1. Introduction to Quantum Shannon Theory Patrick Hayden (McGill University) 12 February 2007, BIRS Quantum Structures Workshop | 

2. Overview  What is Shannon theory?  Why quantum Shannon theory?  Highlights:  The brilliant trivialities  Basic capacity theorems  The grand unified theory

3. Information 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

4. Quantifying uncertainty  Shannon entropy: H(X) = -  x p(x) log 2 p(x)  Term suggested by von Neumann (more on him later)  Can arrive at definition axiomatically:  H(X,Y) = H(X) + H(Y) for independent X, Y, etc.  Operational point of view…

5. X1X1 X 2 … XnXn Compression Source of independent copies of X {0,1} n : 2 n possible strings ~2 nH(X) typical strings 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)

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

7. 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

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. Basic resources |  +  AB =|0 i A |0 i B +|1 i A |1 i B 1 ebit 1 qubit |   span{ |0 , |1  }

11. j 2 {0,1,2,3} Brilliant Triviality # 1: Superdense coding jj Time 1 qubit 1 ebit BW92 j: 2 bits Entanglement allows one qubit to carry two bits of classical data |+|+

12. |  Brilliant Triviality # 2: Teleportation jj Time 1 qubit 2 bits ( j ) 1 ebit BBCJPW93 Reality: Fiction: Two classical bits and one ebit can be used send one qubit |+|+ | 

13. Quantifying uncertainty  Let  =  x p(x) |  x ih  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( tr B  AB )

14. ­­ ­ …­ … ­  Compression Source of independent copies of   : B ­ n dim(Support of  B ­ n ) ~ 2 nH(B) Can compress n copies of B to a system of ~nH(B) qubits while preserving correlations with A No statistical assumptions: Just quantum mechanics! A A A BBB

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

16. H(B)  Quantifying information H(A)  H(B|A)  Information is that which reduces uncertainty I(A;B)  H(A|B)  Uncertainty in A when value of B is known? H(A|B)= H(AB)-H(B) I(A;B) = H(A) – H(A|B) = H(A)+H(B)-H(AB) ¸ 0 H(AB) 

17. 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

18. Sending quantum information through noisy channels Physical model of a noisy channel: (Trace-preserving, completely positive map) |  i 2 C d 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!

19. The family paradigm Many problems in quantum Shannon theory are all versions of the same problem: protocols transform into each other Mother Entanglement distillation Superdense coding with noisy states Teleporting over noisy states Devetak, Harrow, Winter [2003] Father Entanglement-assisted classical capacity Quantum capacity TP SD Stupid

20. Further unification Fully quantum Slepian-Wolf Mother Entanglement distillation Superdense coding with noisy states Teleporting over noisy states Abeyesinghe, Devetak, Hayden, Winter [2006] Father Entanglement-assisted classical capacity Quantum capacity Distributed compression Quantum multiple access capacities Channel simulation Time-reversal Special case Schmidt symmetry TP SD Stupid

21. The art of forgetting

22.  AB 1 B 2 B 3 How can Bob unilaterally destroy his correlation with Alice? What is the minimal number of particles he must discard before the remaining state is uncorrelated? TRASH In this case, by discarding 2 particles, Bob succeeded in eliminating all correlations with Alice’s particle  AB 2 B 3  AB 2 =  A ­  B 2

23. Purification and correlation |  ABCD i Tr BD  ABCD =  A ­  C Purification: When faced with a mixed state, we can always imagine that the state describes part of a larger system on which the state is pure. Purifications are essentially unique. (Up to local transformations of the purifying space.) A ­ σCA ­ σC =(id AC ­ U BD )|  AB i |  CD i |  AB i |  CD i = (id AC ­ U BD -1 )|  ABCD i B D

24. The benefits of forgetting: Applied theology  AB 1 B 2 B 3 TRASH  AB 2 B 3  AB 2 =  A ­  B 2 |  AB 1 B 2 B 3 C i Purification Charlie’s Magical Bucket O’ Particles Watch again: All purifications equivalent up to a local transformation in Charlie’s lab. Charlie holds uncorrelated purifications of both Alice’s particle and Bob’s remaining particles.

25. The benefits of forgetting: Applied theology TRASH |  AB 1 B 2 B 3 C i TRASH |  AC 1 i |  B 2 C 2 C 3 i BeforeAfter Alice never did anything ) Her marginal state  A =  A is unchanged Originally, her purification is held by both Bob and Charlie. Afterwards, entirely by Charlie. Bob transferred his Alice entanglement to Charlie and distilled entanglement with Charlie, just by discarding particles!

26. Fully quantum Slepian-Wolf: How much does Bob need to send? TRASH Before |  ABC i ­ n Uncertainty: von Neumann entropy H(A)  = H(  A ) = - tr[  A log  A ] Correlation: mutual information I(A;B)  = H(A)  + H(B)  – H(AB)  0 if and only if  AB =  A ­  B I(A;B)  = m for m pairs of correlated bits 2m for m ebits (maximal) Initial mutual information: n I(A;B)  Final mutual information:  Each qubit Bob discards has the potential to eliminate at most 2 bits of correlation Bob should (ideally) send around nI(A;B)  /2 qubits to Charlie.

27. How does Bob choose which qubits? TRASH Before |  ABC i ­ n ? At random! (According to the unitarily invariant measure on the typical subspace of B ­ n.) Bob can ignore the correlation structure of his state!

28. Final accounting TRASH |  AC 1 i |  B 2 C 2 C 3 i After Investment: Bob sends Charlie ~n[I(A;B)  ]/2 qubits Rewards: 1) Charlie holds Alice’s purification 2) B and C establish ~n[I(B;C)  ]/2 ebits OK – but what good is it?

29. Entanglement distillation (  BC ) ­ n Bob and Charlie share many copies of a noisy entangled state and would like to convert them to ebits. Only local operations and classical communication are allowed. Forgetting protocol good but uses quantum communication Implement quantum communication using teleportation: Transmit 1 qubit using 2 cbits and 1 ebit. Net rate of ebit production: I(B;C)/2 – I(A;B)/2 = H(C)-H(BC) Optimal [Devetak/Winter 03]

31. Conclusions  Information theory can be generalized to analyze quantum information processing  Yields a rich theory, surprising conceptual simplicity  Compression, data transmission, superdense coding, teleportation, subspace transmission  Capacity zoo, using noisy entanglement, channel simulation: all are closely related  Operational approach to thinking about quantum mechanics