By Ved Prakash Gupta, Prabha Mandayam, V.S. Sunder
ISBN-10: 3319167170
ISBN-13: 9783319167176
ISBN-10: 3319167189
ISBN-13: 9783319167183
This booklet offers readers with a concise advent to present stories on operator-algebras and their generalizations, operator areas and operator structures, with a different specialise in their program in quantum details technology. This easy framework for the mathematical formula of quantum info could be traced again to the mathematical paintings of John von Neumann, one of many pioneers of operator algebras, which types the underpinning of most present mathematical remedies of the quantum conception, in addition to being probably the most dynamic components of 20th century practical research. this day, von Neumann’s foresight reveals expression within the quickly becoming box of quantum info idea. those notes assemble the content material of lectures given through a truly exotic team 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 cloth as a primer at the material. ranging from the elemental definitions of operator areas and operator structures, 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 better examine the summary characterization of operator structures and, encouraged by means of the necessities of other tensor items in quantum details idea, the idea of tensor items in operator platforms is mentioned intimately. at the quantum details facet, the publication deals a rigorous remedy of quantifying entanglement in bipartite quantum structures, and strikes directly to evaluation 4 assorted components within which principles from the speculation of operator structures and operator algebras play a common position: the problem of zero-error conversation over quantum channels, the powerful subadditivity estate of quantum entropy, the several norms on quantum states and the corresponding precipitated norms on quantum channels, and, finally, the functions of matrix-valued random variables within the quantum info setting.
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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
Example text
Complete isomorphism if u is a linear isomorphism such that u and u CB. 2. complete contraction if u is CB with jjujjcb Ä 1. 3. complete isometry if un is an isometry for all n 1. 9 R and C are not isomorphic to each other as operator spaces. Proof Suppose u W R ! C and u 1 W C ! R are both CB maps. Then u and u 1 are Hilbert Schmidt and hence compact on an infinite dimensional Hilbert space, a contradiction. K//. If u is positive, then u is completely positive. Proof We first indicate the proof for finite-dimensional A.
H / for suitable choices of n and H . A proof of this remarkable theorem involves a fair bit of analysis. E/. Let u W E ! K/ for all i 2 I . If jjujjcb Ä 1, then there exists a -representation W A ! K/ such that jE D u. Proof Since u is unital and CB with jjujjcb Ä 1, there is a Hilbert space HO , a representation W A ! a/jK for all a 2 E, where PK WÂHO ! K à is the orthogonal projection. ai bi with respect to HO D K ˚ K ? Ui / D ai . Ui / D ci di Unitarity of Ui implies that ai ai C bi bi D ai ai C ci ci D 1; while the assumed à  ai 0 unitarity of ai is seen to imply that bi D ci D 0.
2 More on CB and CP Maps 21 Ä sup k. E/ ; where we have used the fact that k find that kukcb Ä kuk. E; C/, and we hence t u ! E; F /. Then jjujjcb D jjujj in the following cases: 1. u/ Ä 1. 2. F / is commutative. 3. E D F D R or E D F D C . Proof 1. e/f 8e 2 E. a/k kf k Ä k k kak kf k D kuk kak and hence indeed kukcb D kuk. 2. 5. 3. 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.