Optimal Trading ofClassical Communication,

Quantum Communication, and Entanglement

Mark M. Wilde

arXiv:0901.3038

ERATO-SORST

Min-Hsiu Hsieh

4th 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 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

Dynamic Setting Prior Work

Devetak, Harrow, Winter. IEEE Trans. Inf. Theory, (2008)

Static Setting Prior Work

Devetak, Harrow, Winter. IEEE Trans. Inf. Theory, (2008)

RQ

E

The Unit Resource Capacity Region

Unit resource capacity region consists of rate triples (R,Q,E)

Superdensecoding Teleportation

EntanglementDistribution

(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 <= 0Q + 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 correspondsto 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 theoremand a matching converse proof

General form of resource inequality:

Positive rate

Negative rate

Resource on RHS(as shown)

Resource generated

Resource implicitlyon LHS

Resource consumed

“Grandmother” Protocol

Devetak, Harrow, Winter. IEEE Trans. Inf. Theory, (2008)

Perform Instrument Compressionwith Quantum Side Informationuses 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 state that has purification

Requires rate of classical communication(and some common randomness)

Then perform Quantum State Redistributionconditional on classical information

Classically-Assisted Quantum State Redistribution (for Direct Static)

Direct Static Achievable Region

Combine theClassically-Assisted State Redistribution Protocol

with theUnit Resource Capacity Region

to get theDirect 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 thecapacity region of the octant (+,-,+)

R Q

E

Point goes into (+,-,0) quadrant

Quadrant corresponds toNoisy 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 theoremand a matching converse proof

General form of resource inequality:

Positive rate

Negative rate

Resource on RHS(as shown)

Resource generated

Resource implicitlyon LHS

Resource consumed

Classically-Enhanced Father Protocol

Hsieh and Wilde, arXiv:0811.4227, November 2008.

Direct Dynamic Achievable Region

Combine theClassically-Enhanced Father Protocol

with theUnit Resource Capacity Region

to get theDirect 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?

Why not teleportation?

Why not teleportation?

Why not teleportation?

when quantum channel coding(or EA coding)

and teleportation is best:

Q > |R|/2 > Eor 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 < Eor

Q < E < |R|/2

Current DirectionsInvestigated the abilities of a simultaneous noisy static and noisy dynamic resource

(arXiv:0904.1175)

Investigating the triple trade-off for public communication, private communication, and secret key

(finished dynamic, wrapping up static, posting soon)

THANK YOU!