Random quantum codes from Gaussian ensembles and an uncertainty relation

Using random Gaussian vectors and an information-uncertainty relation, we give a proof that the coherent information is an achievable rate for entanglement transmission through a noisy quantum channel. The codes are random subspaces selected according to the Haar measure, but distorted as a function of the sender’s input density operator. Using large deviations techniques, we show that classical data transmitted in either of two Fourier-conjugate bases for the coding subspace can be decoded with low probability of error. A recently discovered information-uncertainty relation then implies that the quantum mutual information for entanglement encoded into the subspace and transmitted through the channel will be high. The monogamy of quantum correlations finally implies that the environment of the channel cannot be significantly coupled to the entanglement, and concluding, which ensures the existence of a decoding by the receiver.

💡 Research Summary

The paper presents a new proof that the coherent information of a quantum channel is an achievable rate for entanglement transmission. The authors construct random quantum codes by selecting a subspace of the input Hilbert space according to the Haar measure and then “distorting’’ it with the square root of the sender’s input density operator. Concretely, they draw N independent Gaussian vectors |g_j⟩ in the input space, form |γ_j⟩ = √|A| ρ |g_j⟩, and define Γ = Σ_j |γ_j⟩⟨γ_j|. The orthonormal basis of the code subspace is then |φ_j⟩ = Γ^{-1/2}|γ_j⟩.

Two orthogonal bases of this subspace are considered: the “standard’’ basis {|φ_j⟩} and its Fourier‑conjugate {|bφ_k⟩ = N^{-1/2} Σ_j e^{2πijk/N}|φ_j⟩}. By exploiting the unitary invariance of Gaussian vectors and the properties of the discrete Fourier transform, the authors show that the ensembles of output states obtained by sending these basis vectors through the channel have Holevo information arbitrarily close to log N. Large‑deviation bounds for the lengths and inner products of Gaussian vectors guarantee that the average classical decoding error for each ensemble is at most λ = 9√ε + 7√η + 3N exp(−Nε²/6), where ε and η are small parameters that can be chosen during code design.

The central technical tool is an “information‑uncertainty relation’’ (Lemma 7), which states that for any quantum channel M, the sum of the Holevo informations of the two Fourier‑conjugate ensembles is bounded above by the quantum mutual information I(R:B) of the channel’s Choi state. Applying this lemma to the two ensembles yields I(R:B) ≥ 2 log N − O(λ log N). Since for a pure global state I(R:B) + I(R:E) = 2 log N, the mutual information between the reference system R (the logical qubits) and the environment E is at most O(λ log N).

Monogamy of quantum correlations then implies that the environment is essentially decoupled from the transmitted entanglement. Using Pinsker’s inequality, the authors translate the small mutual information into a bound on the trace‑distance error of a recovery map D: P_{q,err} ≤ 2 √{I(R:E)} ≤ O(√λ log N). Thus, with high probability over the random choice of the subspace, there exists a decoding operation that restores the entanglement with arbitrarily low error.

The main results are formalized as two theorems. Theorem 1 shows that for any channel with a given input state ρ and suitable projectors onto the output and environment spaces, one can construct a code of dimension N ≤ min{η D rank P_E, η Δ} with error bounded as above. Theorem 2 (originally due to Lloyd, Shor, and Devetak) follows by applying Theorem 1 to the typical subspace of many channel uses, establishing that the coherent information I_c(ρ;N) is an achievable quantum communication rate. Consequently, the quantum capacity Q(N) equals the regularized coherent information, confirming the well‑known capacity formula.

Overall, the paper provides a conceptually different route to the quantum capacity theorem: rather than directly constructing a decoder, it proves decoupling by showing that both a basis and its Fourier‑conjugate can be reliably distinguished after the channel. The information‑uncertainty relation bridges the classical decoding performance to the quantum mutual information, and monogamy guarantees the existence of a quantum decoder. This approach highlights the power of random Gaussian ensembles and Fourier analysis in quantum coding theory and offers a fresh perspective on the interplay between classical and quantum information in noisy channels.