The Classically Enhanced Father Protocol Mark M. Wilde Centre for Quantum Technologies National University of Singapore 3 Science Drive 2 Singapore 117543 (based on joint work with Min-Hsiu Hsieh: arXiv:0811.4227) Seminar, USC (December 15, 2008)

Singapore Bird….

Outline Briefly review Quantum Shannon theory Entanglement-Assisted Quantum Channel Coding (Father Protocol) The Classically-Enhanced Father Protocol

Shannon Theory Claude Shannon established classical information theory Two fundamental theorems: Noiseless source coding Noisy channel coding Shannon theory gives optimal limits for transmission of bits (really just using the Law of Large Numbers) C. E. Shannon, Bell System Technical Journal, vol. 27, pp. 379-423 and 623-656, July and October, 1948.

Quantum Shannon Theory 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) ???? I. Devetak, A. Harrow, A. Winter, IEEE Trans. Information Theory vol. 54, no. 10, pp. 4587-4618, Oct 2008

Problem Description Given a large number of uses of a noisy quantum channel and some entanglement, How much quantum and classical information can we send? Hsieh and Wilde, arXiv:0811.4227, November 2008.

Entanglement-Assisted Quantum Channel Coding Coding Strategy Use the channel many times so that law of large numbers comes into play Relate the construction to secret-key-assisted private classical coding over a quantum channel (extension of Devetak’s ideas) Show how to construct a secret-key-assisted private classical code and how to perform each of the steps coherently Devetak, Harrow, Winter, IEEE Trans. Information Theory vol. 54, no. 10, pp. 4587-4618, Oct 2008 Hsieh, Luo, Brun, Physical Review A, 78, 042306 (2008). Hsieh and Wilde, arXiv:0811.4227, November 2008.

Father Protocol Devetak, Harrow, Winter, IEEE Trans. Information Theory vol. 54, no. 10, pp. 4587-4618, Oct 2008 Devetak, Harrow, Winter, Phys. Rev. Lett., 93, 230504 (2004).

Can achieve the following resource inequality: Father Protocol Can achieve the following resource inequality: where Devetak, Harrow, Winter, IEEE Trans. Information Theory vol. 54, no. 10, pp. 4587-4618, Oct 2008 Devetak, Harrow, Winter, Phys. Rev. Lett., 93, 230504 (2004).

Father Capacity Region Single-Letter Region: Capacity region of the channel: Devetak, Harrow, Winter, IEEE Trans. Information Theory vol. 54, no. 10, pp. 4587-4618, Oct 2008 Devetak, Harrow, Winter, Phys. Rev. Lett., 93, 230504 (2004).

Father Code Definitions Unencoded State: where Encoded State: Hsieh and Wilde, arXiv:0811.4227, November 2008.

Father Code Definitions (Ctd.) Father Code density operator: Channel input density operator: Hsieh and Wilde, arXiv:0811.4227, November 2008.

code density operator: channel input density operator: Random Father Codes Random father code is an ensemble of father codes: Expected code density operator: Expected channel input density operator: Can make expected input close to a tensor power state! HSW coding theorem accepts tensor power input states! Hsieh and Wilde, arXiv:0811.4227, November 2008.

“Piggybacking” Classical Information Given an ensemble: Given a typical input sequence: Can rewrite typical input sequence as follows: Choose |X| father codes each with Quantum communication rate: Entanglement Consumption rate: Devetak and Shor, Communications in Mathematical Physics, 256, 287-303 (2005) Hsieh and Wilde, arXiv:0811.4227, November 2008.

“Piggybacking” Classical Information (ctd.) “Pasted” random father code has total rates: Total Quantum Communication rate: Total Entanglement Consumption rate: Can piggyback classical information with rate By the HSW coding theorem Devetak and Shor, Communications in Mathematical Physics, 256, 287-303 (2005) Hsieh and Wilde, arXiv:0811.4227, November 2008.

Proof Strategy for Coding Theorem Random Coding Show that expectation of average classical error probability and quantum error over all random classically-enhanced father codes is small Pick one that has small error. Derandomization Expurgation Remove the father codes from the classically-enhanced father code that have the worst classical error probability. Ensures that resulting code has low maximal classical error probability. Hey, that’s my idea!!!! Hsieh and Wilde, arXiv:0811.4227, November 2008.

Proof Strategy for Converse Theorem Proof that bounds quantum communication rate and entanglement consumption rate follows standard techniques Resort to optimality of Shor’s entanglement-assisted classical capacity theorem to prove the bound on classical communication rate Hsieh and Wilde, arXiv:0811.4227, November 2008. Shor, quant-ph/0402129 (2004).

Hsieh and Wilde, arXiv:0811.4227, November 2008. Theorem Statement Single-Letter Capacity Region: Hsieh and Wilde, arXiv:0811.4227, November 2008.

Child Protocols Classically-Enhanced Father Resource Inequality: Combine with entanglement distribution to get the classically-enhanced quantum coding resource inequality: Combine with dense coding to get Shor’s entanglement-assisted classical coding resource inequality: Devetak and Shor, Communications in Mathematical Physics, 256, 287-303 (2005) Hsieh and Wilde, arXiv:0811.4227, November 2008.

The Issue of Time-Sharing Can time-sharing beat the classically-enhanced father protocol? Three time-sharing strategies: Share quantum code with EA classical code Share EA quantum code with classical code Share EA quantum code with EA classical code Time-sharing is NOT optimal when entanglement = 0 Time-sharing IS optimal with infinite entanglement Hsieh and Wilde, arXiv:0811.4227, November 2008.

Structure of Optimal Codes Optimal code does NOT need to encode classical info into ebits Kremsky, Hsieh, and Brun, PRA, 78, 012341 (2008). Hsieh and Wilde, arXiv:0811.4227, November 2008.

The Full Triple Trade-off Q E Unit resource capacity region consists of rate triples (R,Q,E) Entanglement Distribution (0, -t, t) Superdense coding Teleportation (2t, -t, -t) (-2t, t, -t) Combine Classically-Enhanced Father protocol with unit resource inequalities to get Full Triple Trade-off

Conclusion Several open questions remaining: Classically-Enhanced Father Protocol is a step in getting the Full Triple Trade-off Gives insight into Error Correction schemes Several open questions remaining: More resources to include: common randomness, private classical communication, secret key Six-dimensional trade-off regions for multiple-access and broadcast channels