Quantum operation

In quantum mechanics, a quantum operation (also known as quantum dynamical map or quantum process) is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo. This was first discussed as a general stochastic transformation for a density matrix by George Sudarshan.^[1] The quantum operation formalism describes not only unitary time evolution or symmetry transformations of isolated systems, but also the effects of measurement and transient interactions with an environment. In the context of quantum computation, a quantum operation is called a quantum channel.

Note that some authors use the term "quantum operation" to refer specifically to completely positive (CP) and non-trace-increasing maps on the space of density matrices, and the term "quantum channel" to refer to the subset of those that are strictly trace-preserving.^[2]

Quantum operations are formulated in terms of the density operator description of a quantum mechanical system. Rigorously, a quantum operation is a linear, completely positive map from the set of density operators into itself. In the context of quantum information, one often imposes the further restriction that a quantum operation ${\displaystyle {\mathcal {E}}}$ must be physical,^[3] that is, satisfy ${\displaystyle 0\leq \operatorname {Tr} [{\mathcal {E}}(\rho )]\leq 1}$ for any state ${\displaystyle \rho }$.

Some quantum processes cannot be captured within the quantum operation formalism;^[4] in principle, the density matrix of a quantum system can undergo completely arbitrary time evolution. Quantum operations are generalized by quantum instruments, which capture the classical information obtained during measurements, in addition to the quantum information.