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QLogic2005

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Published on January 1, 2008

Author: Laurie

Source: authorstream.com

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Quantum Shannon Theory:  Quantum Shannon Theory Patrick Hayden (McGill) http://www.cs.mcgill.ca/~patrick/QLogic2005.ppt 17 July 2005, Q-Logic Meets Q-Info Overview:  Overview Part I: What is Shannon theory? What does it have to do with quantum mechanics? Some quantum Shannon theory highlights Part II: Resource inequalities A skeleton key Information (Shannon) theory:  Information (Shannon) theory A practical question: How to best make use of a given communications resource? A mathematico-epistemological question: How to quantify uncertainty and information? Shannon: Solved the first by considering the second. A mathematical theory of communication [1948] The Quantifying uncertainty:  Quantifying uncertainty Entropy: H(X) = - x p(x) log2 p(x) Proportional to entropy of statistical physics Term suggested by von Neumann (more on him soon) Can arrive at definition axiomatically: H(X,Y) = H(X) + H(Y) for independent X, Y, etc. Operational point of view… Compression:  X1 X2 … Xn Compression Source of independent copies of X If X is binary: 0000100111010100010101100101 About nP(X=0) 0’s and nP(X=1) 1’s Can compress n copies of X to a binary string of length ~nH(X) Quantifying information:  Quantifying information H(X) H(Y|X) H(X|Y) = H(X,Y)-H(Y) = EYH(X|Y=y) I(X;Y) = H(X) – H(X|Y) = H(X)+H(Y)-H(X,Y) H(X,Y) Sending information through noisy channels:  Sending information through noisy channels Statistical model of a noisy channel: Shannon theory provides:  Shannon theory provides Practically speaking: A holy grail for error-correcting codes Conceptually speaking: A operationally-motivated way of thinking about correlations What’s missing (for a quantum mechanic)? Features from linear structure: Entanglement and non-orthogonality Quantum Shannon Theory provides:  Quantum Shannon Theory provides General theory of interconvertibility between different types of communications resources: qubits, cbits, ebits, cobits, sbits… Relies on a Major simplifying assumption: Computation is free Minor simplifying assumption: Noise and data have regular structure Quantifying uncertainty:  Quantifying uncertainty Let  = x p(x) |xihx| be a density operator von Neumann entropy: H() = - tr [ log ] Equal to Shannon entropy of  eigenvalues Analog of a joint random variable: AB describes a composite system A ­ B H(A) = H(A) = H( trB AB) Compression:   ­  ­ … ­  Compression Source of independent copies of AB: Can compress n copies of B to a system of ~nH(B) qubits while preserving correlations with A A A A B B B [Schumacher, Petz] Quantifying information:  Quantifying information H(A) H(B|A) H(A|B) = H(AB)-H(B) B = I/2 H(A|B) = 0 – 1 = -1 Conditional entropy can be negative! H(AB) Quantifying information:  Quantifying information H(A) H(B|A) H(A|B) = H(AB)-H(B) I(A;B) = H(A) – H(A|B) = H(A)+H(B)-H(AB) ¸ 0 H(AB) Data processing inequality (Strong subadditivity):  Data processing inequality (Strong subadditivity) Alice Bob time U I(A;B)  I(A;B) ¸ I(A;B) Sending classical information through noisy channels:  Sending classical information through noisy channels Physical model of a noisy channel: (Trace-preserving, completely positive map) Sending classical information through noisy channels:  Sending classical information through noisy channels B­ n 2nH(B) X1,X2,…,Xn Sending quantum information through noisy channels:  Sending quantum information through noisy channels Physical model of a noisy channel: (Trace-preserving, completely positive map) Entanglement and privacy: More than an analogy:  Entanglement and privacy: More than an analogy p(y,z|x) x = x1 x2 … xn y=y1 y2 … yn z = z1 z2 … zn How to send a private message from Alice to Bob? AC93 Can send private messages at rate I(X;Y)-I(X;Z) Entanglement and privacy: More than an analogy:  Entanglement and privacy: More than an analogy UA’->BE­ n |xiA’ |iBE = U­ n|xi How to send a private message from Alice to Bob? D03 Can send private messages at rate I(X:A)-I(X:E) Entanglement and privacy: More than an analogy:  Entanglement and privacy: More than an analogy UA’->BE­ n x px1/2|xiA|xiA’ x px1/2|xiA|xiBE How to send a private message from Alice to Bob? SW97 D03 Can send private messages at rate I(X:A)-I(X:E)=H(A)-H(E) Notions of distinguishability:  Notions of distinguishability Basic requirement: quantum channels do not increase “distinguishability” Fidelity Trace distance F(,)=max |h|i|2 T(,)=|-|1 F(,)={Tr[(1/21/2)1/2]}2 F=0 for perfectly distinguishable F=1 for identical T=2 for perfectly distinguishable T=0 for identical T(,)=2max|p(k=0|)-p(k=0|)| where max is over POVMS {Mk} F((),()) ¸ F(,) T(,) ¸ T((,()) Statements made today hold for both measures Conclusions: Part I:  Conclusions: Part I Information theory can be generalized to analyze quantum information processing Yields a rich theory, surprising conceptual simplicity Operational approach to thinking about quantum mechanics: Compression, data transmission, superdense coding, subspace transmission, teleportation Slide23:  Some references: Part I: Standard textbooks: * Cover & Thomas, Elements of information theory. * Nielsen & Chuang, Quantum computation and quantum information. (and references therein) Part II: Papers available at arxiv.org: * Devetak, The private classical capacity and quantum capacity of a quantum channel, quant-ph/0304127 * Devetak, Harrow & Winter, A family of quantum protocols, quant-ph/0308044. * Horodecki, Oppenheim & Winter, Quantum information can be negative, quant-ph/0505062

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