Computational Two-Way Quantum Capacity

• Computational two-way quantum capacity is defined as the maximal quantum information rate achievable via polynomial-time LOCC, setting practical transmission limits.
• It is dual to computational distillable entanglement, linking efficient entanglement purification to the capacity of corresponding Choi states under cryptographic constraints.
• A sharp complexity threshold in dephasing channels reveals a phase transition where increasing channel complexity can collapse the capacity accessible through efficient protocols.

The computational two-way quantum capacity quantifies the maximal asymptotic rate at which quantum information can be transmitted over a family of quantum channels, assuming all encoding and decoding operations are efficiently implementable—specifically, restricted to polynomial-time LOCC (local operations and classical communication). This notion stands in contrast with conventional quantum capacity definitions, which presuppose unbounded computational resources at the endpoints. The computational two-way quantum capacity operationalizes a realistic communication rate by demanding that protocol complexity is polynomial in the number of channel uses and system size, revealing new separations between the information-theoretic and computational limits of quantum communication (Meyer et al., 21 Jan 2026).

1. Formal Definition of Computational Two-Way Capacity

Let $\{\Phi_n\}_{n \in \mathbb{N}}$ be a family of quantum channels, with $\Phi_n$ acting on $O(\mathrm{poly}(n))$-dimensional systems. For a gate-complexity bound $G \in \mathbb{N}$ and error threshold $\varepsilon > 0$, the single-shot G-complexity, $\varepsilon$-error, two-way quantum capacity is defined as

$$Q_{\leftrightarrow}^\varepsilon(\Phi; G) = \log \max\{k \in \mathbb{N} : \exists~ \text{LOCC comb }\mathcal{L}~\text{with}~C(\mathcal{L}) \leq G~\text{such that}~F(\mathcal{L}[\Phi], \mathrm{id}_{2^k}) \geq 1 - \varepsilon\}$$

where $C(\mathcal{L})$ is the number of elementary 2-qubit gates used by $\mathcal{L}$, and $F(\cdot,\cdot)$ is channel fidelity.

The computational two-way quantum capacity is given by

$\Phi_n$0

More precisely,

$\Phi_n$1

where $\Phi_n$2 bounds the total gate complexity for $\Phi_n$3 channel uses (Meyer et al., 21 Jan 2026).

2. Duality with Computational Distillable Entanglement

The computational two-way quantum capacity for channel families that are efficiently Choi-stretchable is exactly equal to the computational distillable entanglement of the corresponding Choi states. For a sequence of bipartite states $\Phi_n$4 on $\Phi_n$5 and gate bound $\Phi_n$6, the computational distillable entanglement is

$\Phi_n$7

$\Phi_n$8

If $\Phi_n$9 can be written as $O(\mathrm{poly}(n))$0 with $O(\mathrm{poly}(n))$1 an LOCC protocol of poly($O(\mathrm{poly}(n))$2) complexity, then

$O(\mathrm{poly}(n))$3

This duality demonstrates that computational capacity is fundamentally tied to whether distillation of entanglement from many Choi states can be performed efficiently (Meyer et al., 21 Jan 2026).

3. Computational–Unbounded Quantum Capacity Separation

A key result is the existence of channels for which the usual (unbounded) two-way quantum capacity is nearly maximal, yet the computational two-way capacity vanishes. Under the existence of quantum-secure one-way functions, there are families of Pauli dephasing channels $O(\mathrm{poly}(n))$4 (with $O(\mathrm{poly}(n))$5 a pseudorandom distribution of low entropy) such that

$O(\mathrm{poly}(n))$6

This separation arises because LOCC protocols restricted to polynomial time cannot distinguish the Choi state $O(\mathrm{poly}(n))$7 from the maximally mixed Choi state, precluding efficient entanglement distillation. These results demonstrate a qualitative divergence between the information-theoretic and efficient quantum Shannon-theoretic regimes, and rely only on standard cryptographic assumptions (Meyer et al., 21 Jan 2026).

4. Sharp Complexity Threshold and Phase Transition

A sharp threshold in computational capacity is observed for dephasing channels as the support size $O(\mathrm{poly}(n))$8 of the dephasing subset transitions from polynomial to super-polynomial in $O(\mathrm{poly}(n))$9. For the uniform dephasing channel on $G \in \mathbb{N}$0,

$G \in \mathbb{N}$1

The following holds:

• If $G \in \mathbb{N}$2, then one can achieve

$G \in \mathbb{N}$3

by efficiently inverting syndrome mappings using Clifford operations and local measurements.

• If $G \in \mathbb{N}$4 grows super-polynomially and $G \in \mathbb{N}$5 is pseudorandom, then

$G \in \mathbb{N}$6

This threshold reflects a computational phase transition: as the channel's (dephasing) complexity increases beyond polynomial, the capacity accessible by efficient protocols abruptly collapses (Meyer et al., 21 Jan 2026).

5. Implications for Quantum Communication and Protocol Design

The concept of computational two-way quantum capacity mandates incorporating computational efficiency—typically polynomial time—in both encoding and decoding protocols. Traditional quantum capacity metrics, by assuming unbounded computational resources, systematically overestimate achievable rates in practice. The demonstrated existence of quantum channels with maximal information-theoretic capacity but vanishing computational capacity suggests the necessity of revising protocol design methodologies for quantum networks and repeaters: noise and information-theoretic error-correction thresholds alone are not sufficient performance predictors; the computational hardness of quantum decoding and entanglement distillation steps can be the bottleneck (Meyer et al., 21 Jan 2026).

The observed phenomenon of "computationally bound" channels, where entanglement is un-distillable by efficient LOCC operations but distillable in principle, constitutes a new resource-theoretic distinction in quantum information—a computational analog of bound entanglement. These findings initiate a broader efficient quantum Shannon theory linking communication complexity, information theory, and cryptographic hardness.

6. Contrasts with Conventional Two-Way and PPTp-Assisted Capacities

Standard two-way quantum capacities $G \in \mathbb{N}$7 (or LOCC-assisted capacity) and PPT-preserving-assisted capacities $G \in \mathbb{N}$8 do not impose computational restrictions. These are typically upper-bounded via semidefinite programming (SDP), specifically by the $G \in \mathbb{N}$9 bound of Wang and Duan: $\varepsilon > 0$0 with $\varepsilon > 0$1 computable via efficient SDPs and often tighter than the Holevo–Werner bound. These upper bounds remain valid in the computationally unconstrained setting, but do not account for the collapse of capacity induced by computational intractability of decoding or entanglement distillation (Wang et al., 2016).

7. Contextualization and Research Directions

The introduction of computational quantum capacities reorients the study of quantum communication from a focus on ultimate (unconstrained) limits to realistic, resource-informed capabilities. This shift foregrounds the alignment of quantum information protocols with cryptographic assumptions and computational complexity theory. A plausible implication is the emergence of new cryptographically motivated channel constructions and complexity-theoretic lower bounds on capacity. Additionally, efficient capacity analysis may inform practical QKD (quantum key distribution) and repeater architectures, influencing the design and deployment of physically realizable quantum networks.

Key references: (Meyer et al., 21 Jan 2026, Wang et al., 2016)