On the undecidability of quantum channel capacities

Abstract: An important distinction in our understanding of capacities of classical versus quantum channels is marked by the following question: is there an algorithm which can compute (or even efficiently compute) the capacity? While there is overwhelming evidence suggesting that quantum channel capacities may be uncomputable, a formal proof of any such statement is elusive. We initiate the study of the hardness of computing quantum channel capacities. We show that, for a general quantum channel, it is QMA-hard to compute its quantum capacity, and that the maximal-entanglement-assisted zero-error one-shot classical capacity is uncomputable.

• The paper establishes that computing quantum channel capacities with a given one-shot input is QMA-hard, making any efficient single-letter formula for quantum capacity improbable unless P equals QMA.
• This research defines a new composition rule for quantum channels, the projective direct sum, and utilizes it to derive exact formulas for quantum capacity.
• The authors prove the first undecidable capacity of quantum channels: the maximally entangled one-shot quantum channel's zero-error classical capacity is undecidable, deriving it from the undecidability of the maximal independent set problem ( also known as the Quantum Independent Set problem ) of graphs.

Motivation and main results

For classical channels, Shannon's noiseless coding theorem yields a single-letter formula: the capacity is the maximum, over input distributions, of the mutual information $I(X;Y)$, which is additive across channel uses and therefore efficiently computable. For general quantum channels no analogous single-letter formula for the quantum capacity $Q(\Phi)$ is known; the best available expression is the regularised coherent information

$$Q(\Phi) = \lim_{n\to\infty} \frac{1}{n} \max_\rho I_c(\Phi^{\otimes n}, \rho),$$

and phenomena such as superadditivity (0905.2634), superactivation (Smith et al., 2008), and channels with positive capacity but zero coherent information at every finite number of uses suggest that no reasonable single-letter formula exists. Despite this evidence, no formal complexity-theoretic lower bound had previously ruled out an efficiently computable characterisation. Bhattacharyya, Mehta, and Zhao initiate a systematic study of the computational hardness and computability of quantum channel capacities and establish two results:

• QMA-hardness: given a quantum channel $\Phi$ specified by a circuit description, it is QMA-hard to decide whether $Q(\Phi) \geq 3/4$ or $Q(\Phi) \leq 1/4$, promised one holds. Consequently, any hypothetical single-letter formula for the quantum capacity must be inefficient to evaluate unless $\mathsf{P} = \mathsf{QMA}$.
• Undecidability: given a quantum channel $\Phi$ and integer $t$, it is undecidable (indeed RE-hard) whether the maximal-entanglement-assisted one-shot zero-error classical capacity satisfies $C^{(1)}_{0,M,E}(\Phi) \geq \log t$. This is the first provably uncomputable capacity notion for quantum channels.

The two results are obtained by entirely separate techniques, and each carries its own caveats discussed below.

QMA-hardness via projective direct sums

The hardness reduction follows directly from the definition of QMA. A verifier for a QMA language produces a circuit $Q(\Phi)$0 implementing a channel $Q(\Phi)$1 from a proof register to a one-qubit decision register, with acceptance probability $Q(\Phi)$2. The natural reduction maps an instance $Q(\Phi)$3 to a channel that applies $Q(\Phi)$4, then routes an auxiliary register through either a full-capacity channel or a zero-capacity channel conditioned on the decision qubit. Making this rigorous requires a new composition rule.

Projective direct sum. The standard direct sum $Q(\Phi)$5 acts on block-diagonal inputs over $Q(\Phi)$6, which is awkward in the circuit model (dimensions need not be powers of two) and does not allow two channels sharing a common input register to be combined while acting on that same register. The authors introduce the projective direct sum: given CPTP maps $Q(\Phi)$7 and POVM elements $Q(\Phi)$8,

$Q(\Phi)$9

where $$Q(\Phi) = \lim_{n\to\infty} \frac{1}{n} \max_\rho I_c(\Phi^{\otimes n}, \rho),$$0 is a state on which the POVM is evaluated. Two properties make this construction effective:

1. Circuit composability: given circuits for $$Q(\Phi) = \lim_{n\to\infty} \frac{1}{n} \max_\rho I_c(\Phi^{\otimes n}, \rho),$$1, $$Q(\Phi) = \lim_{n\to\infty} \frac{1}{n} \max_\rho I_c(\Phi^{\otimes n}, \rho),$$2, and the measurement channel defining $$Q(\Phi) = \lim_{n\to\infty} \frac{1}{n} \max_\rho I_c(\Phi^{\otimes n}, \rho),$$3, a circuit for the projective direct sum is constructible in polynomial time.
2. Complement compatibility: $$Q(\Phi) = \lim_{n\to\infty} \frac{1}{n} \max_\rho I_c(\Phi^{\otimes n}, \rho),$$4, proved via Stinespring dilations.

From these, the paper derives an exact formula for the one-shot quantum capacity of the composite:

$$Q(\Phi) = \lim_{n\to\infty} \frac{1}{n} \max_\rho I_c(\Phi^{\otimes n}, \rho),$$5

using the entropy identity $$Q(\Phi) = \lim_{n\to\infty} \frac{1}{n} \max_\rho I_c(\Phi^{\otimes n}, \rho),$$6 for subnormalised states.

Choice of component channels. The reduction uses the two extremes of the qubit amplitude damping channel $$Q(\Phi) = \lim_{n\to\infty} \frac{1}{n} \max_\rho I_c(\Phi^{\otimes n}, \rho),$$7: $$Q(\Phi) = \lim_{n\to\infty} \frac{1}{n} \max_\rho I_c(\Phi^{\otimes n}, \rho),$$8, which is degradable with $$Q(\Phi) = \lim_{n\to\infty} \frac{1}{n} \max_\rho I_c(\Phi^{\otimes n}, \rho),$$9, and $\Phi$0, which is both degradable and anti-degradable, hence has $\Phi$1. Degradability makes the coherent information additive, so the regularisation collapses and the asymptotic capacity equals the one-shot quantity; anti-degradability forces zero capacity because the environment can simulate the receiver. Combining these with the verification-channel POVM yields: if $\Phi$2 is a yes-instance there exists $\Phi$3 with $\Phi$4, giving $\Phi$5; if $\Phi$6 is a no-instance then $\Phi$7 for all $\Phi$8, giving $\Phi$9.

An important qualification accompanies this result: it concerns channels presented succinctly as circuits. Succinct encodings are known to inflate apparent complexity, and much of quantum information theory works with explicit Kraus-operator or Stinespring descriptions instead. The authors note that if the quantum capacity is ultimately shown uncomputable, this distinction becomes immaterial — but until then, the QMA-hardness result does not preclude an efficient algorithm for channels given by other computable encodings.

Undecidability of the maximal-entanglement-assisted zero-error capacity

The second result proceeds by reduction from the quantum independence number of graphs. The independent set game $Q(\Phi) \geq 3/4$0 asks two non-communicating players to convince a referee that $Q(\Phi) \geq 3/4$1 contains an independent set of size $Q(\Phi) \geq 3/4$2; its perfect-strategy threshold defines $Q(\Phi) \geq 3/4$3 classically and $Q(\Phi) \geq 3/4$4 with entanglement. By the $Q(\Phi) \geq 3/4$5 theorem (LaRose et al., 2020) (building on Slofstra's earlier undecidability of perfect strategies for synchronous games), deciding whether $Q(\Phi) \geq 3/4$6 is RE-hard.

Given a graph $Q(\Phi) \geq 3/4$7, the authors construct a classical–quantum channel $Q(\Phi) \geq 3/4$8 mapping basis states $Q(\Phi) \geq 3/4$9 to pure states $Q(\Phi) \leq 1/4$0 derived from an eigenvector construction on the adjacency matrix, so that $Q(\Phi) \leq 1/4$1 when $Q(\Phi) \leq 1/4$2 and $Q(\Phi) \leq 1/4$3 otherwise, where $Q(\Phi) \leq 1/4$4 is the absolute value of the smallest eigenvalue. The central equivalence is:

$Q(\Phi) \leq 1/4$5

Both directions exploit orthogonality: in the forward direction, PVMs witnessing a perfect strategy for $Q(\Phi) \leq 1/4$6 yield $Q(\Phi) \leq 1/4$7 perfectly distinguishable output states under $Q(\Phi) \leq 1/4$8 applied to half of a maximally entangled state; conversely, any zero-error code with maximal entanglement assistance induces POVMs whose pairwise products vanish exactly on losing question–answer pairs, hence a perfect quantum strategy. Since the map $Q(\Phi) \leq 1/4$9 is computable, deciding whether $\mathsf{P} = \mathsf{QMA}$0 is RE-hard, i.e., undecidable.

Two qualifications matter here. First, the restriction to maximally entangled assistance is an artefact of the synchronous-game reduction rather than essential to the argument; the fully general entanglement-assisted zero-error capacity $\mathsf{P} = \mathsf{QMA}$1 remains open, and notably there exist channels assisted only by non-maximally entangled states. Second, because the problem is undecidable, the conclusion is robust to how the channel is encoded — circuit, Kraus operators, or any other computable description.

Relation to prior work and limitations

The closest antecedent is Beigi and Shor's result that computing the Holevo capacity of a quantum channel is NP-complete (0709.2096); the present QMA-hardness result is, to the authors' knowledge, only the second significant complexity lower bound for a quantum channel capacity. For classical channels, NP-completeness of the zero-error capacity follows from the independence number reduction, and the undecidability result here is the quantum analogue taken to its logical extreme.

The paper is explicit about what it does not achieve. The long-term objective — proving the (asymptotic, unassisted) quantum capacity uncomputable for general channels — remains open, and the authors describe it as extremely challenging. The QMA-hardness result does not rule out a single-letter formula, only an efficiently evaluable one (under the circuit encoding). The undecidability result covers a modified capacity notion restricted to maximally entangled assistance and to the one-shot, zero-error regime. The authors also raise, crediting Mančinska, the finer question of whether a Lovász-type tight bound can be obtained for the Shannon capacity of graphs beyond the five-cycle.

Conclusion

This work supplies the first formal lower bounds in the programme of determining whether quantum channel capacities are computable. Through the projective direct sum — a circuit-friendly composition whose complements decompose compatibly and whose coherent information admits an exact weighted-additive formula — computing the quantum capacity within a constant gap is shown QMA-hard. Independently, an exact correspondence between the maximal-entanglement-assisted one-shot zero-error classical capacity of an explicitly constructed c-q channel and the quantum independence number transfers the undecidability of $\mathsf{P} = \mathsf{QMA}$2 to channel capacities, yielding the first provably uncomputable capacity for quantum channels. The central question left open is whether the unassisted asymptotic quantum capacity itself is uncomputable, and, short of that, whether the general entanglement-assisted zero-error capacity shares the undecidability established here for the maximally entangled case.