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, Northrop Grumman STRL (December 16, 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:
1. Noiseless source coding
2. 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:
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
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).
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.
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 classicallyenhanced father codes is small
Derandomization
Pick one that has small error.
Expurgation
Hey, that’s my
Remove the father codes from the classically-enhanced
idea!!!!
father code that have the worst classical error
probability. Ensures that resulting code has low maximal
classical error probability.
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).
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 classicallyenhanced quantum coding resource inequality:
Combine with dense coding to get Shor’s entanglementassisted 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:
1. Share quantum
code with
EAentanglement
classical code
Time-sharing
is NOT optimal
when
=0
2. Share EA quantum code with classical code
3. Share EA IS
quantum
with EA
classical code
Time-sharing
optimal code
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
Unit resource capacity region consists of rate triples (R,Q,E)
Entanglement
Distribution
E
(0, -t, t)
R
Superdense
coding
Q
(2t, -t, -t)
Teleportation
(-2t, t, -t)
Combine Classically-Enhanced Father protocol with unit
resource inequalities to get Full
Triple Trade-off
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 multipleaccess and broadcast channels