Quantum capacity of the Gaussian random-displacement channel

Problem

What is the unconstrained, unassisted quantum capacity of the single-mode Gaussian random-displacement channel? For a noise standard deviation \(\sigma>0\), define the channel by

\begin{equation} \mathcal N_\sigma(\rho) :=\frac{1}{\pi\sigma^2}\int_{\mathbb C} e^{-\lvert\alpha\rvert^2/\sigma^2} D(\alpha)\rho D(\alpha)^\dagger\,d^2\alpha, \qquad D(\alpha):=e^{\alpha a^\dagger-\alpha^*a}, \tag{1} \end{equation}

where \(a\) is the mode annihilation operator. Equivalently, writing \(\alpha=(x+iy)/\sqrt2\), the channel in Eq. (1) adds independent classical Gaussian shifts \(x,y\sim\mathcal N(0,\sigma^2)\) to the two canonical quadratures. With \(\hat n_j:=a_j^\dagger a_j\), define its energy-unconstrained capacity as

\begin{equation} \begin{aligned} \mathcal Q(\mathcal N_\sigma) &:={\sup}_{0<N_{\rm S}<\infty}\, \lim_{n\to\infty}\frac1n {\sup}_{\substack{\rho_{A^n}:\\ \operatorname{Tr}[\rho_{A^n}\sum_{j=1}^n\hat n_j] \leq nN_{\rm S}}} I_{\rm c}(\rho_{A^n},\mathcal N_\sigma^{\otimes n}),\\ I_{\rm c}(\rho,\mathcal M) &:=S(\mathcal M(\rho))-S(\mathcal M^{\rm c}(\rho)), \end{aligned} \tag{2} \end{equation}

where \(\mathcal M^{\rm c}\) is a complementary channel and logarithms in the entropy are base two. Determine Eq. (2) for every \(\sigma>0\).

Source

Lin and Noh explicitly describe determining \(\mathcal Q(\mathcal N_\sigma)\) as a longstanding open problem [LN25]. Subramanian, Zheng, and Jiang subsequently reaffirm that the exact capacity remains unknown while constructing codes that attain the standard coherent-information rate [SZJ25].

Progress

In the convention of Eq. (1), the standard analytic bounds are

\begin{equation} \left[\log_2\!\frac{1}{e\sigma^2}\right]_+ \leq \mathcal Q(\mathcal N_\sigma) \leq U(\sigma), \qquad U(\sigma):= \begin{cases} \displaystyle\log_2\!\frac{1-\sigma^2}{\sigma^2}, &0<\sigma<1/\sqrt2,\\[1mm] 0,&\sigma\geq1/\sqrt2, \end{cases} \qquad [t]_+:=\max\{t,0\}. \tag{3} \end{equation}

The lower bound in Eq. (3) is the infinite-energy limit of one-use coherent information, while the upper bound follows from a channel decomposition and data processing [HW01], [NAJ19]. Thus the capacity is positive for \(0<\sigma<1/\sqrt e\) and vanishes for \(\sigma\geq1/\sqrt2\), but the bounds do not determine it in the remaining positive-capacity regime; for \(1/\sqrt e\leq\sigma<1/\sqrt2\), even positivity is unknown. Degradable flagged extensions sharpen the upper bound at low noise without closing the gap [FKG21].
Lin and Noh evaluated multimode GKP families with maximum-likelihood decoding. Their largest surface–square instance, of distance \(39\), has a positive computed hashing rate at \(\sigma=0.598\), and its finite-size crossings indicate a threshold close to \(1/\sqrt e\). These are numerical code-performance results below the known positivity threshold, not an exact capacity evaluation or a proof of positivity beyond \(1/\sqrt e\) [LN25].
Subramanian, Zheng, and Jiang constructed concatenated square-GKP and quantum polar codes with analog decoding whose asymptotic rate is

\begin{equation} R_{\rm GKP\text{-}polar}(\sigma) =\left[\log_2\!\frac{1}{e\sigma^2}\right]_+. \tag{4} \end{equation}

Equation (4) removes earlier discreteness and nonconstructiveness restrictions in attaining the lower bound of Eq. (3). It does not prove that this lower bound equals the regularized capacity in Eq. (2) [SZJ25].

Comment

The word “optimal” in the coding result of [SZJ25] refers to attaining the one-use coherent-information lower bound for all noise strengths, not to proving that this rate is the quantum capacity. The exact value remains unknown throughout \(0<\sigma<1/\sqrt2\); in the subinterval \(1/\sqrt e\leq\sigma<1/\sqrt2\), the first unresolved question is whether the capacity is nonzero at all.

References

[HW01]

A. S. Holevo and R. F. Werner, “Evaluating Capacities of Bosonic Gaussian Channels,” Physical Review A 63, 032312 (2001).DOI arXiv

[NAJ19]

K. Noh, V. V. Albert, and L. Jiang, “Quantum Capacity Bounds of Gaussian Thermal Loss Channels and Achievable Rates with Gottesman–Kitaev–Preskill Codes,” IEEE Transactions on Information Theory 65, 2563–2582 (2019).

DOI arXiv

[FKG21]

M. Fanizza, F. Kianvash, and V. Giovannetti, “Estimating Quantum and Private Capacities of Gaussian Channels via Degradable Extensions,” Physical Review Letters 127, 210501 (2021).DOI arXiv

[LN25]

M. Lin and K. Noh, “Exploring the Quantum Capacity of a Gaussian Random-Displacement Channel Using Gottesman–Kitaev–Preskill Codes and Maximum-Likelihood Decoding,” Physical Review A 111, 052445 (2025).DOI arXiv

[SZJ25]

M. Subramanian, G. Zheng, and L. Jiang, “Achievable Rates for Concatenated Square Gottesman–Kitaev–Preskill Codes,” PRX Quantum 6, 040342 (2025).DOI arXiv

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