DEFENSE

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DEFENSE!
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Applications ofCoherent Classical
CommunicationandSchur dualitytoquantum
information theory
Collaborators Dave Bacon, Charles Bennett, Isaac
Chuang, Igor Devetak, Debbie Leung, John Smolin,
Andreas Winter
Committee Isaac Chuang Edward Farhi Peter Shor
Aram Harrow MIT Physics June 28, 2005
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the plan
I. Review of quantum and classical information
II. The Schur transform
III. Coherent classical communication
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Classical Computing, Best of
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
Babbages difference engine
Device-independent fundamentals Information is
reducible to bits (0 or 1). Computation reduces
to logic gates (e.g. NAND and XOR).
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What quantum mechanics says about information
quantum
classical
qubit C2 span0i,1i
bit 0,1
basic unit of information
different non-orthogonal states cannot be
reliably distinguished
states are either the same or they are perfectly
distinguishable
identity and distinguishability
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quantum algorithms
Quantum computers can efficiently simulate
quantum systems. Deutsch 1985
An n-bit number can be factored in poly(n) time
on a quantum computer. Shor 1994
A database of N elements can be searched with
O(pN) quantum queries. Grover 1996
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I. Review of quantum and classical computation
II. The Schur transform
III. Coherent classical communication
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symmetries of (Cd)n
U2Ud ! U U U U
(Cd)4 Cd Cd Cd Cd
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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)
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What can we do with an efficient Schur transform?
Factoring, based on the quantum Fourier transform.
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I. Review of quantum and classical computation
II. The Schur transform
III. Coherent classical communication
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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).

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classical Shannon theory
Alice
Bob
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quantum Shannon theory
Eve
free local operations
free local operations
Alice
Bob
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a zoo of quantum resources
DHW05
cbit one use of a noiseless classical bit
channel ebit the state Fi(0iA0iB
1iA1iB)/p2 qubit one use of a noiseless quantum
bit channel NA!B a noisy quantum channel rAB a
noisy bipartite state UAB a bipartite unitary gate
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problem 1 incomparable resources
Basic resource inequalities 1 qubit gt 1 ebit 1
qubit gt 1 cbit Teleportation (TP) 2 cbits 1
ebit gt 1 qubit BBCJPW93 Super-dense coding
(SD) 1 qubit 1 ebit gt 2 cbits BW92
Why is everything irreversible?
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problem 2 communication with unitary gates
BHLS02
Suppose Alice can send Bob n cbits using a
unitary interaction UxiA0iB ¼ xiByxiAB
for x20,1n 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?
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a zoo of quantum coding theorems

• Entanglement distillation
• C cbits gt E ebits BDSW96/DW03

entanglement-assisted classical communication N
E ebits gt C cbits BSST01
Noisy SD HHHLT01 r Q qubits gt C cbits
quantum capacity N gt Q qubits L96/S02/D03
Noisy TP DHW03 r C cbits gt Q qubits
problem 3 Unify and simplify these.
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problem 4 tradeoff curves
N E ebits gt Q qubits
Q qubits sent per use of channel
E ebits allowed per use of channel
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coherent classical communication(CCC)
H03
cbits seen by the Church of the Larger Hilbert
Space xiA ! xiBxiE for x0,1.
0iB
0iA
a0iA b1iA
1iA
1iB
Give Alice coherent feedback The map xiA !
xiAxiB is called a coherent bit, or
cobit. a0iA b1iA ! a0iA0iB b1iA1iB
yet another quantum resource Alice throws her
output away 1 cobit gt 1 cbit Alice inputs
(0i1i)/p2 or half of Fi 1 cobit gt 1
ebit Alice simulates a cobit locally 1 qubit gt 1
cobit
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the power of CCC
H03
Q 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 teleportation 2 cobits 1 ebit gt 1 qubit 2
ebits Ex super-dense coding 1 qubit 1 ebit gt 2
cobits Implication 2 cobits 1 qubit 1 ebit
More implications -one fewer resource to
remember -problem 1 irreversibility
due to 1 cobit gt 1 cbit
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problem 2 capacities of unitary gates
BHLS02, H03
Theorem For Cgt0, U gt C cbits(!) E ebits
iff U gt C cobits(!) E ebits
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a family of quantum protocols (problem 3)
DHW03
entanglement-assisted classical communication N
E ebits gt C cbits
Entanglement distillation r C cbits gt E ebits
Noisy SD r Q qubits gt C cbits
TP
quantum capacity N gt Q qubits
Noisy TP r C cbits gt Q qubits
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?
Noisy SD
EACC
E. distillation
Q. Cap
Alice
?
Noisy TP
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father trade-off curve (problem 4)
DHW03, DHW05
Q qubits sent per use of channel
father
I(AB)/2
I(AE)/2 I(AB)/2 - Ic(AiB)
E ebits allowed per use of channel
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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
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where next?
theory
classical- quantum protocols
quantum Shannon theory
classical Shannon theory
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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
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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

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Key technical tool use subgroup-adapted bases
Multiplicity-free branching for the chain S1 µ
µ Sn ) subgroup-adapted basis for
Pl pnpn-1p1i s.t. pnl and pj Á pj1.
Similarly, construct a subgroup-adapted basis for
Ql using the chain 1U(0) µ U(1) µµ U(d).
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the Schur transform
i1i
USch
li
i2i
qi
pi
u 2 U(d) p 2 Sn
ini
ql is a U(d)-irrep pl is a Sn-irrep
p
u
USch
USch
u

ql(u)
pl(p)
u
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the Clebsch-Gordan transform
UCG
li
li
Ql
qi
l0i
ii
Q(1) _at_ Cd
M0i
UCG
UCG

ql(u)
ql0(u)
u
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Schur transform iterated CG
(Cd)n
UCG
½i
l1i
i1i
UCG
l2i
l2i
q2i
i2i
l3i
q3i
i3i
ln-1i
UCG
ln-1i
lni
qn-1i
ini
qi
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recursive decomposition of CG
U(d) CG
liqdi
q0di l0i
qd-1i
q0d-1i

q1i
q01i
ii
ji l0 - li
liqdi
q0di l0i
U(d-1) CG
qd-1i
q0d-1i
qd-2i
q0d-2i
q01i
q1i
Wd
ii
kiq0d-1-qd-1i
ji
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normal form of i.i.d. channels
n
B
UN
A

E
lBi
VnN
lAi
lEi
qBi
qEi
qAi
Sn inverse CG
ai
pBi
pAi
pEi
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connections to the Sn QFT
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 CSn in
(Cn)n and do the Schur transform -Based on Howe
duality