ASSESSMENT OF CURRENT RESULTS AND OUTLOOK ON FUTURE EFFORTS
QUANTUM INFORMATION SCIENCE THEORY
NOISY COMMUNICATION CHANNELS
The proper understanding of the capacities of quantum communication channels is at the heart of the study of quantum communication tasks. Of particular importance are the transmission of classical or quantum information, or establishing secret keys. The general framework for distilling classical keys from quantum states have been also established, opening the possibility of secure communication on extremely noisy channels. But it is also known that one can use noise and perfect side communication to implement other cryptographic primitives like bit commitment and oblivious transfer. Channel capacities are of central interest in several different settings, being reflected notably by the classical capacity of quantum channels, quantum capacities, and entanglement-assisted capacities.
The central question is essentially what resources are required for transmitting classical or quantum information using quantum channels, such as optical fibers in a practical realization. An open problem is for example if entanglement can always be used as a source of secret keys; another key problem is whether an increased capacity can be obtained by employing entangled signal states (multiple uses) as opposed to single uses of the channel. This problem is widely known as the additivity problem for the Holevo capacity. There has been recent progress on this question, in particular linking this problem to seemingly unrelated additivity questions. In future work, this link between the different problems must be studied in more detail. For channels of salient interest this question will be directly addressed, using concepts of output purities. Novel methods from global optimization may be helpful here. For Gaussian channels, with practical importance in quantum communication with fibers, it seems within reach to find a complete answer to the above questions.
Finally, it is to be expected that more problems, as well as new perspectives, will arise when one considers multi-user channels, i.e., with more than one sender/receiver. While single-sender-receiver settings serve well to study bipartite correlations, such problems have an immediate impact on understanding multi-partite correlations and their role in quantum communication via noisy channels. Also, quantum analogues of certain basic classical network theory primitives have been identified, and the evidence for new non-classical features, such as negative partial information established. Further investigations will be needed to identify differences and similarities in the classical and quantum network theories.
Key references
[1] C. H. Bennett, G. Brassard, C. Crépeau, and M.-H. Skubiszewska, ‘‘Practical quantum oblivious transfer’’, Lecture Notes in Computer Science 576, 351 (1991).
[2] S. Holevo, ‘‘The capacity of the quantum channel with general signal states’’, IEEE Trans. Inf. Theory 44, 269 (1998).
[3] C. H. Bennett, P. W. Shor, J. A. Smolin, and A. V. Thapliyal, ‘‘Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem’’, Phys. Rev. Lett. 83, 3081 (1999).
[4] G. G. Amosov, A. S. Holevo, and R. F. Werner, ‘‘On some additivity problems in quantum information theory’’, Problems in Information Transmission 36, 305 (2000).
[5] P. W. Shor, ‘‘Equivalence of additivity questions in quantum information theory’’, Commun. Math. Phys. 246, 453 (2004).
[6] M. Horodecki, J. Oppenheim, A. Winter, “Partial information can be negative”, Nature 436, 676 (2005).
[7] K. Horodecki, M. Horodecki, P. Horodecki, J. Oppenheim “Secure key from bound entanglement”, Phys. Rev. Lett. 94, 160502 (2005).
