Dualities in quantum information theory

Igor DevetakCSI, EE-S, USC

quant-ph/0505138

FQRS

FF FQSW

RESOURCEREVERSAL

TIMEREVERSAL

SOURCE-CHANNEL DUALITY

Schumacher (quantum) compression

Alice

Bob

Ref

SIMULATES TO PRECISION …

test state

[Schumacher ’95] !

Schumacher (quantum) compression

Alice

Bob

Ref

Would not work on anyother input!

test state

[Schumacher ’95]

asymptotic

Fully quantum reverse Shannon (FQRS)

cf. Schumacher compression

[Devetak, Hayden, Leung, Shor] !

feedback channel

Quantum reverse Shannon (QRS)

QRS

FQRS

Rule I

[Bennett, Devetak, Harrow, Shor, Winter]

FQRS

Alice

Bob

Ref

SIMULATES TO PRECISION …

[Devetak, Hayden, Leung, Shor]

FQRS

Alice

Bob

Ref

[Devetak, Hayden, Leung, Shor]

FQRS

FF FQSW

RESOURCEREVERSAL

TIMEREVERSAL

SOURCE-CHANNEL DUALITY

Quantum reverse Shannon

The Feedback Father (FF)

Alice

Bob

Alice

Bob

The Feedback Father (FF)

The Feedback Father (FF)

FF

cf. FQRS

Resource reversal duality: FQRS & FF

FQRS

FF FQSW

RESOURCEREVERSAL

TIMEREVERSAL

SOURCE-CHANNEL DUALITY

Quantum reverse Shannon

Father

The Source

!

• Passive protagonist – not allowed to code.• Sends inputs to Alice and/or Bob.• Goal: Redirection of channels originating at the Source (“source coding”)

Fully quantum Slepian Wolf (FQSW)

Alice

Bob

Ref

Source

[Abeyesinghe, Devetak, Hayden, Winter]

FQSW

Bob

Alice

Ref

Source

[Abeyesinghe, Devetak, Hayden, Winter]

FQSW

ED QSW

Mother FQSW

[Abeyesinghe, Devetak, Hayden, Winter]

= state merging[Horodecki, Oppenheim, Winter ’05]

[Devetak, Winter 03]

[Devetak, Harrow, Winter 03]

FQRS

FF FQSW

RESOURCEREVERSAL

TIMEREVERSAL

SOURCE-CHANNEL DUALITY

Quantum reverse Shannon

Father Mother

State merging

Source-channel duality: FQSW & FF

FF

cf. FQSW

Time-reversal duality: FQSW & FQRS

FQRS

cf. FQSW

interchange A & B:

Alice

Bob

Ref

Source

FQSW

FQRS

FQSW

FF

FQRS

Summary: Quantum, distributed, closed & relative

FQRS

FF FQSW

RESOURCEREVERSAL

TIMEREVERSAL

SOURCE-CHANNEL DUALITY

Quantum reverse Shannon

Father Mother

State merging

QuantumDistributedClosed Relative