  The purpose of a phase-preserving linear amplifier is to make a small signal
larger, regardless of its phase, so that it can be perceived by instruments
incapable of resolving the original signal, while sacrificing as little as
possible in signal-to-noise. Quantum mechanics limits how well this can be
done: a high-gain linear amplifier must degrade the signal-to-noise; the noise
added by the amplifier, when referred to the input, must be at least half a
quantum at the operating frequency. This well-known quantum limit only
constrains the second moments of the added noise. Here we derive the quantum
constraints on the entire distribution of added noise: we show that any
phase-preserving linear amplifier is equivalent to a parametric amplifier with
a physical state for the ancillary mode; the noise added to the amplified field
mode is distributed according to the Wigner function of the ancilla state.
