DEFENSE!DEFENSE!

Applications ofCoherent Classical

Communicationand

Schur dualityto

quantum information theory

Aram Harrow

MIT Physics

June 28, 2005

Committee:Isaac

ChuangEdward Farhi

Peter Shor

CollaboratorsDave Bacon, Charles Bennett, Isaac Chuang, Igor Devetak, Debbie Leung, John Smolin, Andreas Winter

I. Review of quantum and classical information

III. Coherent classical communication

II. The Schur transform

the plan

Classical Computing, Best of

Babbage’s difference engine

Device-independent fundamentals:

Information is reducible to bits (0 or 1).

Computation reduces to logic gates (e.g. NAND and XOR).

“Every function which would naturally be regarded as computable can be computed by a Turing machine.”

Alan Turing and Alonzo Church, 1936

“It was designed for developing and tabulating any function whatever. . . the engine [is] the material expression of any indefinite function of any degree of generality and complexity.”

Ada Lovelace, 1843

What quantum mechanics says about information

different non-orthogonal states cannot be reliably distinguished

states are either the same or they are perfectly distinguishable

identity and distinguishabili

ty

collapses the state to the observed outcome

is no problemmeasurement

2n dimensions2n statesn bits

unitary matricesNAND, XOR, etc…basic units of computation

qubit C2 = span{|0i,|1i}

bit: {0,1}basic unit of information

quantumclassical

quantum algorithmsQuantum computers can efficiently simulate quantum systems.

Deutsch 1985

A database of N elements can be searched with O(pN) quantum queries.

Grover 1996

An n-bit number can be factored in poly(n) time on a quantum computer.

Shor 1994

I. Review of quantum and classical computation

II. The Schur transform

III. Coherent classical communication

symmetries of (Cd) n

(Cd) 4 = Cd Cd Cd Cd

U2Ud ! U U U U

(1324)2S4 !

(Cd) n © Q P

Schur duality

Schur duality from 40,000 feet

1. Many known applications to q. info. theory;

analogous to the classical method of types (a.k.a. counting letter frequencies

2. This extends to i.i.d. channels

analogous to classical joint types of two random variables.

3. Efficient circuit for the Schur transform

a) via reduction to Clebsch-Gordan (CG) transform

b) both CG and Schur use subgroup-adapted bases

c) interesting connections to the Sn Fourier transform

[joint work with Bacon and Chuang; quant-ph/0407082, in preparation (x2)]

What can we do with an efficient Schur transform?

Factoring, based on the quantum Fourier transform.

Schur’s algorithm??

I. Review of quantum and classical computation

II. The Schur transform

III. Coherent classical communication

references• [BHLS02]: “On the capacities of bipartite unitary

gates,” Bennett, H., Leung and Smolin, IEEE-IT 2003

• [H03]: “Coherent communication of classical messages,” H., PRL 2003

• [DHW03]: “A Family of Quantum Protocols,” Devetak, H. and Winter, PRL 2003

• [HL05]: “Two-way coherent classical communication,” H. and Leung, QIC 2005

• [DHW05], “Quantum Shannon theory, resource inequalities and optimal tradeoffs for a family of quantum protocols,” Devetak, H. and Winter (in preparation).

classical Shannon theory

Alice Bob

noisy channelX2XN

N(X)

noisy correlations

P(X,Y)

YX

perfect bit channelcbit

X2{0,1}

X

Coding theorems are resource inequalities.

e.g. N > C(N) cbits, where C(N) is the capacity of N.

Asymptotic and approximate: N n can send n(C-n) bits with error n such that n,n!0 as n!1.

quantum Shannon theory

Alice

Eve

Bob

free

local

operations

free

local

operations

noisy quantum channel

NA!B

noisy shared entanglement

AB

noiseless quantum channel

qubit

unitary gateUAB

cbit one use of a noiseless classical bit channel

ebit the state |i=(|0iA|0iB + |1iA|1iB)/p2

qubit one use of a noiseless quantum bit channel

NA!B a noisy quantum channel

a noisy bipartite state

Ua bipartite unitary gate

a zoo of quantum resources

[DHW05]

problem #1: incomparable resources

Basic resource inequalities

1 qubit > 1 ebit

1 qubit > 1 cbit

Teleportation (TP): 2 cbits + 1 ebit > 1 qubit [BBCJPW93]

Super-dense coding (SD): 1 qubit + 1 ebit > 2 cbits [BW92]

Why is everything irreversible?

problem #2: communication with

unitary gatesSuppose Alice can send Bob n cbits using a unitary interaction:

U|xiA|0iB ¼ |xiB|xiAB for x2{0,1}n

This must be more powerful than an arbitrary noisy interaction, because it implies the ability to create n ebits. But what exactly is its power?

[BHLS02]

a zoo of quantum coding theorems

problem #3: Unify and simplify these.

Noisy SD [HHHLT01]

+ Q qubits > C cbitsNoisy TP: [DHW03]

+ C cbits > Q qubits

Entanglement distillation

+ C cbits > E ebits [BDSW96/DW03]

quantum capacity

N > Q qubits

[L96/S02/D03]

entanglement-assisted classical communication

N + E ebits > C cbits [BSST01]T

P

I(A:B)/2 [BSST]

H(A)+I(A:B)

problem #4: tradeoff curves

Q: q

ub

its sen

t per u

se o

f ch

an

nel

E: ebits allowed per use of channel

Ic =H(B) - H(AB)

[L/S/D]

qubit > ebit bound

45o

N + E ebits > Q qubits

coherent classical communication

(CCC)cbits seen by the Church of the Larger Hilbert Space

|xiA ! |xiB|xiE

for x={0,1}.Give Alice coherent feedback:

The map |xiA ! |xiA|xiB is called a coherent bit, or cobit.

a|0iA + b|1iA ! a|0iA|0iB + b|1iA|1iB

[H03]

|0iA

|1iA |1iB

|0iB

a|0iA + b|1iA|a|2

|b|2

yet another quantum resource:

Alice throws her output away: 1 cobit > 1 cbit

Alice inputs (|0i+|1i)/p2 or half of |i: 1 cobit > 1 ebit

Alice simulates a cobit locally: 1 qubit > 1 cobit

the power of CCCQ: When can cobits generate both cbits and ebits?A: When the cbits used/created are uniformly random and decoupled from all other quantum systems, including the environment.Ex: teleportation2 cobits + 1 ebit > 1 qubit + 2 ebitsEx: super-dense coding1 qubit + 1 ebit > 2 cobitsImplication:2 cobits = 1 qubit + 1 ebit

[H03]

More implications

-one fewer resource to remember

-problem #1: irreversibility

due to 1 cobit > 1 cbit

problem #2: capacities of unitary gates

Theorem: For C>0,

U > C cbits(!) + E ebits

iff U > C cobits(!) + E ebits

[BHLS02, H03]

iff there exists an ensemble E={pi,|iiABA’B’} such that

(U(E)) - (E) > C

E(U(E)) - E(E) > EE = Holevo information between i and trAA’i.

E(E) = average entanglement of E

a family of quantum protocols (problem #3)

Noisy SD

+ Q qubits > C cbits

Noisy TP: + C cbits > Q qubits

Entanglement distillation

+ C cbits > E ebits

quantum capacity

N > Q qubits

entanglement-assisted classical communication

N + E ebits > C cbits

TP

+ Q qubits > E ebits

TP

TP

SD

: N + E ebits > Q qubits

1qubit > 1 ebit

SD

[DHW03]

Noisy

SD

E. distillation

Noisy

TP

EACC

Q. Cap

Alice

father trade-off curve (problem #4)

Q: q

ub

its sen

t per u

se o

f ch

an

nel

E: ebits allowed per use of channel

Ic(AiB)

[L/S/D]

45o

I(A:E)/2 = I(A:B)/2 - Ic(AiB)

I(A:B)/2

[DHW03, DHW05]

father

information theory recap

new formalism: resource inequalities, purifications

new tool: coherent classical communication

new results: a family of quantum protocols,

2-D tradeoff curves, unitary gate capacities,

and a better understanding of the role of classical information in quantum communication.

references: [BHLS02], [H03], [DHW03], [HL05], [DHW05]

where next?theory

pra

ctic

e

classicalShannontheory

classicalShannontheory

classical-quantumprotocols

classical-quantumprotocols

quantumShannontheory

quantumShannontheory

HSW codingteleportationsuper-dense codingnoisy SD, etc..

CCCfamilyunitary gatesmore?

information

technology

information

technology

Brady Bunch

broadcasts

Brady Bunch

broadcasts

cryptographycryptography

practical

codes

practical

codes

QECC

QECC

distributed

QC

distributed

QC

FTQC

FTQC ??

thanks!

Ike Chuang, Eddie Farhi, Peter Shor

IBM: Nabil Amer, Charlie Bennett, David DiVincenzo, Igor Devetak, Debbie Leung, John Smolin, Barbara Terhal

Hospitality of Caltech IQI and UQ QiSci group.

many collaborators, including Dave Bacon and Andreas Winter

NSA/ARDA/ARO for three years of funding

references[BHLS02]: “On the capacities of bipartite unitary gates,” Bennett, H.,

Leung and Smolin, IEEE-IT 2003

[H03]: “Coherent communication of classical messages,” H., PRL 2003

[DHW03]: “A Family of Quantum Protocols,” Devetak, H. and Winter, PRL 2003

[HL05]: “Two-way coherent classical communication,” H. and Leung, QIC 2005

[DHW05], “Quantum Shannon theory, resource inequalities and optimal tradeoffs for a family of quantum protocols,” Devetak, H. and Winter (in preparation).

[BCH04] “Efficient circuits for Schur and Clebsch-Gordan transforms,” Bacon, Chuang and H., quant-ph/0407082

[BCH05a] “The quantum Schur transform: I. Efficient qudit circuits,” Bacon, Chuang and H., in preparation

[BCH05b] “The quantum Schur transform: II. Connections to the quantum Fourier transform,” Bacon, Chuang and H., in preparation

Key technical tool: use subgroup-adapted bases

Multiplicity-free branching for the chain S1 µ … µ Sn

) subgroup-adapted basis for P

|pnpn-1…p1i s.t. pn= and pj Á pj+1.

Similarly, construct a subgroup-adapted basis for Q using the chain: {1}=U(0) µ U(1) µ…µ U(d).

uuuu

uu

|i1i

|i2i

|ini

USc

h

USc

h

|i

|qi

|pi

USc

h

USc

h

= USc

h

USc

h

q(u)q(u)

p()p()

u 2 U(d)

2 Snq is a U(d)-irrep

p is a Sn-irrep

the Schur transform

UCGUCG|qi

|i

|ii

|i

|0i

|M0i

UCGUCG

q(u)

q(u)

uu= UCGUCG

q0(u)

q0(u)

Q

Q(1) Cd

the Clebsch-Gordan transform

UCGUCG|i1i

|½i

|i2i

|ini

|1i

|2i

|q2i

|i3i

UCGUCG

|2i

|3i

|q3i

|n-1i|qn-1i UCGUCG

|n-1i

|ni

|qi

(Cd) n

Schur transform = iterated CG

recursive decomposition of CG

U(d)CG

|i=|qdi

|qd-1i

|q1i

|ii

|q0di = |

0i|q0d-1i

|q01i

|ji = |0 - i

=

U(d-1)CG

|i=|qdi

|qd-1i

|q1i

|ii

|q0di = |

0i|q0d-1i

|q01i

|ki=|q0d-1-qd-1i

|qd-2i |q0d-2i

Wd |ji

normal form of i.i.d. channels

UNA

B

E

|Bi|Ei|qBi|qEi

|i

VnN

|Ai

|qAi

|pAi

Sn

inverse

CG

|pBi

|pEi

n

=

1) Sn QFT ! Schur transform: Generalized Phase Estimation-Only permits measurement in Schur basis, not full Schur transform.-Similar to [abelian QFT ! phase estimation].

1) Sn QFT ! Schur transform: Generalized Phase Estimation-Only permits measurement in Schur basis, not full Schur transform.-Similar to [abelian QFT ! phase estimation].

2) Schur transform ! Sn QFT-Just embed C[Sn] in (Cn) n and do the Schur transform-Based on Howe duality

2) Schur transform ! Sn QFT-Just embed C[Sn] in (Cn) n and do the Schur transform-Based on Howe duality

connections to the Sn QFT