Quantum channel–state duality represents a quantum channel as a bipartite state (a Choi state), and Liu and Wang (2025) recently introduced quantum capacities for channels whose information sources are Choi states, proving capacity theorems for the resulting Choi capacity Q^\hat Q and its quantum-control-assisted variant Q^CA\hat Q_{CA}. These capacities are expressed as regularized limits, their relation to entanglement-generation capacities is only bounded, and the higher-order capacity hierarchy obtained by iterating the duality has unquantified gaps. The surviving open core is threefold: (1) prove or refute the conjectured identities Q^(Φ)=E^(Φ)\hat Q(\Phi)=\hat E(\Phi) and Q^CA(Φ)=E^CA(Φ)\hat Q_{CA}(\Phi)=\hat E_{CA}(\Phi) between Choi quantum capacities and Choi entanglement-generation capacities; (2) classify the channels for which the regularization in Q^(Φ)\hat Q(\Phi) is unnecessary; (3) determine under what conditions the gaps in the higher-order capacity chain collapse.

Background

A quantum channel is a completely positive trace-preserving map Φ\Phi that models noise acting on quantum information. The quantum capacity Q(Φ)Q(\Phi) is the supremum rate at which qubits can be transmitted through many uses of Φ\Phi with vanishing error, and the capacity theorem expresses it as a regularized limit Q(Φ)=lim⁡n→∞1nmax⁡ρIc(ρ,Φ⊗n)Q(\Phi)=\lim_{n\to\infty}\frac{1}{n}\max_{\rho} I_c(\rho,\Phi^{\otimes n}) of the coherent information Ic(ρ,Φ)=S(Φ(ρ))−S(Φc(ρ))I_c(\rho,\Phi)=S(\Phi(\rho))-S(\Phi^{c}(\rho)), where SS is the von Neumann entropy, ρ\rho the input state, and Φc\Phi^{c} the complementary channel obtained by tracing out the output of the channel's Stinespring dilation (its isometric extension to a larger output space that includes an environment) instead of the ancilla. Because IcI_c is not additive — coherent information is superadditive for generic channels, as shown for the dephrasure channel (a concatenation of a dephasing and an erasure channel) and for the platypus channels (a two-parameter family combining erasure and dephasing behaviour, engineered to violate additivity of coherent information) — the regularization is in general unavoidable and QQ is uncomputable for arbitrary channels. The main exception is the class of degradable channels (channels whose complementary output can be reconstructed from the direct output, introduced by Devetak and Shor), for which the regularization collapses to a single-letter formula. If the sender and receiver share entanglement in advance, the entanglement-assisted quantum capacity QEA(Φ)=max⁡ρ12I(ρ,Φ)Q_{EA}(\Phi)=\max_{\rho}\frac{1}{2}I(\rho,\Phi), with quantum mutual information I(ρ,Φ)=S(ρ)+Ic(ρ,Φ)I(\rho,\Phi)=S(\rho)+I_c(\rho,\Phi), is also single-letter. The entanglement-generation capacity E(Φ)E(\Phi) — the supremum rate at which shared ebits (maximally entangled pairs, the unit of bipartite entanglement) can be established between sender and receiver — equals Q(Φ)Q(\Phi) in this standard setting, since quantum transmission and entanglement generation simulate each other.

The channel–state duality (Jamiołkowski–Choi isomorphism) represents a channel Φ\Phi as the bipartite Choi state ωΦ=(Φ⊗1)(ω)\omega_{\Phi}=(\Phi\otimes\mathbb{1})(\omega), where ω\omega is a normalized maximally entangled state. The operations on channels that preserve Choi-state form are superchannels (pre- and post-operations connected by a noise-free memory). Liu and Wang (2025) asked which capacities arise when Choi states themselves are the information sources, motivated by stored-program quantum computing, where programs are stored as Choi states, and by the resource theory of coding. A Choi coding protocol uses 2n2n instances of Φ\Phi arranged as nn Choi pairs, a super-isometric encoding (a superchannel whose pre- and post-operators are isometries) acting on the Choi states, and a decoding channel, and demands high entanglement fidelity (squared state overlap) between the output and 2k2k halves of ebits; the Choi capacity Q^(Φ)\hat Q(\Phi) is the supremum achievable rate. Their main theorem gives Q^(Φ)=lim⁡n→∞12nmax⁡ωEIc(ωE,Φ⊗2n)\hat Q(\Phi)=\lim_{n\to\infty}\frac{1}{2n}\max_{\omega_{\mathcal{E}}} I_c(\omega_{\mathcal{E}},\Phi^{\otimes 2n}), the maximum running over Choi states of encoding superchannels, and, for the quantum-control-assisted (CA) setting in which a noiseless control register plays the role entanglement assistance plays in standard coding, Q^CA(Φ)=max⁡ωE14I(ωE,Φ⊗2)\hat Q_{CA}(\Phi)=\max_{\omega_{\mathcal{E}}}\frac{1}{4}I(\omega_{\mathcal{E}},\Phi^{\otimes 2}). The Choi entanglement-generation capacities E^(Φ)\hat E(\Phi) and E^CA(Φ)\hat E_{CA}(\Phi) are defined in the source (Definition 4) as the suprema of achievable rates α\alpha, with k=⌊αn⌋k=\lfloor\alpha n\rfloor, for the corresponding entanglement-generation tasks: the source is a purified Choi state ξ\xi, the decoding is a general channel D\mathcal{D} without any super-isometric encoding constraint, and the target is kk copies of the Bell state ω(2)\omega_{(2)} built from the Bell basis of the doubled (d2d^2-dimensional) space, with fidelity condition F(ω(2)⊗k,(D Φ⊗2n⊗1⊗2k)(ξ))≥1−ϵF(\omega_{(2)}^{\otimes k},(\mathcal{D}\,\Phi^{\otimes 2n}\otimes\mathbb{1}^{\otimes 2k})(\xi))\ge 1-\epsilon; in the CA variant E^CA\hat E_{CA} the channel becomes (Φ⊗1)⊗2n(\Phi\otimes\mathbb{1})^{\otimes 2n} acting with 2n2n noiseless control registers. The source proves only the bounds Q^(Φ)≤E^(Φ)\hat Q(\Phi)\le\hat E(\Phi) and Q^CA(Φ)≤E^CA(Φ)\hat Q_{CA}(\Phi)\le\hat E_{CA}(\Phi) and conjectures the equalities by analogy with the standard identity Q(Φ)=E(Φ)Q(\Phi)=E(\Phi). Since Choi protocols are more restrictive than standard ones, Q≥Q^Q\ge\hat Q and QEA≥Q^CAQ_{EA}\ge\hat Q_{CA}. Iterating the channel–state duality (treating superchannels as channels and applying the duality again) generates higher-order capacities defined, for tt-th order Choi states ωE(t)\omega_{\mathcal{E}^{(t)}}, by Icr(t)(Φ):=lim⁡n→∞max⁡E(t)1n2tIc(ωE(t),Φ⊗n2t)I_{c_r}^{(t)}(\Phi):=\lim_{n\to\infty}\max_{\mathcal{E}^{(t)}}\frac{1}{n2^{t}}I_c(\omega_{\mathcal{E}^{(t)}},\Phi^{\otimes n2^{t}}), which form a chain Ic(π,Φ)≤⋯≤Icr(t)(Φ)≤⋯≤Icr(1)(Φ)≤Icr(Φ)=Q(Φ)I_c(\pi,\Phi)\le\cdots\le I_{c_r}^{(t)}(\Phi)\le\cdots\le I_{c_r}^{(1)}(\Phi)\le I_{c_r}(\Phi)=Q(\Phi), where π\pi is the maximally mixed input; for degradable channels the regularization is unnecessary and the chain reduces to the single-letter chain Ic(π,Φ)≤⋯≤Ic(t)(Φ)≤⋯≤Ic(1)(Φ)≤Ic(Φ)I_c(\pi,\Phi)\le\cdots\le I_c^{(t)}(\Phi)\le\cdots\le I_c^{(1)}(\Phi)\le I_c(\Phi), where Ic(t)(Φ):=max⁡E(t)12tIc(ωE(t),Φ⊗2t)I_c^{(t)}(\Phi):=\max_{\mathcal{E}^{(t)}}\frac{1}{2^{t}}I_c(\omega_{\mathcal{E}^{(t)}},\Phi^{\otimes 2^{t}}), but in general the paper states the gaps, especially between Ic(Φ)I_c(\Phi) and Ic(1)(Φ)I_c^{(1)}(\Phi), remain unexplored. Numerical optimization of the relevant coherent informations (a nonconvex problem feasible only for small systems) on the qubit Pauli channel (random Pauli errors) and the dephrasure channel exhibits both coincidence and separation between Choi-restricted and usual coherent information, so superadditivity behaves differently under the Choi-source constraint in a way that is not yet understood.

Problem Statement

For a finite-dimensional quantum channel Φ\Phi, characterize the Choi capacities introduced by Liu and Wang (2025) beyond their regularized-limit expressions, by resolving three linked questions: (1) Prove or refute the conjectured identities between the Choi quantum capacities and the corresponding Choi entanglement-generation capacities, Q^(Φ)=E^(Φ)\hat Q(\Phi)=\hat E(\Phi) and Q^CA(Φ)=E^CA(Φ)\hat Q_{CA}(\Phi)=\hat E_{CA}(\Phi), for all channels Φ\Phi (only the ≤\le bounds are proven); (2) determine for which channels the regularization in Q^(Φ)=lim⁡n→∞12nmax⁡ωEIc(ωE,Φ⊗2n)\hat Q(\Phi)=\lim_{n\to\infty}\frac{1}{2n}\max_{\omega_{\mathcal{E}}} I_c(\omega_{\mathcal{E}},\Phi^{\otimes 2n}) is unnecessary, i.e., give a (single-letter or otherwise computable) characterization of Q^\hat Q and the class of channels for which it collapses, beyond the degradable channels already known to collapse the chain; (3) quantify the gaps in the higher-order capacity chain Ic(π,Φ)≤⋯≤Icr(t)(Φ)≤⋯≤Icr(1)(Φ)≤Icr(Φ)=Q(Φ)I_c(\pi,\Phi)\le\cdots\le I_{c_r}^{(t)}(\Phi)\le\cdots\le I_{c_r}^{(1)}(\Phi)\le I_{c_r}(\Phi)=Q(\Phi) and determine under what conditions on Φ\Phi and on the admissible encoding families those inequalities become equalities, in particular for the gap between Ic(Φ)I_c(\Phi) and Ic(1)(Φ)I_c^{(1)}(\Phi) singled out in the source's open remark — in the notation of the degradable single-letter chain defined in the Background, i.e. Ic(1)(Φ)=max⁡E(1)12Ic(ωE(1),Φ⊗2)I_c^{(1)}(\Phi)=\max_{\mathcal{E}^{(1)}}\frac{1}{2}I_c(\omega_{\mathcal{E}^{(1)}},\Phi^{\otimes 2}), a different object from the regularized t=1t=1 term Icr(1)(Φ)=Q^(Φ)I_{c_r}^{(1)}(\Phi)=\hat Q(\Phi) of question (2). These three questions jointly amount to one objective — replacing the regularized definition of the Choi capacities by a characterized, checkable quantity. Each is also independently resolvable: settling either of the two conjectured identities in (1) is a self-contained result while the equality claim as a whole requires both, and questions (2) and (3) may each be answered on their own. The auxiliary directions of scalable optimization algorithms and a predictive theory of superadditivity under Choi-source constraints are not part of the completion boundary.

References

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