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Quantum Shannon Theory Patrick Hayden (McGill) http://www.cs.mcgill.ca/~patrick/QLogic2005.ppt 17 July 2005, Q-Logic Meets Q-Info 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 A practical question: A mathematico-epistemological question: How to best make use of a given communications resource? How to quantify uncertainty and information? Shannon: Solved the first by considering the second. A mathematical theory of communication [1948] The 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 Source of independent copies of X X …n X21X If X is binary: 0000100111010100010101100101 About nP(X=0) 0’s and nP(X=1) 1’s {0,1}n: 2n possible strings 2nH(X) typical strings Can compress n copies of X to a binary string of length ~nH(X) Quantifying information H(X) Uncertainty in X when value of Y is known H(X|Y) H(X,Y) I(X;Y) H(Y) H(Y|X) H(X|Y) = H(X,Y)-H(Y) = EYH(X|Y=y) Information is that which reduces uncertainty I(X;Y) = H(X) – H(X|Y) = H(X)+H(Y)-H(X,Y) Sending information through noisy channels ´ Statistical model of a noisy channel: m Encoding Decoding m’ Shannon’s noisy coding theorem: In the limit of many uses, the optimal rate at which Alice can send messages reliably to Bob through is given by the formula Shannon theory provides Practically speaking: Conceptually speaking: A holy grail for error-correcting codes 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 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 Let = x p(x) |xihx| 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 Source of independent copies of AB: No statistical assumptions: Just quantum mechanics! … (aka typical subspace) A B A B A B Bn dim(Effective supp of B n ) ~ 2nH(B) Can compress n copies of B to a system of ~nH(B) qubits while preserving correlations with A [Schumacher, Petz] Quantifying information H(A) Uncertainty in A when value of B is known? H(AB) H(B) H(B|A) H(A|B) H(A|B)= H(AB)-H(B) H(A|B) = 0 – 1 = -1 |iAB=|0iA|0iB+|1iA|1iB B = I/2 Conditional entropy can be negative! Quantifying information H(A) Uncertainty in A when value of B is known? H(A|B)= H(AB)-H(B) H(A|B) H(AB) I(A;B) H(B) H(B|A) Information is that which reduces uncertainty I(A;B) = H(A) – H(A|B) = H(A)+H(B)-H(AB) ¸ 0 Data processing inequality (Strong subadditivity) Alice AB I(A;B) U I(A;B) I(A;B) ¸ I(A;B) Bob time Sending classical information through noisy channels Physical model of a noisy channel: (Trace-preserving, completely positive map) m Encoding ( state) Decoding (measurement) m’ HSW noisy coding theorem: In the limit of many uses, the optimal rate at which Alice can send messages reliably to Bob through is given by the (regularization of the) formula where Sending classical information through noisy channels m Encoding ( state) Decoding (measurement) 2nH(B) 2nH(B|A) X1,X2,…,Xn 2nH(B|A) 2nH(B|A) m’ Bn Sending quantum information through noisy channels Physical model of a noisy channel: (Trace-preserving, completely positive map) |i 2 Cd Encoding (TPCP map) Decoding (TPCP map) ‘ LSD noisy coding theorem: In the limit of many uses, the optimal rate at which Alice can reliably send qubits to Bob (1/n log d) through is given by the (regularization of the) formula where Conditional entropy! Entanglement and privacy: More than an analogy x = x1 x2 … xn p(y,z|x) y=y1 y2 … yn z = z1 z 2 … z n How to send a private message from Alice to Bob? Sets of size 2n(I(X;Z)+) All x Random 2n(I(X;Y)-) x Can send private messages at rate I(X;Y)-I(X;Z) AC93 Entanglement and privacy: More than an analogy |xiA’ UA’->BE n |iBE = U n|xi How to send a private message from Alice to Bob? Sets of size 2n(I(X:E)+) All x Random 2n(I(X:A)-) x Can send private messages at rate I(X:A)-I(X:E) D03 Entanglement and privacy: More than an analogy x px1/2|xiA|xiA’ UA’->BE n x px1/2|xiA|xiBE How to send a private message from Alice to Bob? All x Random 2n(I(X:A)-) x Sets of size 2n(I(X:E)+) H(E)=H(AB) SW97 Can send private messages at rate I(X:A)-I(X:E)=H(A)-H(E) D03 Notions of distinguishability Basic requirement: quantum channels do not increase “distinguishability” Fidelity F(,)={Tr[(1/21/2)1/2]}2 F=0 for perfectly distinguishable F=1 for identical F(,)=max |h|i|2 F((),()) ¸ F(,) Trace distance T(,)=|-|1 T=2 for perfectly distinguishable T=0 for identical T(,)=2max|p(k=0|)-p(k=0|)| where max is over POVMS {Mk} T(,) ¸ T((,()) Statements made today hold for both measures 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 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