Amplification uncertainty relation for probabilistic amplifiers

Traditionally, quantum amplification limit refers to the property of inevitable noise addition on canonical variables when the field amplitude of an unknown state is linearly transformed through a quantum channel. Recent theoretical studies have determined amplification limits for cases of probabilistic quantum channels or general quantum operations by specifying a set of input states or a state ensemble. However, it remains open how much excess noise on canonical variables is unavoidable and whether there exists a fundamental trade-off relation between the canonical pair in a general amplification process. In this paper we present an uncertainty-product form of amplification limits for general quantum operations by assuming an input ensemble of Gaussian-distributed coherent states. It can be derived as a straightforward consequence of canonical uncertainty relations and retrieves basic properties of the traditional amplification limit. In addition, our amplification limit turns out to give a physical limitation on probabilistic reduction of an Einstein-Podolsky-Rosen uncertainty. In this regard, we find a condition that probabilistic amplifiers can be regarded as local filtering operations to distill entanglement. This condition establishes a clear benchmark to verify an advantage of non-Gaussian operations beyond Gaussian operations with a feasible input set of coherent states and standard homodyne measurements.

Figure 1

(a) Solid curves represent general lower bounds on the product of the mean-square deviations (MSD) of Eq. (18) given by ${\overline{V}}_x{\overline{V}}_p\ge ({\eta }^{\prime }+|{\eta }^{\prime }\mp 1{\left|\right)}^2/4$ [see Eq. (26)]. Here, an effective gain factor is set to be ${\eta }^{\prime }=\eta /\left(1+\lambda \right)=1.3$ with the constraint on the gain pair $\sqrt{{\eta }_x{\eta }_p}=\eta$. This constraint parameterizes the gain pair as $({\eta }_x,{\eta }_p)=(\eta e^R,\eta e^{-R})$, with $R\in \mathbb{R}$. The upper solid curve is for the phase-conjugation task, and the lower solid curve is for the normal amplification task. Each of them can be determined by taking the intersection of our amplification limits in Eq. (19) with various ratios between the gain pair ${\eta }_x/{\eta }_p=e^{2R}$. Dotted curves represent the cases for ${\eta }_x/{\eta }_p=1$ and ${\eta }_x/{\eta }_p=2$. (b) Typical behavior of the MSDs in the cases of ${\overline{V}}_x={\overline{V}}_p=\overline{V}$ as a function of the gain $\eta$. The upper solid line represents the limitation on the phase-conjugation process [Eq. (30)], and the lower kinked solid line represents the limitation on the normal amplification and attenuation process [Eq. (28)]. The dashed curve for $\eta \in [1,{(1+\eta )}^2]$ shows the minimum of the MSD due to Gaussian channels ${\overline{V}}^{☆}$ [Eq. (33)]. Dash-dotted lines are due to the traditional form of amplification limits for completely unknown coherent states [Eqs. (A10) and (A11) with $G=\eta ]$. They can be retrieved by our bounds in the limit of $\lambda \to 0$. In this figure we set $\lambda =0.4$ so that the structure around $\eta =1+\lambda$ is displayed clearly. When $\eta =0$, all lines indicate the minimum value of $\overline{V}=1/2$ due to canonical uncertainty relations as it corresponds to the trivial case of $g_x=g_p=0$ in Eq. (20).