User:Hendra I. Nurdin/Sandbox

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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 $\scriptstyle \rho '={\frac {P_{i}\rho P_{i}}{{\rm {tr}}(P_{i}\rho P_{i})}}$, where $\scriptstyle P_{i}$ is the projection operator $\scriptstyle P_{i}\,=\,\psi _{i}\psi _{i}^{*}$. The quantum operation associated with this measurement is a linear map $\scriptstyle T$ acting on an unnormalized density operator on $\scriptstyle \mathbb {C} ^{n}$ as $\scriptstyle T:\,d\,\mapsto \,{\rm {tr}}(P_{i}dP_{i})$. Therefore, the density operator $\scriptstyle \rho '$ after the measurement is just a normalized version of $\scriptstyle T(\rho )$.

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 $\scriptstyle Q$ and a projective measurement of X is performed on each copy of $\scriptstyle Q$. 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 $\scriptstyle \lambda _{1}$ or $\scriptstyle \lambda _{n}$. Now, if each system in the ensemble has been identically prepared in a state with density operator $\rho$ then the density operator of the reduced ensemble after selective measurement can be described via the quantum operation $\scriptstyle T$ given by $\scriptstyle T:\,d\,\mapsto \,P_{1}dP_{1}+P_{n}dP_{n}$. That is, $\scriptstyle T(\rho )=P_{1}\rho P_{1}+P_{n}\rho P_{n}$ and the density operator $\rho '$ of the post-measurement ensemble is simply $\scriptstyle \rho '\,=\,{\frac {T(\rho )}{{\rm {tr}}(\rho )}}$. Descriptions of more complex transformation of the state of ensembles of quantum systems can be conveniently be given via the formalism of quantum operation valued measures.

mixed-state is a concept introduced for describing a quantum mechanical system whose state is not known precisely, but can possibly be in a collection of pure states (i.e., elements of the Hilbert space of the quantum mechanical system with unity norm) with a certain probability of being in one of the pure states in this collection. For example, consider a finite-dimensional quantum system that has been prepared in a pure state $\scriptstyle \phi$ and measurement of an observable X on this system with, say, eigenvalue-eigenvector pairs $\scriptstyle (\lambda _{i},\psi _{i})$ for i = 1, ..., n, where $\scriptstyle \lambda _{i}$ and $\scriptstyle \psi _{i}$ is an eigenvalue and eigenvector of X, respectively. For a simple illustration, consider the situation where an (external) observer is told that a projective measurement of X has been carried out, but the particular outcome of the measurement is not revealed to this observer. Now, when a projective measurement is performed and the measurement outcome recorded is $\scriptstyle \lambda _{i}$ according to quantum mechanics the probability $\scriptstyle p_{i}$ of obtaining this measurement result is $\scriptstyle p_{i}\,=\,|\langle \phi ,\psi _{i}\rangle \rangle |^{2}$ (here $\scriptstyle \langle \cdot ,\cdot \rangle$ denotes the inner product between two vectors in the associated Hilbert space), and immediately after measurement the state of the system becomes (or, in popular terminology, “collapses” to) $\scriptstyle \psi _{i}$. However, to our observer, from whom the measurement outcome has been kept secret, based on the information that measurement has been performed alone the best description that he or she has of the current (post-measurement) state of the system is that it can be any of the states in the collection $\scriptstyle \{\psi _{i};\,i\,=\,1,\ldots ,n\}$ with probability $\scriptstyle p_{i}$ of being in state $\scriptstyle \psi _{i}$. Thus, to the observer, the system after the measurement would be in a mixed state.

The mixed state is an extremely useful extension of the original (but limited) notion of “state” in quantum mechanics (that is, pure states) that allows an effective description of various scenarios that are encountered in theory and experiments that cannot be described with the pure state formalism, such as selective measurements (on ensembles). An intuitively obvious way of representing a mixed state would be to write it as (in this instance, for the case of the finite dimensional quantum system in the above paragraph) $\scriptstyle \{(p_{i},\psi _{i});\;i\,=\,1,\ldots ,n)\}$, physicists have found that a more effective operational way of representing a mixed state is via a so-called density operator. In the density operator formalism, a pure state $\scriptstyle \phi$ would represented by a projection operator $\scriptstyle P\,=\,\phi \phi ^{*}$ (here $\scriptstyle \phi ^{*}$ denotes the linear functional $\scriptstyle \phi ^{*}(\cdot )\,=\,\langle \phi ,\cdot \rangle$, which for quantum mechanical systems described on $\scriptstyle \mathbb {C} ^{n}$ can be identified with the Hermitian transpose of $\scriptstyle \phi$), while a mixed state $\scriptstyle \{(p_{i},\psi _{i});\;i\,=\,1,\ldots ,n)\}$ in the notation above is described by the density operator $\scriptstyle \rho \,=\,\sum _{i=1}^{n}p_{i}P_{i}$ with $\scriptstyle P_{i}\,=\,\psi \psi ^{*}$ (note that each $\scriptstyle P_{i}$ is also a density operator). Density operators are trace class operators on the Hilbert space of the system with unity trace. The density operator corresponding to a pure state satisfied $\scriptstyle \{{\rm {tr(\rho ^{2})\,=\,1}}$, while the density operator of a mixed state satisfies $\scriptstyle \{{\rm {tr(\rho ^{2})\,<\,1}}$, where $\scriptstyle {\rm {tr}}(\cdot )$ denotes the trace of the operator.