  In an ideal linear amplifier, the output signal is linearly related to the
input signal with an additive noise that is independent of the input. The
decoherence of a quantum-mechanical state as a result of optical amplification
is usually assumed to be due to the addition of quantum noise. Here we show
that entanglement between the input signal and the amplifying medium can
produce an exponentially-large amount of decoherence in an ideal optical
amplifier even when the gain is arbitrarily close to unity and the added noise
is negligible. These effects occur for macroscopic superposition states, where
even a small amount of gain can leave a significant amount of which-path
information in the environment. Our results show that the usual input/output
relation of a linear amplifier does not provide a complete description of the
output state when post-selection is used.
