Quantum operation

In physics, in particular in mathematical physics, a quantum operation is a mathematical formalism used to describe general transformations of states of a quantum (mechanical) system. The state of a quantum system on a Hilbert space $\scriptstyle {\mathcal {H}}$ is represented by a non-negative definite trace class operator on $\scriptstyle {\mathcal {H}}$ with trace equal to one. Such operators are called density operators. However, the quantum operation formalism is not defined on density operators, but rather on more general class of non-negative definite trace class operators that need not have trace one, that is the class that is sometimes referred to as unnormalized density operators.

Suppose that the class of unnormalized density operators is denoted by $\scriptstyle {\mathcal {D}}$ then a quantum operation T is a linear map that takes any element of $\scriptstyle {\mathcal {D}}$ and sends it to another element of $\scriptstyle {\mathcal {D}}$ with the property that $\scriptstyle {\rm {tr}}(T(\rho ))\leq {\rm {tr}}(\rho )$ for all $\scriptstyle \rho \,\in \,{\mathcal {D}}$, where $\scriptstyle {\rm {tr}}(\rho )$ denotes the trace of $\scriptstyle \rho$.

To illustrate, consider the projective measurement of an observable (i.e., a self-adjoint, densely defined operator) X of a quantum system $\scriptstyle Q$ with Hilbert space $\scriptstyle \mathbb {C} ^{n}$, and suppose that X has a finite set of eigenvalues $\lambda _{k}$ and a corresponding set of orthonormal eigenvectors $\scriptstyle \psi _{k}$, $\scriptstyle k\,=\,1,\ldots ,n$. Say that the density operator of the system prior to measurement is $\scriptstyle \rho$, then after a projective measurement of X is performed and the outcome observed is $\scriptstyle \lambda _{i}$ the state transforms $\scriptstyle \rho$ to a new state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \rho'=\frac{P_i \rho P_i}{{\rm tr}(P_i \rho P_i)}} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle P_i} is the projection operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle P_i\,=\,\psi_i \psi_i^*} . The quantum operation associated with this measurement is a linear map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle T} acting on an unnormalized density operator on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbb{C}^n} as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle T:\, d \,\mapsto\, {\rm tr}(P_i d P_i) } . Therefore, the density operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \rho'} after the measurement is just a normalized version of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle T(\rho)} (that is, normalized to have trace one).

To look at a slightly more complicated example than described in the previous paragraph, imagine that we now have an infinite ensemble of identical copies of the quantum system Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle Q} and a projective measurement of X is performed on each copy of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle Q} . Furthermore, suppose that we perform a selective measurement on this ensemble by discarding, after the measurements have been made, all systems in the ensemble who measurement outcome is {\em} not Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \lambda_1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \lambda_n } . Now, if each system in the ensemble has been identically prepared in a state with density operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho} then the density operator of the reduced ensemble after selective measurement can be described via the quantum operation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle T} given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle T:\, d \,\mapsto \, P_1 d P_1 + P_n d P_n} . That is, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle T(\rho)= P_1 \rho P_1 + P_n \rho P_n} and the density operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho'} of the post-measurement ensemble is simply Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \rho' \,=\, \frac{T(\rho)}{{\rm tr}(T(\rho))}} .

Descriptions of more complex transformation of the state of ensembles of quantum systems can be conveniently given using the language of quantum operation valued measures.