Classical capacity of quantum non-Gaussian attenuator and amplifier channels

TL;DR

The paper addresses the classical capacity of quantum non-Gaussian attenuator and amplifier channels formed by coupling an input mode to an arbitrary environment via a Gaussian unitary. It leverages the Gaussian-equivalent channel 𝓜_G to bound the capacity, proving C(𝓜_G) ≤ C(𝓜) ≤ C(𝓜_G) + Δ, with Δ determined by the non-Gaussianity of the environment through MOE-related quantities and relative entropies. The authors derive exact Gaussian capacities, establish lower bounds using Gaussian encodings, and present upper bounds tied to the minimum output entropy of 𝓜 and 𝓜_G, culminating in Δ_max as a key figure that captures potential non-Gaussian gains. Through extensive numerical MOE studies and symmetry-based conjectures, they show that coherent states often minimize MOE for phase-invariant environments, while non-Gaussian MOE states respecting environmental symmetries can arise in more structured environments. These results provide a practical pathway to access the capacity of broad classes of non-Gaussian bosonic channels and motivate future work on exact MOE characterizations and multimode extensions.

Abstract

We consider a quantum bosonic channel that couples the input mode via a beam splitter or two-mode squeezer to an environmental mode that is prepared in an arbitrary state. We investigate the classical capacity of this channel, which we call a non-Gaussian attenuator or amplifier channel. If the environment state is thermal, we of course recover a Gaussian phase-covariant channel whose classical capacity is well known. Otherwise, we derive both a lower and an upper bound to the classical capacity of the channel, drawing inspiration from the classical treatment of the capacity of non-Gaussian additive-noise channels. We show that the lower bound to the capacity is always achievable and give examples where the non-Gaussianity of the channel can be exploited so that the communication rate beats the capacity of the Gaussian-equivalent channel (i.e., the channel where the environment state is replaced by a Gaussian state with the same covariance matrix). Finally, our upper bound leads us to formulate and investigate conjectures on the input state that minimizes the output entropy of non-Gaussian attenuator or amplifier channels. Solving these conjectures would be a main step towards accessing the capacity of a large class of non-Gaussian bosonic channels.

Figures (9)

• Figure 1: Left panel : Classical non-Gaussian additive-noise channel, where $X$, $Y$, and $N$ denote respectively the input, output, and noise random variables. The noise $N$ admits an arbitrary probability density. Right panel : quantum non-Gaussian attenuator channel. The input state $\hat{\rho}$ is coupled with an arbitrary environment state $\hat{\sigma}$ through a beam splitter of transmittance $\eta$, resulting in an output state ${\cal M}[\hat{\rho}]$ while tracing out the other output mode. Here ${\cal M}$ denotes the corresponding trace-preserving completely-positive map acting on $\hat{\rho}$.
• Figure 2: Commutation properties of the beam splitter. For each column, the upper and lower setups yield the same two-mode unitary operation (up to a global phase). In the first column (displacement), the relation holds provided $\beta_1=\sqrt{\eta}\,\alpha_1+\sqrt{1-\eta}\,\alpha_2$ and $\beta_2=-\sqrt{1-\eta}\,\alpha_1+\sqrt{\eta}\,\alpha_2$. In the second column (rotation), the commutation relation holds provided all rotation operators have the same angle $\theta$. In the third column (reflection), the commutation relation holds provided all reflection operators have the same angle $\theta$. $(^\ast$ Note that reflections are not physically implementable since they are related to the phase-conjugation operator, which is anti-unitary.) In the fourth column (squeezing), the commutation relation holds provided all squeezing operators have the same squeezing parameter $r$.
• Figure 3: Capacity interval width $\Delta$ for several Fock attenuator channels (with the environment in state $\hat{\sigma}=\left| n \newline \right>\left< n \newline \right|$) as a function of the transmittance $\eta$. We note that $\Delta$ is independent of the constraint on the photon number $\nu$ at the input.
• Figure 4: Upper and lower bounds on the classical capacity $\mathcal{C}$ of a non-Gaussian quantum channel as a function of the photon number $\nu$ at the input. The illustrated example is a Fock attenuator with environment $\hat{\sigma}=\ket{1}\bra{1}$ and transmittance $\eta=1/2$. The classical capacity $\mathcal{C}$ must lie in the blue area, between the lower bound $\mathcal{C}_G$ and the upper bound $\mathcal{C}_G+\Delta$. In the regime of low $\nu$, the capacity $\mathcal{C}$ goes to zero so that $\mathcal{C}\rightarrow\mathcal{C}_G$ for $\nu\ll 1$. In the regime of high $\nu$, the capacity $\mathcal{C}$ tends to the upper bound $\mathcal{C}\rightarrow\mathcal{C}_G+\Delta$ for $\nu\ll 1$ as the bound becomes asymptotically achievable.
• Figure 5: Wigner function of the state $\ket{\psi}=(\ket{0}+\ket{3})/\sqrt{2}$. The state $\ket{\psi}$ has a zero mean displacement vector and a covariance matrix proportional to the identity. It has 2 rotation symmetries, namely $\hat{R}_\theta$ for $\theta\in\lbrace{2\pi/3,4\pi/3\rbrace}$, and 3 reflection symmetries, namely $\hat{M}_\theta$ for $\theta\in\lbrace 0,2\pi/3,4\pi/3\rbrace$. We are considering a quantum attenuator channel whose environment is in state $\ket{\psi}$.

Theorems & Definitions (4)

• Lemma 1: Channel covariance
• Conjecture 1: Fock attenuator
• Conjecture 2: Phase-covariant attenuator
• Conjecture 3: Phase-space symmetries