ASSESSMENT OF CURRENT RESULTS AND OUTLOOK ON FUTURE EFFORTS
QUANTUM INFORMATION SCIENCE THEORY
SPIN-OFF TO OTHER FIELDS
A very exciting aspect of theoretical work in QIS is the impact that it is beginning to make on other fields of science. In the case of classical computing such insights include the first exponential bounds on certain locally decodable codes, classical proof systems for lattice problems, bounds on the query complexity of local search problems, an efficient classical cryptographic scheme whose security is based on quantum considerations, and a quantum method to compute how many Toffoli gates are required to realize a reversible classical computation. The potential that QIS is offering for classical computing and mathematics may be understood by the following analogy. Real analysis is a very successful discipline but it contained a number of unsolved problems that were only solved by considering complex numbers, i.e. going to a larger space in which to describe the problem. By analogy we expect that moving from classical state space into the much larger quantum mechanical state space we will find novel approaches towards the solution of problems that ostensibly lie entirely within the classical realm. As the enormous size of the quantum mechanical state space is due to entanglement, one may view this as a further consequence of entanglement and a further justification for the importance of the study of entanglement (see section 4.3.6).
Relatively recently the study of the role of entanglement in infinitely extended quantum many-body systems and quantum field theories has attracted considerable interest. Many of the questions that are now being asked in this area can only be answered or even formulated correctly because of the many insights and techniques gained in the research in entanglement theory in recent years. These results have already born fruits in the development of novel simulation techniques for quantum many-body systems - generalization of the Density Matrix Renormalization Group (DMRG) method -, novel facets of correlations and phase transitions in spin systems and quantum field theories and solutions of longstanding open questions.
This demonstrates that the research into entanglement, its characterization, manipulation and quantification will not only continue to have impact within quantum information but is now reaching the stage where its insights are being applied to other areas of physics, with potentially enormous benefits, both intellectually but perhaps also commercially.
Key references
[1] I. Kerenidis and R. de Wolf, Exponential lower bound for 2-query locally decodable codes via a quantum argument, quant-ph/0208062, http://xxx.arxiv.org.
[2] S. Popescu, B. Groisman and S. Massar, Lower bound on the number of Toffoli gates in a classical reversible circuit through quantum information concept, quant-ph/0407035, http://xxx.arxiv.org.
[3] J. I. Latorre, E. Rico, and G. Vidal, Ground state entanglement in quantum spin chains, Quant. Inf. Comp. 4, 048 (2004).
[4] F. Verstraete, D. Porras and J. I. Cirac, DMRG and periodic boundary conditions: a quantum information perspective, Phys. Rev. Lett. 93, 227205 (2004).
[5] M. B. Plenio, J. Eisert, J. Dreissig and M. Cramer, Entropy, Entanglement, and Area: Analytical Results for Harmonic Lattice Systems, Phys. Rev. Lett. 94, 060503 (2005).
