The Classically Enhanced Father Protocol Mark M. Wilde Seminar, USC (December 15, 2008) Centre for...
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Transcript of The Classically Enhanced Father Protocol Mark M. Wilde Seminar, USC (December 15, 2008) Centre for...
The Classically Enhanced Father Protocol
Mark M. Wilde
Seminar, USC (December 15, 2008)
Centre for Quantum TechnologiesNational University of Singapore
3 Science Drive 2Singapore 117543
(based on joint work with Min-Hsiu Hsieh: arXiv:0811.4227)
Singapore Bird….
Outline
Briefly review Quantum Shannon theory
Entanglement-Assisted Quantum Channel Coding(Father Protocol)
The Classically-Enhanced Father Protocol
Shannon TheoryClaude Shannon established classical information theory
Two fundamental theorems:
1. Noiseless source coding2. 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 TheoryQuantum 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. 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
Use the channel many times so that law of large numbers comes into play
Coding Strategy
Show how to construct a secret-key-assisted private classical code and how to perform each of the steps
coherently
Relate the construction to secret-key-assisted private classical coding over a quantum channel (extension of Devetak’s ideas)
Devetak, Harrow, Winter, IEEE Trans. Information Theory vol. 54, no. 10, pp. 4587-4618, Oct 2008Hsieh, 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 2008Devetak, Harrow, Winter, Phys. Rev. Lett., 93, 230504 (2004).
Father Protocol
Can achieve the following resource inequality:
where
Devetak, Harrow, Winter, IEEE Trans. Information Theory vol. 54, no. 10, pp. 4587-4618, Oct 2008Devetak, Harrow, Winter, Phys. Rev. Lett., 93, 230504 (2004).
Father Capacity Region
Capacity region of the channel:
Single-Letter Region:
Devetak, Harrow, Winter, IEEE Trans. Information Theory vol. 54, no. 10, pp. 4587-4618, Oct 2008Devetak, Harrow, Winter, Phys. Rev. Lett., 93, 230504 (2004).
Father Code DefinitionsUnencoded 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.
Random Father Codes
Hsieh and Wilde, arXiv:0811.4227, November 2008.
Random father code is an ensemble of father codes:
Expectedcode density operator:
Expectedchannel input density operator:
Can make expected input close to a tensor power state!
HSW coding theorem accepts tensor power input states!
“Piggybacking” Classical Information
Devetak and Shor, Communications in Mathematical Physics, 256, 287-303 (2005)Hsieh and Wilde, arXiv:0811.4227, November 2008.
Given an ensemble:
Choose |X| father codes each with
Given a typical input sequence:
Can rewrite typical input sequence as follows:
Quantum communication rate: Entanglement Consumption rate:
“Piggybacking” Classical Information (ctd.)
Devetak and Shor, Communications in Mathematical Physics, 256, 287-303 (2005)Hsieh and Wilde, arXiv:0811.4227, November 2008.
“Pasted” random father code has total rates:
Can piggyback classical information with rate
By the HSW coding theorem
Total Entanglement Consumption rate:
Total Quantum Communication rate:
Proof Strategy for Coding Theorem
Hsieh and Wilde, arXiv:0811.4227, November 2008.
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.
Random Coding
Show that expectation of average classical error probability and quantum error over all random classically-enhanced father codes is small
Hey, that’s my idea!!!!
Proof Strategy for Converse Theorem
Hsieh and Wilde, arXiv:0811.4227, November 2008.Shor, quant-ph/0402129 (2004).
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
Theorem Statement
Hsieh and Wilde, arXiv:0811.4227, November 2008.
Single-Letter Capacity Region:
Child Protocols
Devetak and Shor, Communications in Mathematical Physics, 256, 287-303 (2005)Hsieh and Wilde, arXiv:0811.4227, November 2008.
Classically-Enhanced Father Resource Inequality:
Combine with dense coding to get Shor’s entanglement-assisted classical coding resource inequality:
Combine with entanglement distribution to get the classically-enhanced quantum coding resource inequality:
The Issue of Time-Sharing
Hsieh and Wilde, arXiv:0811.4227, November 2008.
Can time-sharing beat the classically-enhanced father protocol?
Three time-sharing strategies:
1. Share quantum code with EA classical code2. Share EA quantum code with classical code3. Share EA quantum code with EA classical code
Time-sharing is NOT optimal when entanglement = 0
Time-sharing IS optimal with infinite entanglement
Structure of Optimal Codes
Kremsky, Hsieh, and Brun, PRA, 78, 012341 (2008).Hsieh and Wilde, arXiv:0811.4227, November 2008.
Optimal code does NOT need to encode classical info into ebits
RQ
E
The Full Triple Trade-off
Unit resource capacity region consists of rate triples (R,Q,E)
Superdensecoding Teleportation
EntanglementDistribution
Combine Classically-Enhanced Father protocol with unit
resource inequalities to get Full Triple Trade-off
(2t, -t, -t)(-2t, t, -t)
(0, -t, t)
Conclusion
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