Gaussian capacity of the quantum bosonic memory channel with additive correlated Gaussian noise

We present an algorithm for calculation of the Gaussian classical capacity of a quantum bosonic memory channel with additive Gaussian noise. The algorithm, restricted to Gaussian input states, is applicable to all channels with noise correlations obeying certain conditions and works in the full input energy domain, beyond previous treatments of this problem. As an illustration, we study the optimal input states and capacity of a quantum memory channel with Gauss-Markov noise [J. Schäfer, Phys. Rev. A 80, 062313 (2009)]. We evaluate the enhancement of the transmission rate when using these optimal entangled input states by comparison with a product coherent-state encoding and find out that such a simple coherent-state encoding achieves not less than 90$\%$ of the capacity.

Figure 1

Stacked bar plot: optimal eigenvalues of both quadratures. Above and below the threshold the optimal input state is squeezed and the less noisy quadrature is more modulated. (a) $\lambda >{\lambda }_{\mathrm{thr}}$. The optimal eigenvalues are determined by the quantum water-filling solution. ${\overline{\nu }}_{\mathrm{WF}}$ denotes the water-filling level. (b) $\lambda <{\lambda }_{\mathrm{thr}}$. The modulation ${\gamma }_{\mathrm{mod}}^q=0$ since ${\gamma }_{\mathrm{env}}^q>{\gamma }_{\mathrm{env}}^p$. ${\overline{\nu }}_{\mathrm{WF}}$ denotes the “virtual” water-filling level.

Figure 2

Stacked area plot: optimal input and modulation eigenvalue spectra ${\gamma }_{\mathrm{in}}^q\left(x\right),{\gamma }_{\mathrm{mod}}^q\left(x\right)$ and noise spectrum ${\gamma }_{\mathrm{env}}^q\left(x\right)$ (for $p$ in dashed and dotted curves) for a particular $\phi$ and $N$. (a) Global quantum water-filling solution, where $\lambda >{\lambda }_{\mathrm{thr}}$. ${\overline{\nu }}_{\mathrm{WF}}$ (solid bar) denotes the water-filling level. (b) Below threshold: Modes with $x\in [\alpha ,\pi -\alpha ]$ belong to ${\mathcal{N}}_3$ with water-filling level ${\overline{\nu }}_{\mathrm{WF}}$ (solid bar). Modes with $x\in [0,\alpha ]$ and $x\in [\pi -\alpha ,\pi ]$ belong to set ${\mathcal{N}}_2$, where the modulation is below ${\overline{\nu }}_{\mathrm{WF}}$.

Figure 3

Functions ${\mu }_0\left(x\right)$ (upper curve) ${\mu }_{\mathrm{thr}}\left(x\right)$ (lower curve) for (a) $\phi =0.5$, (b) $\phi =0.7$, (c) $\phi =0.9$, and (d) $\phi =0.99$. For all plots we took $N=1$.

Figure 4

Functions ${\mu }_{\mathrm{thr}}\left(x\right)$ (lower dashed curve) and ${\mu }_0\left(x\right)$ (upper dashed curve) and values for $\mu$ (solid bars) for different input energies $\lambda$, and noise parameters $\phi =0.85$, $N=1$. From top to bottom the values are $\mu =1.45(\lambda =1.006),1.34(\lambda =1.04),0.42(\lambda =3)$, and 0.04 $(\lambda =35)$. The numbers indicate the intervals on the $x$ axis that belong to set ${\mathcal{N}}_1$, ${\mathcal{N}}_2$, or ${\mathcal{N}}_3$.

Figure 5

Optimal input ${\gamma }_{\mathrm{in}}^q\left(x\right)$ (solid curve) and modulation ${\gamma }_{\mathrm{mod}}^q\left(x\right)$ (dashed curve) eigenvalue spectra of the $q$ quadrature ($p$-quadrature spectra are the same but mirrored with respect to a vertical line at $\pi /2$) vs. the spectral parameter $x$, for $\phi =0.85$, $N=1$, and $\lambda <{\lambda }_{\mathrm{thr}}$. The partitioning in sets is taken from Fig. 4. (a) $\lambda =3$, which corresponds to $\mu =0.42$; (b) $\lambda =1.04$, which corresponds to $\mu =1.34$.

Figure 6

(a) Capacity $C$ (in bits) vs. correlation $\phi$, where, from top to bottom, $N=1$, 2, and 3. (b) Capacity $C$ (in bits) vs. noise variance $N$, where, from top to bottom, $\phi =0.9$, 0.7, and 0.5. The input energy is $\lambda =3$ for both plots. The dashed part of the curves corresponds to the global water-filling solution with $\lambda >{\lambda }_{\mathrm{thr}}$. One observes that the capacity for full correlations $\phi \to 1$ tends to the capacity of the ideal noiseless channel $N=0$.

Figure 7

Gain $G$ vs. $\overline{n}$ for an infinite number of modes and for two modes (inset). For both plots we took $\mathrm{SNR}=3$ and $\phi =0.7$ (solid curve), $\phi =0.9$ (dashed curve), and $\phi =0.99$ (dotted curve).

Figure 8

Contour plot of the maximal gain ${\max }_{\overline{n}}G$ vs. $\phi$, SNR for two modes. In the area above the dotted line the quantum water-filling solution holds.

Figure 9

Contour plot of the maximal gain ${\max }_{\overline{n}}G$ vs. $\phi ,\mathrm{SNR}$ for an infinite number of modes. In the area left of the dotted line the global quantum water-filling solution holds.