Albert, Noh, Duivenvoorden et al. (Phys. Rev. A 97, 032346, 2018) compared the four primary single-mode bosonic qubit codes — cat, binomial, numerically optimized, and Gottesman–Kitaev–Preskill (GKP) — against the pure-loss channel using optimal (channel-adapted) recovery under fixed photon-number budgets, and listed the open issues of that comparison. The experimental item (practical GKP realization) has since been resolved in trapped ions and superconducting cavities, and the loss–dephasing channel structure is now substantially understood; the surviving open core is the performance theory of single-mode bosonic codes as the remaining idealizations are removed: (i) the optimal growth of binomial-code parameters and the limiting optimal-recovery entanglement fidelity when no energy constraint is imposed, including whether unit fidelity is attainable by any single-mode code; and (ii) optimal-recovery performance under the source's two-parameter loss+Kerr noise, where a universal large-Kerr failure was observed. The source reports evidence for convergence to a fidelity strictly below one and deems unit fidelity unlikely, leaving the optimal parameter growth and the limiting fidelity as the stated open question.

Background

In bosonic (continuous-variable) quantum error correction, a logical qubit is encoded in a single harmonic-oscillator mode — a microwave cavity, a trapped ion's motion, or an optical field — whose infinite-dimensional Fock space (number states ∣n⟩|n\rangle, eigenstates of n^=a^†a^\hat n=\hat a^\dagger\hat a) supplies the redundancy. The primary single-mode encodings compared by Albert et al. (2018) are: cat codes, superpositions of coherent states ∣α⟩|\alpha\rangle and ∣−α⟩|{-\alpha}\rangle protected by photon-number parity; binomial codes bin(N,S)\mathtt{bin}(N,S), whose logical words are superpositions of Fock states ∣m(S+1)⟩|m(S{+}1)\rangle with binomial-coefficient weights and which exactly correct the error set {I,a^,…,a^L,a^†,…,(a^†)G,n^,…,n^D}\{I,\hat a,\dots,\hat a^L,\hat a^\dagger,\dots,(\hat a^\dagger)^G,\hat n,\dots,\hat n^D\} with S=L+GS=L{+}G and N=max⁡(L,G,2D)N=\max(L,G,2D); and Gottesman–Kitaev–Preskill (GKP, or grid) codes, whose logical words are superpositions of Gaussian peaks on a periodic phase-space lattice and which correct small displacements. Binomial codes are also called number-phase codes in later literature. The dominant noise is photon loss, modeled by the pure-loss channel Nγ\mathcal N_\gamma with loss rate γ=1−η\gamma=1-\eta: each photon survives with probability η\eta, and the loss of ℓ\ell photons acts through Kraus operators proportional to a^ℓ\hat a^{\ell}. Performance is measured by the entanglement fidelity FEF_{\mathcal{E}} of the composed encoding–channel–recovery map acting on half of a maximally entangled pair, maximized over all recovery operations (channel-adapted optimal recovery, computable by semidefinite programming on a truncated Fock space), with codes compared under equal mean-photon-number budgets nˉcode≤2,5,10\bar n_{\mathrm{code}}\le 2,5,10. Under these conditions Albert et al. found GKP codes outperform cat and binomial codes for most γ\gamma and, on numerically relaxing the energy constraint, that binomial codes along rays S=ξNS=\xi N (ξ≥1\xi\ge 1 depending on γ\gamma, e.g. S≈2NS\approx 2N) show monotonically increasing FEF_{\mathcal{E}} — verified only up to N≤14N\le 14, S≤36S\le 36 at γ=0.1\gamma=0.1; they deemed it unlikely that FE→1F_{\mathcal{E}}\to 1 for any single-mode code as nˉ→∞\bar n\to\infty, and posed as an open question which parameter should grow faster for optimal fidelity. Adding a coherent error generated by the Kerr Hamiltonian H^K=K2a^†2a^2=K2n^(n^−1)\hat H_K=\tfrac{K}{2}\hat a^{\dagger 2}\hat a^2=\tfrac{K}{2}\hat n(\hat n-1) (a cavity nonlinearity that rotates each Fock state ∣n⟩|n\rangle by a phase quadratic in nn), they studied the resulting two-parameter loss+Kerr channel Nχ,Kt(⋅)=e−iKt[12n^(n^−1),⋅]+χD(⋅)\mathcal N_{\chi,Kt}(\cdot)=e^{-iKt[\frac{1}{2}\hat n(\hat n-1),\cdot]+\chi\mathcal D(\cdot)} acting over an evolution time tt, where D\mathcal D is the pure-loss Lindbladian with cavity decay rate κ\kappa, so that the two dimensionless scales of the problem are χ=κt\chi=\kappa t and the accumulated Kerr phase KtKt; at a fixed energy budget (nˉ≈2\bar n\approx 2) they observed oscillations at small KtKt and failure of all codes at about the same rate at sufficiently large KtKt. The same study compared achievable quantum-communication rates via the hashing bound (the rate at which the effective channel still distills entanglement, computed under optimal recovery), and all its recoveries are adapted to a perfectly known γ\gamma. Later work established key structure of the combined loss–dephasing channel (Leviant et al. 2022; Mele et al. 2024): it is non-degradable (neither the receiver's output nor the environment's can be reconstructed from the other exactly), and for any loss rate sufficiently strong dephasing makes it antidegradable, in which case no code helps; Wen et al. (2026) then estimated the GKP-versus-number-phase performance boundary under loss–dephasing with optimized code parameters. These advances, however, all work at finite energy and exclude coherent (Kerr) errors — precisely the idealizations whose removal defines the open problem below.

Problem Statement

This is the open code-performance core of Albert et al. (2018, Secs. VI.2, VIII.1, and IX.1): determine the optimal-recovery entanglement-fidelity limits of the primary single-mode bosonic encodings (cat, binomial/number-phase, GKP) as the remaining idealizations of that comparison — the finite energy budget and the exclusion of coherent errors — are removed. Two linked components serve the one objective of completing the single-mode performance theory against realistic bosonic noise. Throughout, FEF_{\mathcal{E}} denotes the entanglement fidelity of the encoding–noise-channel–recovery chain, maximized over all recovery operations (optimal recovery), and bin(N,S)\mathtt{bin}(N,S) the binomial code correcting the error set {I,a^,…,a^L,a^†,…,(a^†)G,n^,…,n^D}\{I,\hat a,\dots,\hat a^L,\hat a^\dagger,\dots,(\hat a^\dagger)^G,\hat n,\dots,\hat n^D\} with S=L+GS=L{+}G and N=max⁡(L,G,2D)N=\max(L,G,2D). (1) No energy constraint, pure loss. For each loss rate γ\gamma, determine the optimal growth of the binomial-code parameters bin(N,S)\mathtt{bin}(N,S) as N,S→∞N,S\to\infty: which ray S=ξNS=\xi N (if any) maximizes the optimal-recovery entanglement fidelity in the limit; whether the numerically observed monotone increase along rays such as S≈2NS\approx 2N persists to the limit or reverses at finite N,SN,S; the limiting value of, or tight bounds on, the fidelity FEbin(∞)(γ)F_{\mathcal{E}}^{\mathrm{bin}(\infty)}(\gamma) and its analogues for the cat and GKP families; and whether FE→1F_{\mathcal{E}}\to 1 is attainable by any single-mode code as nˉ→∞\bar n\to\infty — the source deems this unlikely, so either prove the impossibility for all single-mode codes (in particular in the regime γ<1/2\gamma<1/2 where antidegradability does not already forbid it) or refute it with a concrete family. (2) Coherent Kerr error combined with loss. Under the source's two-parameter loss+Kerr channel Nχ,Kt(⋅)=e−iKt[12n^(n^−1),⋅]+χD(⋅)\mathcal N_{\chi,Kt}(\cdot)=e^{-iKt[\frac{1}{2}\hat n(\hat n-1),\cdot]+\chi\mathcal D(\cdot)}, with dimensionless scales χ=κt\chi=\kappa t and KtKt and at a fixed finite energy budget as in the source's comparison, characterize the optimal-recovery entanglement fidelity of these single-mode families, and decide whether the observed failure of all single-mode codes at sufficiently large KtKt is a fundamental limitation of single-mode encodings or an artifact of the specific codes and recoveries studied. A solution is a rigorous resolution of at least one of the two components, which address a common objective but can be resolved independently. Component (1) is resolved only when its sub-questions are settled jointly — the optimal ray S=ξNS=\xi N, the persistence or reversal of the monotone increase, the limiting fidelity FEbin(∞)(γ)F_{\mathcal{E}}^{\mathrm{bin}(\infty)}(\gamma) or tight bounds on it together with the cat and GKP analogues, and the attainability of FE→1F_{\mathcal{E}}\to 1 by some single-mode code — with a result that settles only some of these counting as partial progress and stating which sub-questions remain open; component (2) is resolved when the optimal-recovery entanglement fidelity under the loss+Kerr channel is characterized and the large-KtKt failure is classified as fundamental or artifactual.

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