on mode B. We demonstrate in Fig. \[fig\_hetopt\] that this is true for a typical choice of parameters. Since a heterodyne measurement on mode B collapses mode A into a distribution of coherent states, $\chi_c$ and $A_c$ were calculated by integrating the information quantities of single coherent probe ($\chi_s$, $A_s$) as as function of energy. The computed upper and lower bounds for $A_c$ are equal to within six significant figures.\
*Case 3*. $\chi_s$ and $A_s$: The Holevo information $\chi_s$ is plotted in Fig. \[fig\_plot01\]. Fuchs lower and upper bounds for $A_s$ were calculated and are equal to within seven significant figures, and are indistinguishable in Fig. \[fig\_plot01\]. Unlike case 1, when using a coherent state, the Holevo and accessible information differ by a small amount, only $0.4 \%$.\

![The Holevo information obtained when an EPR state is used for illumination, but a local Gaussian measurement is first performed on mode B. The measurement consists of beam splitter with transmissivity $t$, followed by conjugate homodyne measurements on both outputs. The maximum gives $\chi_c$, which occurs up to numerical precision at $t=0.5$, which corresponds to a heterodyne measurement. Parameters are $\epsilon=0.1$, $\bar n=0.5$, and $\bar n_{\rm env}=4$. []{data-label="fig_hetopt"}](hetopt.pdf){width="8cm"}

![image](plot2a.pdf){width="15.5cm"} ![image](plot2b.pdf){width="15.5cm"}

From Figs. 