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A Family of Quantum Protocols

A Family of Quantum Protocols. quant-ph/0308044. IEEE Symposium on Information Theory June 28, 2004. Igor Devetak, IBM Aram Harrow, MIT Andreas Winter, Bristol. outline. Introduction basic concepts and resource inequalities. A family of protocols Rederive and connect old protocols

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A Family of Quantum Protocols

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  1. A Family of Quantum Protocols quant-ph/0308044 IEEE Symposium on Information Theory June 28, 2004 Igor Devetak, IBM Aram Harrow, MIT Andreas Winter, Bristol

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

  3. Information processing resources may be: • classical / quantum c / q • noisy / noiseless (unit) { } / [ ] • dynamic / static ! / ¢ examples of bipartite resources

  4. Church of the larger Hilbert space Channel N:HA’!HB) isometric extension UN:HA’!HB­HE s.t. N(r) = trEUN(r). Use a test source |fiAA’ and define |yiABE = (IA­ UN)|fiAA’ A |fiAA’ |yiABE A0 UN B E static rAB) purification |yiABE s.t. rAB = trEyABE.

  5. information theoretic quantities von Neumann entropy: H(A)y = -tr [yA log yA] mutual information: I(A:B) = H(A) + H(B) – H(AB) coherent information: Ic(AiB) = H(B) – H(AB)

  6. resource inequalities Example: quantum channel coding {q!q}N> Ic(AiB)y [q!q] Meaning there exists an asymptotic and approximate protocol transforming the LHS into the RHS. For any e>0 and any R<Ic(AiB) and for sufficiently large n there exist encoding and decoding maps E: H2­nR! HA’­n and D: HB­n!H2­nR such that for any input |fi2H2­n, (D ¢ N­n ¢ E)|fi ¼e|fi The capacity is given by limn!1 (1/n) maxy Ic(AiB)y, where the maximization is over all y arising from N­n.

  7. 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]

  8. 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] SD TP SD {qq} + H(A) [q!q] > I(A:B) [c!c] H3LT, [QIC 1, 2001], noisy SD {q!q} + H(A) [qq] > I(A:B) [c!c] BSST, [IEEE IT 48, 2002], E-assisted cap. QE {qq} + I(A:E) [c!c] > Ic(AiB) [q!q] DW, entanglement distillation {q!q} > Ic(AiB) [q!q] L/S/D, quantum channel cap. TP TP {qq} + I(A:B) [c!c] > Ic(AiB) [q!q] DHW, noisy TP (TP) 2[c!c] + [qq] > {q!q} (SD) [q!q] + [qq] > 2[c!c] (QE) [q!q] > [qq]

  9. 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, 097902 (2004)

  10. 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] O O SD TP SD I {qq} + H(A) [q!q] > I(A:B) [c!c] H3LT, [QIC 1, 2001], noisy SD {q!q} + H(A) [qq] > I(A:B) [c!c] BSST, [IEEE IT 48, 2002], E-assisted cap. {qq} + I(A:E) [c!c] > Ic(AiB) [q!q] DW, entanglement distillation QE {q!q} > Ic(AiB) [q!q] L/S/D, quantum channel cap. TP TP {qq} + I(A:B) [c!c] > Ic(AiB) [q!q] DHW, noisy TP

  11. qubit > ebit bound I(A:B)/2 [BSST; quant-ph/0106052] 45o Ic(A>B) [L/S/D] H(A)+I(A:B) main result #2: tradeoff curves example: quantum channel capacity with limited entanglement Q: qubits sent per use of channel E: ebits allowed per use of channel

  12. I(A:B)/2 father 45o Ic(AiB) [L/S/D] I(A:E)/2 = I(A:B)/2 - Ic(AiB) father trade-off curve Q: qubits sent per use of channel E: ebits allowed per use of channel

  13. tradeoff techniques measurement compression: [Winter, IEEE IT 45, 1999] An instrument T:A ! AEXAXB can be simulated on |fiAR 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 Px with probability px allows an extra output of I(X:B) [c!c]. [DS, quant-ph/0311131]

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

  15. what’s left • In quant-ph/0308044, 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

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