By Ved Prakash Gupta, Prabha Mandayam, V.S. Sunder
This e-book presents readers with a concise advent to present stories on operator-algebras and their generalizations, operator areas and operator platforms, with a distinct specialise in their program in quantum details technological know-how. This simple framework for the mathematical formula of quantum details might be traced again to the mathematical paintings of John von Neumann, one of many pioneers of operator algebras, which varieties the underpinning of most modern mathematical remedies of the quantum concept, along with being the most dynamic parts of 20th century useful research. at the present time, von Neumann’s foresight unearths expression within the quickly starting to be box of quantum info concept. those notes assemble the content material of lectures given by means of a truly exclusive workforce of mathematicians and quantum details theorists, held on the IMSc in Chennai a few years in the past, and nice care has been taken to give the fabric as a primer at the material. ranging from the fundamental definitions of operator areas and operator structures, this article proceeds to debate a number of very important theorems together with Stinespring’s dilation theorem for thoroughly confident maps and Kirchberg’s theorem on tensor items of C*-algebras. It additionally takes a better examine the summary characterization of operator platforms and, stimulated via the necessities of other tensor items in quantum info concept, the idea of tensor items in operator structures is mentioned intimately. at the quantum details part, the e-book bargains a rigorous remedy of quantifying entanglement in bipartite quantum platforms, and strikes directly to evaluate 4 assorted components during which rules from the idea of operator structures and operator algebras play a average position: the problem of zero-error communique over quantum channels, the powerful subadditivity estate of quantum entropy, the various norms on quantum states and the corresponding brought on norms on quantum channels, and, finally, the purposes of matrix-valued random variables within the quantum info surroundings
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Extra info for The functional analysis of quantum information theory : a collection of notes based on lectures by Gilles Pisier, K. R. Parthasarathy, Vern Paulsen and Andreas Winter
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K L Y D : M N Since TrŒY X D 0 for any X 2 S and Y 2 S ? , we have, " Tr K M L N ! L/ D 0: Thus, every element of the orthogonal complement of S is of the form, ! Y / S ?. m; n/. m; n/ D S ˚ S ? m; n/, into a direct sum of subspace S and its orthogonal complement S ? Y / s; 8 0 ¤ Y 2 S ? : Finally, we return to the question of the maximum dimension of the subspace S whose elements have Schmidt rank greater than or equal to k. n k/. k C 1. m; n/. m C n k/. X / k C 1. Note that, Sk \ L D 0.
The joint state of HAB is then transformed to j ˛ i. Alice then sends across her half of the state to Bob. To decode the message, Bob has to perform a measurement and obtain a classical output. As discussed in Sect. 1, corresponding to the orthonormal basis fj ˛ ig, we have a projective measurement characterized by the collection of one-dimensional projections fj ˛ ih ˛ jg. When Bob performs this measurement on any state , the probability of obtaining outcome ˛ is TrŒ j ˛ ih ˛ j. Thus, given the state j ˛ i, Bob will correctly decode the message ˛ with probability one.
Zd /. x/ıx ; where fıx j 8x 2 Zd g is the standard orthonormal basis for H and b is a character of the group Zd . The products fUa Vb j a; b 2 Zd g, which are the Weyl operators of the abelian group Zd , form a unitary error basis. Superdense coding is a simple yet important example of a quantum communication task that is made possible using EPR states. The goal of the task is for one party A (Alice) to communicate d 2 classical messages to another party B (Bob), by physically sending across just one quantum state.