In physics, a quantum amplifier is an amplifier that uses quantum mechanical methods to amplify a signal; examples include the active elements of lasers and optical amplifiers.

The main figures of merit of a quantum amplifier are its gain and added noise. These parameters are not independent. In a linear amplifier, the higher the gain, the higher the added noise. In lasers, the added noise corresponds to the amplified spontaneous emission of the active medium. The unavoidable noise of quantum amplifiers is one of the reasons for the use of digital signals in optical communications and can be deduced from the fundamentals of quantum mechanics.^{[citation needed]}

An amplifier increases the amplitude of its input. While classical amplifiers take in classical signals, quantum amplifiers take in quantum signals, such as coherent states. This does not necessarily mean that the output is a coherent state; indeed, typically it is not. The form of the output depends on the specific amplifier design. Besides amplifying the intensity of the input, quantum amplifiers can also increase the quantum noise present in the signal.

The physical electric field in a paraxial single-mode pulse can be approximated with superposition of modes; the electric field ${\displaystyle ~E_{\rm {phys}}~}$ of a single mode can be described as

where

The analysis of the noise in the system is made with respect to the mean value^{[clarification needed]} of the annihilation operator. To obtain the noise, one solves for the real and imaginary parts of the projection of the field to a given mode ${\displaystyle ~M({\vec {x}})~}$. Spatial coordinates do not appear in the solution.

Denote the expectation value of the initial field as ${\displaystyle ~{\left\langle {\hat {a}}\right\rangle _{\rm {initial}}}~}$. Physically, the initial state corresponds to the coherent pulse at the input of the optical amplifier; the final state corresponds to the output pulse. The amplitude-phase behavior of the pulse must be known, although only the quantum state of the corresponding mode is important. The pulse may be treated in terms of a single-mode field.

The action of a quantum amplifier is a unitary transform ${\displaystyle {\hat {U}}}$, acting on the initial state ${\displaystyle ~|{\rm {initial}}\rangle ~}$ and producing the amplified state ${\displaystyle ~|{\rm {final}}\rangle ~}$. In the Schrödinger representation, this is