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Optimal Trading of Classical Communication , Quantum Communication , and Entanglement

Optimal Trading of Classical Communication , Quantum Communication , and Entanglement. Min-Hsiu Hsieh. ERATO-SORST. Mark M. Wilde. SAIC. arXiv:0901.3038. 4 th Workshop on the Theory of Quantum Computation, Communication and Cryptography, Wednesday, May 13, 2009. Overview.

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Optimal Trading of Classical Communication , Quantum Communication , and Entanglement

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  1. Optimal Trading ofClassical Communication,Quantum Communication, and Entanglement Min-Hsiu Hsieh ERATO-SORST Mark M. Wilde SAIC arXiv:0901.3038 4th Workshop on the Theory of Quantum Computation, Communication and Cryptography, Wednesday, May 13, 2009

  2. Overview • Quantum Shannon theory • Prior research • Unit Resource Capacity Region • Dynamic Setting • Static Setting • Implications for Quantum Coding Theory

  3. Quantum information has three fundamentally different resources: • Quantum bit (qubit) • Classical bit (cbit) • Entangled bit (ebit) Quantum Shannon theory—consume or generate these different resources with the help of • Noisy quantum channel (dynamic setting) • Shared noisy quantum state (static setting) Quantum Shannon Theory I. Devetak, A. Harrow, A. Winter, IEEE Trans. Information Theory vol. 54, no. 10, pp. 4587-4618, Oct 2008

  4. Dynamic Setting Prior Work Devetak, Harrow, Winter. IEEE Trans. Inf. Theory, (2008)

  5. Static Setting Prior Work Devetak, Harrow, Winter. IEEE Trans. Inf. Theory, (2008)

  6. Entanglement Distribution E R Q Superdense coding Teleportation The Unit Resource Capacity Region What if only noiseless resources? Unit resource capacity region consists of rate triples (R,Q,E) (0, -t, t) (2t, -t, -t) (-2t, t, -t)

  7. Converse Proof for Unit Capacity Region Show that region given by R + Q + E <= 0 Q + E <= 0 ½ R + Q <= 0 is optimal for all octants Use as postulates: (1) Entanglement alone cannot generate classical communication or quantum communication or both. (2)Classical communicationalone cannot generate entanglementor quantum communication or both.

  8. Example of Proof Strategy Consider a point (R,-|Q|,E) in the octant (+,-,+) E Q R Holevo bound applies R + 2E <= |Q| + E Suppose this point corresponds to a protocol Then use all entanglementand more quantum comm. with super-dense coding: (R,-|Q|,E) + (2E,-E,-E) = (R+2E,-|Q|-E,0) New point corresponds to a protocol.

  9. Static Setting General form of resource inequality: Positive rate Resource on RHS (as shown) Negative rate Resource implicitly on LHS Resource generated Resource consumed Need a direct coding theorem and a matching converse proof

  10. “Grandmother” Protocol Devetak, Harrow, Winter. IEEE Trans. Inf. Theory, (2008)

  11. Classically-Assisted Quantum State Redistribution (for Direct Static) Begin with state that has purification Perform Instrument Compression with Quantum Side Information uses techniques from Winter’s Instrument Compression and Devetak-Winter Classical Compression with Quantum Side Information (Also see Renes and Boileau “Optimal state merging without decoupling”) rate of classical communication (and some common randomness) Requires Then perform Quantum State Redistribution conditional on classical information

  12. Classically-Assisted Quantum State Redistribution (for Direct Static)

  13. Direct Static Achievable Region Combine the Classically-Assisted State Redistribution Protocol with the Unit Resource Capacity Region to get the Direct Static Achievable Region Need to show that this strategy is Optimal Off we go, octant by octant…

  14. Example of Converse Proof(Direct Static) Consider a point (R,-|Q|,E) in the capacity region of the octant (+,-,+) E Q R Can use all entanglementand more quantum comm. with super-dense coding Point goes into (+,-,0) quadrant Quadrant corresponds to Noisy Super-dense Coding Essentially resort to the optimality of noisy super-dense coding (NSD special case of CASR with unit protocols)

  15. Dynamic Setting Positive rate Resource on RHS (as shown) Negative rate Resource implicitly on LHS General form of resource inequality: Resource generated Resource consumed Need a direct coding theorem and a matching converse proof

  16. Classically-Enhanced Father Protocol Hsieh and Wilde, arXiv:0811.4227, November 2008.

  17. Direct Dynamic Achievable Region Combine the Classically-Enhanced Father Protocol with the Unit Resource Capacity Region to get the Direct Dynamic Achievable Region Need to show that this strategy is Optimal Again, octant by octant…, similarly Except!

  18. Octant (-,+,-) Why not teleportation? Why not teleportation? Why not teleportation? Quantum Shannon theory now states: Why not teleportation? when teleportation alone is best: Q < |R|/2 < E or Q < E < |R|/2 Relevant for the theory of entanglement-assisted coding when quantum channel coding (or EA coding) and teleportation is best: Q > |R|/2 > E or Q > |R|/2 and E > |R|/2 or E < Q < |R|/2

  19. Current Directions Investigated the abilities of a simultaneous noisy static and noisy dynamic resource (arXiv:0904.1175) Investigating the triple trade-off for public communication, private communication, and secret key (finished dynamic, wrapping up static, posting soon) THANK YOU!

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