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

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

Quantum Shannon Theory Quantum information has three fundamentally different resources: 1.Quantum bit (qubit) 2.Classical bit (cbit) 3.Entangled bit (ebit) Quantum Shannon theory— consume or generate these different resources with the help of 1.Noisy quantum channel (dynamic setting) 2.Shared noisy quantum state (static setting) I. Devetak, A. Harrow, A. Winter, IEEE Trans. Information Theory vol. 54, no. 10, pp , Oct 2008

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

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

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

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 communication alone cannot generate entanglement or quantum communication or both.

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

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

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

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”) Classically-Assisted Quantum State Redistribution (for Direct Static) Begin with statethat has purification Requires rate of classical communication (and some common randomness) Then perform Quantum State Redistribution conditional on classical information

Classically-Assisted Quantum State Redistribution (for Direct Static)

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…

Example of Converse Proof (Direct Static) Can use all entanglement and more quantum comm. with super-dense coding Consider a point (R,-|Q|,E) in the capacity region of the octant (+,-,+) R Q E 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)

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

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

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!

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

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