Random and Nonrandom Coding for Quantum Information
Lead Research Organisation:
University of Bristol
Department Name: Mathematics
Abstract
Quantum information theory represents a cross-disciplinary blend of physics and information theory. Recent years have seen dramatic progress in many operational quantum information tasks: the basic question in all of them is after the optimal rate of compression, transmission, etc., and the crucial element in any solution is the construction of a coding scheme. Virtually all of them are obtained by random coding'', the key method of information theory, where the real art is the setting-up and average performance analysis of a family of codes.This motivates the proposed research, at whose starting point are the various random codes used with such success in recent work. The present state of knowledge, however, is still sorely incomplete: there are a number of fundamental problems that cannot be approached with the available random codes, which means we need to develop new ones. And on the other hand we still cannot compute the quantum capacity of a channel, in spite of random selection in principle describing asymptotically optimal codes - which means that we still do not have the correct code family for quantum error correction.To focus the work, four larger themes are outlined: Characterisation of random codes; Failure of additivity, and nonrandom'' codes for channels; Reverse coding and normal forms; Covering and colouring. Under these headings more specific problems are formulated, which will be pivotal for future progress.
Organisations
Publications
Abeyesinghe A
(2009)
The mother of all protocols: restructuring quantum information's family tree
in Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Hayden P
(2008)
Counterexamples to the Maximal p-Norm Multiplicativity Conjecture for all p > 1
in Communications in Mathematical Physics
Pawlowski M
(2009)
Information causality as a physical principle.
in Nature
Popescu S
(2006)
Entanglement and the foundations of statistical mechanics
in Nature Physics
| Description | In the five years of the fellowship, I published 50 papers, not counting preprints still awaiting publication. Hence here only a summary of the most significant results. 1) Following the Nature paper on negative information [with M. Horodecki and J. Oppenheim], we produced a long version [Comm. Math. Phys. 269(1):107-136, 2007], and I participated in three papers exploring different random codes for the quantum capacities [Open Syst. Inf. Dyn. 15(1), 2008]. The by now canonical formulation of all these ideas was given in a paper with A. Abeyesinghe, I. Devetak and P. Hayden, Proc. Roy. Soc. Lond. A 465:2537-2563, 2009]. 2) The random techniques used also had applications in a programme we developed in Bristol on the foundations of statistical mechanics [with S. Popescu and A.J. Short, Nature Phys. 2:754-758, 2006; with N. Linden, S. Popescu and A.J. Short, Phys Rev. E 79:061103, 2009]. This is still an active direction we are following. 3) Since the quantum capacity is still elusive, we considered an upper bound assisted by the class of symmetric channels [with G. Smith and J.A. Smolin, IEEE Trans. Inf. Theory 54(8):4208-4217, 2008]. This turns out to be additive and can be upper bounded efficiently by other additive quantities. 4) Thanks to a series of papers [arXiv:0707.0402; with P. Hayden, Comm. Math. Phys. 284(1):263-280, 2008; with T.S. Cubitt et al., Comm. Math. Phys. 284(1):281-290, 2008] we disproved additivity conjectures for so-called minimum output (Renyi-)entropies. Based on this work, Hastings eventually found counterexamples to the original additivity conjecture for the Holevo capacity. Following Smith/Yard's demonstration that the quantum capacity is not additive, we showed the analogous statement for the private capacity [with K. Li et al., Phys. Rev. Lett. 103:120501, 2009]. 5) With M. Christandl and N. Schuch [Phys. Rev. Lett. 104:240405, 2010] we bounded several entanglement measures of interest for the dxd-antisymmetric states, most notably showing that the entanglement cost is between 0.415 and 0.5 (independent of d) and that the regularised relative entropy of entanglement is likewise lower bounded by a constant. 6) With C.H. Bennett, I. Devetak, A. Harrow and P.W. Shor [arXiv:0912.5537] we proved the long-conjectured Quantum Reverse Shannon Theorem. 7) In various collaborations [with T.S. Cubitt et al., Phys. Rev. Lett. 104:230503, 2010 and IEEE Trans. Inf. Theory 57(8):5509-5523, 2011; with R. Duan and S. Severini, arXiv:1002.2514] I started investigating zero-error capacities of quantum and classical channels, assisted by various resources, which is still an ongoing programme. Among the many outcomes is, in the latter paper, a definition of "non-commutative graphs" and a Lovasz number for them, extending the eponymous theta parameter. 8) I started getting into various problems related to the foundations of quantum mechanics, from quantum pseudo-telepathy [with P.J. Cameron et al., Elec. J. Comb. 14(1), 2007], to Bell inequalities [with N. Linden, S. Popescu and A.J. Short, Phys. Rev. Lett. 99:180502, 2007], to "Information Causality" [with M. Pawlowski et al., Nature 461:1101-1104, 2009]. 9) I'm continuing to be interested in entropy inequalities: with B. Ibinson and N. Linden [Comm. Math. Phys. 269(1):223- 238, 2007; Comm. Math. Phys. 277(2):289-304, 2008] we showed that the relative entropy only satisfies monotonicity and that constrained entropy inequalities cannot easily made unconstrained. With J. Cadney and N. Linden [arXiv:1107.0624] we demonstrate an infinite family of constrained entropy inequalities. |
| Exploitation Route | Many of the results from the project were ground breaking advances in quantum information science, which by now have become part of the canon of results and techniques and which continue to be used by my fellow researchers. |
| Sectors | Digital/Communication/Information Technologies (including Software),Culture, Heritage, Museums and Collections,Other |
| Description | European Research Council |
| Amount | £1,400,000 (GBP) |
| Organisation | European Research Council (ERC) |
| Sector | Public |
| Country | Belgium |
| Start | |
| Description | European Research Council |
| Amount | £1,400,000 (GBP) |
| Organisation | European Research Council (ERC) |
| Sector | Public |
| Country | Belgium |
| Start | |