Computability of memoryless quantum capacity
Given a finite-dimensional memoryless quantum channel through an explicit rational/algebraic Choi matrix and a rational , does a single algorithm always terminate and output satisfying ? Here
Problem statement
Given a finite-dimensional memoryless quantum channel through an explicit rational/algebraic Choi matrix and a rational , does a single algorithm always terminate and output satisfying ? Here is the ordinary unassisted quantum capacity, in qubits per channel use.
Lean 4
No Lean statement or proof is attached yet. Have a formalization? Share its source to add it to this problem.
Background
This question asks for computability, not an efficient formula or a polynomial-time algorithm. Finite-block coherent-information optimizations provide lower bounds through the regularized capacity formula. The absence of a known effective uniform convergence bound is an obstacle, not a proof of uncomputability.
What is known: Uncomputability results for channels with memory concern a different model. Likewise, zero-error capacity, assisted capacities and capacity-achieving optimizers are different tasks. Noncomputability of zero-error capacity on computable-real channel descriptions cannot simply be transferred to ordinary capacity on finite rational/algebraic input.
The March 2026 v3 preprint by Bhattacharyya–Mehta–Zhao proves QMA-hardness for computing quantum capacity and uncomputability of a restricted entanglement-assisted zero-error capacity. It explicitly leaves general capacity uncomputability unresolved. Version 3 corrects an error in the zero-error part of v2; neither its title nor an earlier abstract should be read as proving the headline problem settled.
C1 treats formulas and tractable channel families; C6 treats zero-error questions; C9 treats memory. Classical and private memoryless capacity computability are companion questions requiring their own precise representations and assistance conventions. Related: C1, C6, C9.
Progress & source assessment
Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.
preprint: QMA-hardness of quantum capacity and a restricted assisted zero-error uncomputability theorem; ordinary memoryless quantum-capacity computability is not settled. v3 corrects v2. Evidence
- [1]Bhattacharyya, Mehta & Zhao, 'On the undecidability of quantum channel capacities' (2026 preprint, corrected v3) link
- [2]Elkouss & Pérez-García, 'Memory effects can make the transmission capability of a communication channel uncomputable', Nat. Commun. 9, 1149 (2018) link
- [3]Boche & Deppe, 'Computability of the Zero-Error Capacity of Noisy Channels' (2020; revised 2024) link
Recorded progress
Add yours →Nothing recorded yet. A pointer to a paper the entry is missing counts.
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