By Ved Prakash Gupta, Prabha Mandayam, V.S. Sunder
This publication presents readers with a concise advent to present reviews on operator-algebras and their generalizations, operator areas and operator platforms, with a unique concentrate on their software in quantum info technology. 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 types the underpinning of most modern mathematical remedies of the quantum conception, along with being some of the most dynamic parts of 20th century sensible research. this present day, von Neumann’s foresight unearths expression within the swiftly transforming into box of quantum info thought. those notes assemble the content material of lectures given via a truly amazing staff 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 cloth as a primer at the subject material. ranging from the elemental definitions of operator areas and operator structures, this article proceeds to debate a number of vital 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 structures and, stimulated through the necessities of other tensor items in quantum info conception, the speculation of tensor items in operator structures is mentioned intimately. at the quantum info facet, the booklet deals a rigorous remedy of quantifying entanglement in bipartite quantum platforms, and strikes directly to assessment 4 diversified components within which rules from the idea of operator structures and operator algebras play a typical position: the problem of zero-error verbal exchange over quantum channels, the robust subadditivity estate of quantum entropy, different 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 surroundings
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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
C /. 6, we find that jj. cj /jj1=2 Ä 1. As and were arbitrary, we find thus that jjtjjmax Ä 1. So, we indeed have jjujj Ä 1, thus completing the proof of Kirchberg’s theorem. u t References 1. D. Blecher, Tensor products of operator spaces II. Can. J. Math. 44, 75–90 (1992) 2. D. Blecher, V. Paulsen, Tensor products of operator spaces. J. Funct. Anal. 99, 262–292 (1991) 3. E. Kirchberg, On non-semisplit extensions, tensor products and exactness of group C -algebras. Invent. Math. 112, 449–489 (1993) 4.
This is true because for any inclusion of operator spaces E E1 , we have O D L1 . X; /˝E O 1: L1 . M / /. For an inclusion of operator spaces E E1 , only when M is injective, there is an isometric embedding M ˝bE M ˝bE1 . 4 A Passing Remark on the Haagerup Tensor Product Apart from the above two tensor products of operator spaces, there is another extremely important tensor product of operator spaces, namely, the Haagerup tensor product, usually detoned E ˝h F . Unlike the above two tensor products, the Haagerup tensor product has no analogy in Banach space theory.
C; `2 / thus: C. C. u0 / and u00 . Œ ij k D kŒ u0 u00 60 6 ij k D k 6 : 4 :: 0 u00 :: : : : : 0 0 0 0 :: : 3 7 7 7Œ 5 ij k Ä ku00 k kŒ ij k : u00 The case of R is proved similarly. ’s fe1j g and fei1 g respectively. C; R/, jjujjcb D jjujjHS . R; C /. xn /kMn Ä kukcb . ukj ek1 /k 2 j D1 i;kD1 Dk 1 n X X 1 uij ukj e1i ek1 k 2 j D1 i;kD1 Dk 1 n X X 1 uij ukj ıik e11 k 2 j D1 i;kD1 D. 1 n X X 1 jukj j2 / 2 : j D1 kD1 Letting n ! 1, we see that kukHS Ä kukcb . uij // as above. C / j D1 Dk m X xj ˝ j D1 1 X !