Theory of entanglement




Secret correlations are an important resource already in classical cryptography where, for perfect secrecy, sender and receiver hold two identical and therefore perfectly correlated code-books whose contents are only known to them. Such secret correlations can neither be created nor enhanced by public discussion. Entanglement represents a novel and particularly strong form of such secret correlations. Therefore, entanglement is a key resource in quantum information science. Its role as a resource becomes even clearer when one is considering a communication scenario between distant laboratories. Then, experimental capabilities are constrained to local operations and classical communication (LOCC) as opposed to general non-local quantum operations affecting both laboratories. This is an important setting in quantum communication but also distributed quantum computation and general quantum manipulations. The resulting theory of entanglement aims to answer three basic questions.

Firstly, we wish to characterize and verify entangled resources to be able to decide, ideally in an efficient way, when a particular state that has been created in an experimental set-up or a theoretical consideration contains the precious entanglement resource. For the experimental verification of this resource, the tool of entanglement witnesses allows to detect entanglement with local measurements only, and thus is easily implementable with present technology. Secondly, we wish to determine how entangled state may be manipulated under LOCC. In many situations an experimental setting will yield a certain type of entangled state that may suffer certain deficiencies. It may not be the correct type of state or it may have suffered errors due to experimental imperfections and be entangled. Once characterization methods have determined that the resulting state contains entanglement one can then aim to transform the initial state into the desired final state. Thirdly, it will be important to quantify the efficiency of all the processes and procedures as well as the entanglement resources that have been identified in the above two areas of research. If we have found entanglement in a state, then one will need to know how much of it there is.

Considerable progress in this area has been made in recent years, in particular in the case of bi-partite entanglement, but we are still far away from a comprehensive understanding of this key resource for quantum information processing. Research in this area will continue to play a central role in the field, and we expect that an increasing effort will be undertaken towards the classification and quantification of entanglement in multi-party entangled states. It is worth pointing out that insights in the theory of entanglement are not only important the field of QIS itself, but they have now reached the stage where they are being applied to other areas of physics (see the subsection 4.3.10).

Key references

[1] R.F. Werner, ‘‘Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model’’, Phys. Rev. A 40, 4277 (1989).

[2] M. Horodecki and P. Horodecki and R. Horodecki, ‘‘Separability of mixed states: necessary and sufficient conditions’’, Phys. Lett. A 1, 223 (1996).

[3] C.H. Bennett, H.J. Bernstein, S. Popescu and B. Schumacher, ‘‘Concentrating partial entanglement by local operations’’, Phys. Rev. A 53, 2046 (1996).

[4] V. Vedral and M.B. Plenio, ‘‘Entanglement measures and purification procedures’’, Phys. Rev. A 57, 1619 (1998).

[5] M.A. Nielsen, ‘‘Conditions for a Class of Entanglement Transformations’’, Phys. Rev. Lett. 83, 436 (1999).

[6] Recent tutorial reviews include M.B. Plenio and S. Virmani, “An introduction to entanglement measures” Quant. Inf. Comp. 7, 1 (2007); R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, “Quantum entanglement”, arXiv:quant-ph/0702225,

[7] M. Bourennane et al, “Experimental detection of multipartite entanglement using witness operators”, Phys. Rev. Lett. 92, 087902 (2004).