By Ved Prakash Gupta, Prabha Mandayam, V.S. Sunder
This booklet offers readers with a concise advent to present reviews on operator-algebras and their generalizations, operator areas and operator platforms, with a distinct specialize in their software in quantum info technology. This uncomplicated framework for the mathematical formula of quantum details will be traced again to the mathematical paintings of John von Neumann, one of many pioneers of operator algebras, which kinds the underpinning of most present mathematical remedies of the quantum thought, along with being the most dynamic components of 20th century practical research. this present day, von Neumann’s foresight reveals expression within the speedily becoming box of quantum info conception. those notes assemble the content material of lectures given through a truly unusual 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 offer the fabric as a primer at the material. ranging from the fundamental definitions of operator areas and operator platforms, this article proceeds to debate numerous vital theorems together with Stinespring’s dilation theorem for thoroughly optimistic maps and Kirchberg’s theorem on tensor items of C*-algebras. It additionally takes a more in-depth examine the summary characterization of operator platforms and, stimulated through the necessities of alternative tensor items in quantum details idea, the idea of tensor items in operator platforms is mentioned intimately. at the quantum details aspect, the ebook deals a rigorous therapy of quantifying entanglement in bipartite quantum structures, and strikes directly to overview 4 diverse components within which rules from the idea of operator platforms and operator algebras play a ordinary function: the difficulty of zero-error conversation over quantum channels, the powerful subadditivity estate of quantum entropy, the various norms on quantum states and the corresponding precipitated norms on quantum channels, and, finally, the purposes of matrix-valued random variables within the quantum info environment
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Additional resources 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
Sample text
Furthermore, each j ˛ i is maximally entangled, that is, Tr1;2 Œj ˛ ih ˛ j D I ; 8˛; d whether the partial trace is taken over the first or the second Hilbert space. 2 Entanglement 47 A remark on uniqueness and existence of unitary orthogonal bases: note that a given unitary orthogonal basis is invariant under scalar multiplication (with a scalar of unit modulus), permutation, and conjugation by a fixed operator (W˛ ! W˛ ). It is an interesting question to classify the distinct unitary orthogonal bases upto this equivalence.
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. n k/. P / Ä p k 1. zp //, where z1 ; z2 ; : : : ; zp are all distinct.
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. Say HA is the d -dimensional Hilbert space associated with Alice’s system and HB the d -dimensional Hilbert space corresponding to Bob’s system.
