the classically enhanced father protocol

The Classically Enhanced Father Protocol

Mark M. Wilde

Seminar, Northrop Grumman STRL (December 16, 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


Father Capacity Region

Capacity region of the channel:

Single-Letter Region:


Father Code DefinitionsUnencoded State:

where

Encoded State:


Father Code Definitions (Ctd.)

Father Code density operator:

Channel input density operator:


Random Father Codes


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.)


“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


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


Single-Letter Capacity Region:

Child Protocols


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


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

the classically enhanced father protocol

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