizes over channel uses.Alice may prepend a classical-to-quantum channel before multiple uses of N, inducing a classical-quantum wiretap channel.Source: §3 · p. 9
• The private information Ip(ρ,N) is at least the coherent information Ic(ρ,N), and the equality case is relevant for entanglement generation.The paper notes that some inputs can satisfy Ip(ρ,N) > Ic(ρ,N).Source: §3 · p. 9

Entanglement-generation task

• E(N) is defined as the supremum of rates at which Alice and Bob can share nearly maximally entangled states over many uses of N.An (n, ε) code targets a maximally entangled state on a 2^nR × 2^nR space.Source: §4 · pp. 9, 11

Direct coding theorem

• The construction relies on fidelity properties, purifications, and phase alignment to transform reliable private decoding and environmental decoupling into an entanglement-generating code.Uhlmann’s theorem and controlled unitaries connect the environment’s nearly fixed state to the desired coherent superposition.Source: §4 · pp. 10, 12, 13

Relation to key generation

• K(N) ≥ E(N) because measuring shared pure entanglement in a fixed basis produces a secret key, although the supplied discussion leaves strictness unresolved.The paper explicitly notes that the formulas do not determine whether the inequality can be strict.Source: §4 · p. 11

Converse theorem

• The converse uses quantum data processing because post-processing cannot increase coherent information, with fidelity and entropy continuity controlling approximation errors.The proof combines the data-processing inequality with Lemma 6, whose bound follows from fidelity monotonicity and Fannes’ inequality.Source: §4 · p. 11

Direct coding theorem

• The direct coding theorem achieves Ic(ρ, N) by decomposing ρ into pure states, applying a private classical code, and using Bob’s decoding operation to construct entanglement generation.The construction uses an isometric extension of N, controls phases through Uhlmann’s theorem, and produces a final state close to the target entangled state.Source: §4 · pp. 12, 14

Scope and limitation

• The private-code-to-entanglement conversion appears to require pure-state decompositions, because mixed decompositions can leave the relevant states shared with Eve and demand unavailable joint operations.The authors also note a structural similarity to CSS codes while distinguishing the classical-quantum construction from purely classical codes.Source: §4 · p. 14

5 Quantum information transmission over quantum channels.

• Quantum capacity Q(N) is defined as the supremum of all rates achievable by encoding and transmitting quantum states over many channel uses.The alternative entanglement-fidelity formulation defines eQ(N), and prior work established Q(N) = eQ(N).Source: §5 · p. 14
• E(N) = eQ(N), so entanglement generation and entanglement transmission have equal capacities.The equality yields a simpler converse proof for the quantum channel capacity theorem.Source: §5 · pp. 14–15
• Entanglement generation can achieve at least entanglement-transmission capacity because the channel can transmit half of a maximally entangled state.The reverse inequality uses forward classical communication and teleportation, giving E(N) ≤ eQ(N).Source: §5 · p. 15
• Bilateral twirling converts the generated state into a Werner state, whose entanglement fidelity supports transmission with fidelity at least 1 − ǫ.A suitable bilateral unitary can be absorbed into the entanglement-generation protocol, completing the equality argument.Source: §5 · p. 15
• Random entanglement-transmission codes can approach Ic(ρ, N) while having average codeword density arbitrarily close to ρ^⊗n.This direct coding construction is given as an alternative proof route for entanglement transmission.Source: §5 · p. 15

6 Discussion

• Cp(W) = K(W), Cp(N) = K(N), and E(N) = Q(N), establishing an operational connection between quantum privacy and quantum coherence.The protocols use coherent subclasses to obtain entanglement generation and quantum information transmission capacities.Source: §6 · p. 15
• K(N) ≥ E(N), but it remains open whether some channels can generate secret keys without transmitting quantum information.The discussion relates this possible separation to states with secret key but zero two-way distillable entanglement.Source: §6 · p. 15
• Single-letter formulas remain open for Cp(W) and Cp(N), while counterexamples to single-letterization are known for Q(N).For degradable channels, Q(N) and Cp(W) are efficiently computable, but the corresponding property is unknown for Cp(N).Source: §6 · pp. 15–16
• The convergence rates of the asymptotic limits defining private capacities remain unresolved.The discussion identifies this as a broader computability question beyond the single-letterization issue.Source: §6 · p. 16
• Allowing two-way public or classical communication can improve both key-generation and entanglement-generation capacities, although general formulas seem unlikely.This is presented as a natural extension beyond the forward-communication setting.Source: §6 · p. 16

A Definitions of typical sequences and subspaces

• Typical sequences are defined by constraining empirical symbol frequencies to remain close to a distribution p.The count N(x|x^n) records how often symbol x occurs in a length-n sequence.Source: A Definitions of typical sequences and… · p. 16
• Conditionally typical sequences constrain empirical frequencies according to a stochastic map P: U → X and a reference sequence u^n.The notation T^n_X|U,δ(u^n) denotes this conditional typical set.Source: A Definitions of typical sequences and… · p. 16
• For a density operator ρ, the quantum typical projector is defined from its eigen-decomposition and selects the typical subspace across n tensor factors.The associated notation is used when ρ corresponds to a quantum system Q.Source: A Definitions of typical sequences and… · p. 16
• Conditionally typical projectors are defined for collections of states ρu on tensor positions selected by a classical sequence u^n.These projectors provide the quantum analogue of conditional typicality for a classical-quantum system Q|U.Source: A Definitions of typical sequences and… · p. 16

B The modified HSW theorem

• ν = 2^{n[I(X;Q)−2(c+c′δ)δ]} random variables are sampled from a pruned distribution to form the code.The construction targets an HSW code for Q|X.Source: B The modified HSW theorem · p. 17
• A decoding POVM {Y_s} is constructed so that the random code has low probability of error.The argument applies the Hayashi-Nagaoka operator inequality to control decoding errors.Source: B The modified HSW theorem · p. 17
• The final error bound follows by averaging the decoding expression over independently sampled codewords.The chosen value of ν supplies the last inequality in the bound.Source: B The modified HSW theorem · p. 17

• The example uses a four-dimensional Hilbert space split into the subspaces spanned by {|1⟩, |2⟩} and {|3⟩, |4⟩}.The channel is defined on B(H4), with projectors Π12 and Π34 onto these two subspaces.Source: C An example of Ip(ρ, N ) > Ic(ρ, N ) · p. 17
• The channel applies complete depolarization on the {|3⟩, |4⟩} subspace through D34.The construction was suggested by J. A. Smolin [4].Source: C An example of Ip(ρ, N ) > Ic(ρ, N ) · pp. 17–18
• The perturbation is defined by π34 = (1 −ǫ)π34 + ǫπ12 for small positive ǫ.The supplied expression is presented as the continuity-based step leading to the strict information inequality.Source: C An example of Ip(ρ, N ) > Ic(ρ, N ) · p. 18
• The supplied passage reports the strict inequality “12, N) > Ic(π′ 12, N) > 0.”By continuity, this holds for sufficiently small ǫ.Source: C An example of Ip(ρ, N ) > Ic(ρ, N ) · p. 18

D The average density operator of random quantum codes

• Entanglement-generation codes can be converted into entanglement-transmission codes at the same rate Ic(ρ, N) −δ.The conversion applies to a subclass of the generation codes described earlier.Source: D The average density operator of… · p. 18
• The encoding isometry maps the reference-entangled input basis |k⟩A′ to mutually orthogonal quantum codewords |φk⟩P.This transfers Alice’s entanglement with the reference system into the channel input code.Source: D The average density operator of… · p. 18
• Choosing the input state to be maximally entangled yields an achievable entropy rate of 1 through the decoding construction.The decoding follows the operation specified earlier in the proof.Source: D The average density operator of… · pp. 18, 20
• Random quantum codes are constructed so their average code density operator becomes arbitrarily close to ρ⊗n for sufficiently large blocklength.The construction uses a double randomization of the original protocol.Source: D The average density operator of… · p. 19
• The first randomization samples Fourier-state codewords uniformly from the set of ǫ-good codewords.This layer concludes after establishing the corresponding average code density operator.Source: D The average density operator of… · p. 19
• A second randomization samples the code construction according to p′ and then applies expurgation, producing a doubly randomized protocol.The expurgation has only a small effect on the average code density operator.Source: D The average density operator of… · pp. 19–20