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Kraus matrices are not uniquely determined by the quantum operation in general. For example, different Cholesky factorizations of the Choi matrix might give different sets of Kraus operators. The following theorem states that all systems of Kraus matrices representing the same quantum operation are related by a unitary transformation:
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.
The representation he used for these maps is now known as the Kraus Representation, Kraus Operator Formalism or Operator-Sum Formalism, and is now frequently used in the field of quantum information. The Kraus representation is based on a theorem of WF Stinespring about completely positive images of finite-dimensional C*-algebras. [6]
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 ...
Quantum decoherence is the loss of quantum coherence. Quantum decoherence has been studied to understand how quantum systems convert to systems which can be explained by classical mechanics. Beginning out of attempts to extend the understanding of quantum mechanics, the theory has developed in several directions and experimental studies have ...
Quantum depolarizing channel. A quantum depolarizing channel is a model for quantum noise in quantum systems. The -dimensional depolarizing channel can be viewed as a completely positive trace-preserving map , depending on one parameter , which maps a state onto a linear combination of itself and the maximally mixed state, The condition of ...
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.