DEFENSE!

Applications of Coherent Classical Communication and Schur duality to quantum information theory Aram Harrow MIT Physics June 28, 2005 Committee: Isaac Chuang Edward Farhi Peter Shor Collaborators Dave 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 distinguishability collapses the state to the observed outcome is no problemmeasurement 2 n dimensions2 n statesn bits unitary matrices NAND, XOR, etc… basic units of computation qubit C 2 = span{|0 i,|1 i } bit: {0,1} basic unit of information quantumclassical

quantum algorithms Quantum computers can efficiently simulate quantum systems. Deutsch 1985 A database of N elements can be searched with O( p N) 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 ( C d ) ­ n ( C d ) ­ 4 = C d ­ C d ­ C d ­ C d U 2U d ! U ­ U ­ U ­ U (1324) 2S 4 ! ( C d ) ­ 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 S n Fourier transform [joint work with Bacon and Chuang; quant-ph/ , 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 A lice B ob noisy channel X2XX2X N N(X) noisy correlations P(X,Y) Y X perfect bit channel cbit X 2 {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 A lice E ve B ob free local operations free local operations noisy quantum channel NA!BNA!B noisy shared entanglement  AB noiseless quantum channel qubit unitary gate U AB

cbit one use of a noiseless classical bit channel ebit the state |  i =(|0 i A |0 i B + |1 i A |1 i B )/ p 2 qubit one use of a noiseless quantum bit channel N A ! B a noisy quantum channel   a noisy bipartite state U  a 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 gates Suppose Alice can send Bob n cbits using a unitary interaction: U|x i A |0 i B ¼ |x i B |  x i AB for x 2 {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 cbits Noisy 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] TP

I(A:B)/2 [BSST] H(A)+I(A:B) problem #4: tradeoff curves Q : qubits sent per use of channel E: ebits allowed per use of channel I c =H(B) - H(AB) [L/S/D] qubit > ebit bound 45 o N + E ebits > Q qubits

coherent classical communication (CCC) cbits seen by the Church of the Larger Hilbert Space |x i A ! |x i B |x i E for x={0,1}. Give Alice coherent feedback: The map |x i A ! |x i A |x i B is called a coherent bit, or cobit. a|0 i A + b|1 i A ! a|0 i A |0 i B + b|1 i A |1 i B [H03] |0 i A |1 i A |1 i B |0 i B a|0 i A + b|1 i A |a| 2 |b| 2 yet another quantum resource: Alice throws her output away: 1 cobit > 1 cbit Alice inputs (|0 i +|1 i )/ p 2 or half of |  i : 1 cobit > 1 ebit Alice simulates a cobit locally: 1 qubit > 1 cobit

the power of CCC 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 > 1 qubit + 2 ebits Ex: super-dense coding 1 qubit + 1 ebit > 2 cobits Implication: 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 iffU > C cobits( ! ) + E ebits [BHLS02, H03] iff there exists an ensemble E ={p i,|  i i ABA’B’ } such that  (U( E )) -  ( E ) > C E(U( E )) - E( E ) > E  E  = Holevo information between i and tr AA’  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 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 : qubits sent per use of channel E : ebits allowed per use of channel I c (A i B) [L/S/D] 45 o I(A:E)/2 = I(A:B)/2 - I c (A i B) 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 practice classical Shannon theory classical Shannon theory classical- quantum protocols classical- quantum protocols quantum Shannon theory quantum Shannon theory HSW coding teleportation super-dense coding noisy SD, etc.. CCC family unitary gates more? information technology information technology Brady Bunch broadcasts Brady Bunch broadcasts cryptography practical codes practical codes QECC distributed QC distributed QC 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/ [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 S 1 µ … µ S n ) subgroup-adapted basis for P |p n p n-1 …p 1 i s.t. p n = and p j Á p j+1. Similarly, construct a subgroup-adapted basis for Q using the chain: {1}=U(0) µ U(1) µ … µ U(d).

u u u u u u   |i 1 i |i 2 i |i n i U Sch | i |q i |p i U Sch = q (u) p (  ) u 2 U(d)  2 S n q is a U(d)-irrep p is a S n -irrep the Schur transform

U CG |q i | i |i i | i | 0 i |M 0 i U CG q ( u ) u u = U CG q 0 (u) Q Q (1)  C d the Clebsch-Gordan transform

U CG |i 1 i |½ i |i 2 i |i n i | 1 i | 2 i |q 2 i |i 3 i U CG | 2 i | 3 i |q 3 i | n-1 i |q n-1 i U CG | n-1 i | n i |q i (Cd)­n(Cd)­n Schur transform = iterated CG

recursive decomposition of CG U(d) CG | i =|q d i |q d-1 i |q 1 i |i i |q 0 d i = | 0 i |q 0 d-1 i |q 0 1 i |j i = | 0 - i = U(d-1) CG | i =|q d i |q d-1 i |q 1 i |i i |q 0 d i = | 0 i |q 0 d-1 i |q 0 1 i |k i =|q 0 d-1 -q d-1 i |q d-2 i |q 0 d-2 i WdWd |j i

normal form of i.i.d. channels UNUN A B E | B i | E i |q B i |q E i |i|i VnNVnN | A i |q A i |p A i S n inverse CG |p B i |p E i ­n­n =

1) S n QFT ! Schur transform: Generalized Phase Estimation - Only permits measurement in Schur basis, not full Schur transform. -Similar to [abelian QFT ! phase estimation]. 1) S n 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 ! S n QFT -Just embed C [S n ] in ( C n ) ­ n and do the Schur transform -Based on Howe duality 2) Schur transform ! S n QFT -Just embed C [S n ] in ( C n ) ­ n and do the Schur transform -Based on Howe duality connections to the S n QFT