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1. Quantum Shannon Theory Patrick Hayden (McGill) http://www.cs.mcgill.ca/~patrick/QLogic2005.ppt 17 July 2005, Q-Logic Meets Q-Info

2. 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

3. 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  The

4. Quantifying uncertainty • Entropy: H(X) = - xp(x) log2p(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…

5. {0,1}n: 2n possible strings 2nH(X)typical strings Compression Source of independent copies of X If X is binary: 0000100111010100010101100101 About nP(X=0) 0’s and nP(X=1) 1’s X2 … X1 Xn Can compress n copies of X to a binary string of length ~nH(X)

6. H(Y) Uncertainty in X when value of Y is known H(X|Y) I(X;Y) Information is that which reduces uncertainty Quantifying information H(X) H(X,Y) 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)

7. ´ m’ m Decoding Encoding 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 Sending information through noisy channels Statistical model of a noisy channel:

8. 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

9. 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

10. 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)

11. No statistical assumptions: Just quantum mechanics! B­ n (aka typical subspace) dim(Effective supp of B­ n ) ~ 2nH(B) Compression Source of independent copies of AB: ­ ­  ­… A A A B B B Can compress n copies of B to a system of ~nH(B) qubits while preserving correlations with A [Schumacher, Petz]

12. H(B) Uncertainty in A when value of B is known? H(A|B) |iAB=|0iA|0iB+|1iA|1iB Quantifying information H(A) H(AB) H(B|A) H(A|B) = H(AB)-H(B) H(A|B) = 0 – 1 = -1 Conditional entropy can be negative! B = I/2

13. H(B) Uncertainty in A when value of B is known? H(A|B) I(A;B) Information is that which reduces uncertainty Quantifying information H(A) H(AB) 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

14. I(A;B) Data processing inequality(Strong subadditivity) Alice Bob time U  I(A;B) I(A;B)¸ I(A;B)

15. Encoding ( state) Decoding (measurement) m’ 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 Physical model of a noisy channel: (Trace-preserving, completely positive map)

16. Encoding ( state) Decoding (measurement) m’ m 2nH(B|A) 2nH(B|A) 2nH(B|A) Sending classical information through noisy channels B­ n 2nH(B) X1,X2,…,Xn

17. Encoding (TPCP map) Decoding (TPCP map) ‘ |i2 Cd 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 Conditional entropy! where Sending quantum information through noisy channels Physical model of a noisy channel: (Trace-preserving, completely positive map)

18. Sets of size 2n(I(X;Z)+) All x Random 2n(I(X;Y)-) x Entanglement and privacy: More than an analogy y=y1 y2 … yn x = x1 x2 … xn p(y,z|x) z = z1 z2 … zn How to send a private message from Alice to Bob? Can send private messages at rate I(X;Y)-I(X;Z) AC93

19. Sets of size 2n(I(X:E)+) All x Random 2n(I(X:A)-) x Entanglement and privacy: More than an analogy |iBE = U­ n|xi UA’->BE­ n |xiA’ How to send a private message from Alice to Bob? Can send private messages at rate I(X:A)-I(X:E) D03

20. Sets of size 2n(I(X:E)+) All x Random 2n(I(X:A)-) x H(E)=H(AB) Entanglement and privacy: More than an analogy x px1/2|xiA|xiBE UA’->BE­ n x px1/2|xiA|xiA’ 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)

21. Notions of distinguishability Basic requirement: quantum channels do not increase “distinguishability” Fidelity Trace distance F(,)={Tr[(1/21/2)1/2]}2 T(,)=|-|1 F=0 for perfectly distinguishable F=1 for identical T=2 for perfectly distinguishable T=0 for identical F(,)=max |h|i|2 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

22. 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

23. 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