Quantum communication: a discrimination approach

Abstract

We develop a state discrimination approach to unassisted quantum communication through a noisy quantum channel. Starting from the operational duality between entanglement recovery and state discrimination, we construct entanglement-generation codes with explicit decoders constructed from Petz recovery maps. The resulting error bounds hold for every number of channel uses and are expressed in terms of Petz-Rényi coherent information. For product sources, they yield exponential error decay at rates below coherent information.

The paper studies whether unassisted quantum communication can be analyzed directly through state discrimination, with explicit decoders rather than a decoder whose existence follows from decoupling. It starts from a known operational duality: the optimal squared purified-distance error for recovering a rank-$d$ maximally entangled state equals both the discrimination error of $d^2$ equiprobable Weyl-rotated output states and an optimized environmental decoupling error. Using random-subspace preparations and discrimination measurements, it constructs decoders corresponding to ordinary Petz and averaged rotated Petz recovery. For a source output $\rho_{RB}$, every integer $d\geq2$, and $\alpha\in[1/2,1)$, the construction gives an average entanglement-generation error bounded by $\bar\varepsilon(d)\leq c_\alpha 2^{-\frac{1-\alpha}{\alpha}[I_\alpha(R\rangle B)_\rho-\log_2d]}$, where $I_\alpha$ is Petz–Rényi coherent information and $c_\alpha\in[1/2,1]$. Stronger finite-dimensional bounds include subtractive corrections; a noiseless-channel example establishes sharpness of the prefactor. Product sources give exponential error decay below coherent information. A separate binary-hypothesis-testing bound is $\varepsilon(d)\leq\operatorname{Tr}[\rho_{RB}\wedge dI_R\otimes\rho_B]$, also obtained in concurrent independent work. The analysis combines the decoder–measurement correspondence, a sharp operator tilting inequality, operator concavity, and Haar second moments; exact unitary $2$-designs suffice. Standard conversions preserve the achievable exponent for entanglement and strong subspace transmission. The setting is finite dimensional, and random coding proves existence of a fixed code rather than an efficient implementation. The resulting exponent agrees with an earlier decoupling exponent when both use squared purified distance; the second-order consequence likewise reproduces an existing achievable rate.