One of the best known applications of the principles of quantum theory is quantum cryptography,
a subject founded by ideas of Wiesner (1983) and Bennett and Brassard (1984). It applies these
principles to securely establish a secret key between two distant parties. Quantum cryptography
gained public attention after Shor (1994) showed that quantum computers can break most of the
cryptographic schemes currently used. Based upon ideas of Ekert (1991), who has shown a relation
between quantum cryptography and entanglement purification, it is possible to compute upper and
lower bounds on the efficiency of quantum cryptography in various schemes, e.g. ideal or practical
qubit and higher-dimensional systems as well as for continuous-variable schemes.

In our group, we currently investigate basic aspects of quantum cryptography and its relation to entanglement:

### Quantum cryptography

The peculiar statistical properties of entangled quantum states can be exploited practically for distributing securely
private random keys between various parties. Using quantum theory for solving the classically impossible key distribution
problem is at the heart of quantum cryptography. The first secure quantum key distribution protocol was propsed in 1984
by Bennett and Brassard (BB84 protocol). In recent years also a thorough theoretical understanding of what the security of such a
quantum protocol means has also been obtained. Recently, we analyzed a generalization of the original BB84 protocol which is valid
not only for qubits but for arbitrary-dimensional elementary quantum mechanical data carriers and we explored its
robustness, i.e. the maximum tolerable error probability which still guarantees the generation of a private random key.

Oliver Kern and Joseph M. Renes, Quantum Information and Computation 8, 756-772 (2008)

J. M. Renes and G. Smith, Phys. Rev. Lett. 98, 020502 (2007)

K. S. Ranade and G. Alber, J. Phys. A 40, 139 (2007)

A. Khalique, Georgios M. Nikolopoulos, and G. Alber, Eur. Phys. J. D (2006)

G. Nikolopoulos, A. Khalique, and G. Alber, Eur. Phys. J. D 37, 441 (2006)

Georgios M. Nikolopoulos, K. S. Ranade, and G. Alber, Phys. Rev. A 73, 032325 (2006)

K.S. Ranade and G. Alber, J. Phys. A 39, 1701 (2006)

G. Nikolopoulos, A. Khalique, and G. Alber, Eur. Phys. J. D 37, 441 (2006)

G. Nikolopoulos and G. Alber, Phys. Rev. A. 72, 032320 (2005)

G. Alber and Th. Walther, Thema Forschung 1, 44 (2004)

### Quantum state purification

With the help of nonlinear quantum transformations it is possible to develop purification processes. Typically, in such
a purification process one aims at producing a pure entangled quantum state starting from a mixed one by means of local
operations only. From the point of view of practical applicability it is desirable to develop purification protocols
which are not only efficient but which also involve local operations which can be implemented easily. Such a bipartite
purification protocol which purifies arbitrary-dimensional quantum states is developed in

G. Alber, A. Delgado, N. Gisin, and I. Jex, J. Phys. A 34, 8821 (2001).