A Family of Quantum Protocols Igor Devetak, IBM Aram Harrow, MIT Andreas Winter, Bristol quant-ph/ IEEE Symposium on Information Theory June 28, 2004

outline Introduction – basic concepts and resource inequalities. A family of protocols –Rederive and connect old protocols –Prove new protocols (parents) Optimal trade-off curves

cbit[c ! c]1 noiseless bit channel ebit[qq]the state (|0 i A |0 i B + |1 i A |1 i B )/ p 2 qubit[q ! q]1 noiseless qubit channel noisy state {qq}  noisy bipartite quantum state  AB noisy channel {q ! q} N noisy cptp map: N : H A’ !H B Information processing resources may be: classical / quantumc / q noisy / noiseless (unit){ } / [ ] dynamic / static ! / ¢ examples of bipartite resources

Church of the larger Hilbert space static  AB ) purification |  i ABE s.t.  AB = tr E  ABE. |  i AA’ UNUN A A0A0 B E |  i ABE Channel N : H A’ !H B ) isometric extension U N : H A’ !H B ­H E s.t. N (  ) = tr E U N (  ). Use a test source |  i AA’ and define |  i ABE = (I A ­ U N )|  i AA’

information theoretic quantities von Neumann entropy: H(A)  = -tr [  A log  A ] mutual information: I(A:B) = H(A) + H(B) – H(AB) coherent information: I c (A i B) = H(B) – H(AB)

resource inequalities Example: quantum channel coding {q ! q} N > I c (A i B)  [q ! q] Meaning there exists an asymptotic and approximate protocol transforming the LHS into the RHS. For any  >0 and any R<I c (A i B) and for sufficiently large n there exist encoding and decoding maps E : H 2 ­ nR ! H A’ ­ n and D : H B ­ n ! H 2 ­ nR such that for any input |  i2H 2 ­ n, ( D ¢ N ­ n ¢ E )|  i ¼  |  i The capacity is given by lim n !1 (1/n) max  I c (A i B) , where the maximization is over all  arising from N ­ n.

main result #1: parent protocols father: {q ! q} + ½ I(A:E) [qq] > ½ I(A:B) [q ! q] mother: {qq} + ½ I(A:E) [q ! q] > ½ I(A:B) [qq] Basic protocols combine with parents to get children. (TP)2[c ! c] + [qq] > {q ! q} (SD)[q ! q] + [qq] > 2[c ! c] (QE)[q ! q] > [qq]

the family tree  {q ! q} + ½ I(A:E) [qq] > ½ I(A:B) [q ! q]  {qq} + ½ I(A:E) [q ! q] > ½ I(A:B) [qq] {q ! q} + H(A) [qq] > I(A:B) [c ! c] BSST, [IEEE IT 48, 2002], E-assisted cap. {q ! q} > I c (A i B) [q ! q] L/S/D, quantum channel cap. {qq} + H(A) [q ! q] > I(A:B) [c ! c] H 3 LT, [QIC 1, 2001], noisy SD {qq} + I(A:B) [c ! c] > I c (A i B) [q ! q] DHW, noisy TP SD QE TPSD TP {qq} + I(A:E) [c ! c] > I c (A i B) [q ! q] DW, entanglement distillation TP (TP) 2[c ! c] + [qq] > {q ! q} (SD)[q ! q] + [qq] > 2[c ! c] (QE)[q ! q] > [qq]

coherent classical communication rule I: X + C [c ! c] > Y ) X + C/2 ( [q ! q] – [qq] ) > Y rule O: X > Y + C [c ! c] ) X > Y + C/2 ( [q ! q] + [qq] ) Whenever the classical message in the original protocol is almost uniformly distributed and is almost decoupled from the remaining quantum state of Alice, Bob and Eve. based on PRL 92, (2004)

generating the parents  {q ! q} + ½ I(A:E) [qq] > ½ I(A:B) [q ! q]  {qq} + ½ I(A:E) [q ! q] > ½ I(A:B) [qq] {q ! q} + H(A) [qq] > I(A:B) [c ! c] BSST, [IEEE IT 48, 2002], E-assisted cap. {q ! q} > I c (A i B) [q ! q] L/S/D, quantum channel cap. {qq} + H(A) [q ! q] > I(A:B) [c ! c] H 3 LT, [QIC 1, 2001], noisy SD {qq} + I(A:B) [c ! c] > I c (A i B) [q ! q] DHW, noisy TP SD QE TPSD TP {qq} + I(A:E) [c ! c] > I c (A i B) [q ! q] DW, entanglement distillation TP O O I

I(A:B)/2 [BSST; quant-ph/ ] H(A)+I(A:B) main result #2: tradeoff curves Q : qubits sent per use of channel E : ebits allowed per use of channel I c (A>B) [L/S/D] qubit > ebit bound 45 o example: quantum channel capacity with limited entanglement

father trade-off curve 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 father

tradeoff techniques measurement compression: [Winter, IEEE IT 45, 1999] An instrument T:A ! AEX A X B can be simulated on |  i AR using I(X:R) [c ! c] + H(X|R) [cc]. derandomization: If the output state is pure, [cc] inputs are unnecessary. piggybacking: Time-sharing protocol P x with probability p x allows an extra output of I(X:B) [c ! c]. [DS, quant-ph/ ]

mother trade-off curve {qq} + ½ I(A:E) [q ! q] > ½ I(A:B)[qq] preprocessing instrument T:A ! AE’X {qq} + ½ I(A:EE’|X) [q ! q] + H(X)[c ! c] > ½ I(A:B|X)[qq] H(X) [c ! c] measurement compression I(X:BE) [c ! c] + H(X|BE) [cc] I(X:BE) [c ! c] derandomization ½ I(X:BE) ( [q ! q] – [qq] ) rule I {qq} + ½ (I(A:EE’|X) + I(X:BE)) [q ! q] > ½ (I(A:B|X) + I(X:BE)) [qq]

what’s left In quant-ph/ , we prove similar tradeoff curves for the rest of the resource inequalities in the family. Remaining open questions include – Finding single-letter formulae (i.e. additivity) – Reducing the optimizations over instruments – Addressing two-way communication – Multiple noisy resources