  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.
