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In mathematics, Choi's theorem on completely positive maps is a result that classifies completely positive maps between finite-dimensional (matrix) C*-algebras. An infinite-dimensional algebraic generalization of Choi's theorem is known as Belavkin 's "Radon–Nikodym" theorem for completely positive maps.
Mathematically, a quantum operation is a linear map Φ between spaces of trace class operators on Hilbert spaces H and G such that. If S is a density operator, Tr (Φ (S)) ≤ 1. Φ is completely positive, that is for any natural number n, and any square matrix of size n whose entries are trace-class operators and which is non-negative, then is ...
Quantum channel. In quantum information theory, a quantum channel is a communication channel which can transmit quantum information, as well as classical information. An example of quantum information is the general dynamics of a qubit. An example of classical information is a text document transmitted over the Internet.
The Choi-Jamiołkowski isomorphism is a mathematical concept that connects quantum gates or operations to quantum states called Choi states. It allows us to represent a gate's properties and behavior as a Choi state. In the generalised gate teleportation scheme, we can teleport a quantum gate from one location to another using entangled states ...
Stinespring's theorem says that all completely positive maps are compositions of *-homomorphisms and these special maps. Every positive functional (in particular every state) is automatically completely positive. Given the algebras and of complex-valued continuous functions on compact Hausdorff spaces , every positive map is completely positive.
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose —that is, the element in the i -th row and j -th column is equal to the complex conjugate of the element in the j -th row and i -th column, for all indices i and j: or in matrix form:
Freudenthal magic square. In mathematics, the Freudenthal magic square (or Freudenthal–Tits magic square) is a construction relating several Lie algebras (and their associated Lie groups ). It is named after Hans Freudenthal and Jacques Tits, who developed the idea independently. It associates a Lie algebra to a pair of division algebras A, B.
Entanglement witness. In quantum information theory, an entanglement witness is a functional which distinguishes a specific entangled state from separable ones. Entanglement witnesses can be linear or nonlinear functionals of the density matrix. If linear, then they can also be viewed as observables for which the expectation value of the ...