Equivalence Relations for the Classical Capacity of Single-Mode Gaussian Quantum Channels

We prove the equivalence of an arbitrary single-mode Gaussian quantum channel and a newly defined fiducial channel preceded by a phase shift and followed by a Gaussian unitary operation. This equivalence implies that the energy-constrained classical capacity of any single-mode Gaussian channel can be calculated based on this fiducial channel, which is furthermore simply realizable with a beam splitter, two identical single-mode squeezers, and a two-mode squeezer. In a large domain of parameters, we also provide an analytical expression for the Gaussian classical capacity, exploiting its additivity, and prove that the classical capacity cannot exceed it by more than $1/\ln 2$ bits.

Figure 1

Admissible regions in the parameter space ($\tau$, $y$) for Gaussian quantum channels. Each thermal channel ${\Phi }_{(\tau ,y)}^{\mathrm{TH}}$ is associated with a particular point ($\tau ,y$). The vertical line $\tau =0$ corresponds to the zero-transmission channel as well as the classical signal channel ${\Phi }^{\mathrm{CS}}$. The vertical line $\tau =1$ corresponds to the classical additive-noise channel. Both the perfect transmission channel and the single-quadrature classical noise channel ${\Phi }^{\mathrm{SQ}}$ correspond to ($\tau =1$, $y=0$). The Gaussian capacity of ${\Phi }_{(\tau ,y,s)}^F$ is additive if $\overline{N}\ge {\overline{N}}_{\mathrm{thr}}$. This is equivalent to $y\le y_{\mathrm{thr}}=|\tau |[e^{-2|s|}(1+2\overline{N})-1]/(1-e^{-4|s|})$. An example of $y_{\mathrm{thr}}$ is given by the dashed line, where $\overline{N}=0.5$ and $s=0.12$.

Figure 2

Realization of (a) the thermal channel ${\Phi }^{\mathrm{TH}}$ and (b) the fiducial channel ${\Phi }^F$ by a beam splitter with transmissivity $T$, a two-mode squeezer with gain $G$, and a single-mode squeezer $S$. Here $|0⟩$ stands for the vacuum state, and “$⊣$” denotes “tracing out” the mode.