In physics, a quantum amplifier uses quantum-mechanical processes to amplify a signal; examples include the active elements of lasers and optical amplifiers. Key properties are the amplification coefficient and the uncertainty (noise), which are not independent: higher gain entails higher noise. For lasers, this uncertainty corresponds to amplified spontaneous emission from the active medium. The unavoidable noise of quantum amplifiers, which follows from quantum mechanics, is one reason digital signals are used in optical communications.

An amplifier increases the amplitude of whatever passes through it. Classical amplifiers handle classical signals, while quantum amplifiers handle quantum signals such as coherent states. The output is not necessarily a coherent state—its form typically depends on the amplifier's design. In addition to amplifying intensity, quantum amplifiers can increase the quantum noise present in the signal. The physical electric field in a paraxial single‑mode pulse can be approximated as a superposition of modes. The electric field of a single mode can be written in terms of the spatial coordinate r (with z denoting the direction of propagation), the polarization vector e, the longitudinal wave number kz, and the annihilation operator a for that mode. Noise analysis is performed with respect to the mean value of the annihilation operator; to obtain the noise one solves for the real and imaginary parts of the field's projection onto the given mode. Spatial coordinates do not appear in the solution. Assume that the mean value of the initial field is ⟨a⟩. Physically, the initial state corresponds to the coherent pulse at the input of the optical amplifier and the final state to the amplified output pulse. Only the quantum state of the corresponding mode matters, so the pulse may be treated as a single‑mode field. A quantum amplifier is a unitary transformation that maps the initial state to the amplified state. The amplification depends on the mean value ⟨a⟩ of the field operator and its variance. A coherent state has minimal uncertainty; under the amplifier transformation the uncertainty may increase. This increase can be interpreted as added noise in the amplifier. The gain may be defined in several equivalent ways; in the Heisenberg representation the field operator evolves while the state vector remains unchanged, so amplification is attributed to the evolution of the operator. In general the gain can be complex and may depend on the initial state. For laser applications the amplification of coherent states is most important, so one usually assumes an initial coherent state characterized by a complex amplitude. Even with this restriction the gain may depend on the amplitude or phase of the initial field.

We use the Heisenberg representation and evaluate all expectation values with respect to the initial coherent state. The relevant quantity characterizes the increase of the field uncertainty due to amplification. Because the intrinsic uncertainty of the field operator does not depend on the coherent-state parameter, this quantity indicates how much the output field deviates from a coherent state.

Linear phase-invariant amplifiers may be described as follows. Assume a unitary operator amplifies the field so that the input and output operators are related by a linear relation with c-number coefficients and an amplifier creation operator. Without loss of generality the coefficients may be taken as real. The commutator of the field operators is invariant under the unitary transformation. From unitarity it follows that the amplifier mode satisfies the canonical Bose commutation relations, which imposes constraints on the c-number coefficients. Hence a phase-invariant amplifier acts by introducing an additional bosonic mode that stores energy. The gain and noise of this amplifier can be calculated; the coefficient is sometimes called the intensity amplification coefficient, and the amplifier necessarily adds a minimum amount of noise. A useful property of the linear amplifier is that if several modes are amplified by the same factor, the noise in each mode is determined independently. To obtain large gain with minimal added noise one can use homodyne detection to prepare a field state with known amplitude and phase corresponding to a linear phase-invariant amplifier. The uncertainty principle sets a lower bound on quantum noise in an amplifier. In particular, the outputs of laser systems and optical generators are not coherent states. Nonlinear amplifiers do not have a linear relation between input and output, and their minimum noise cannot be much smaller than that of an idealized linear amplifier. This limit is determined by the derivatives of the mapping function: larger derivatives imply greater uncertainty. Examples include most lasers, which can act as near-linear amplifiers when operating close to threshold and thus show large uncertainty and nonlinear behavior. As with linear amplifiers, they may preserve phase and keep uncertainty low, but there are exceptions, such as parametric oscillators, which amplify while shifting the input phase.