The concept of quantum computation can be traced back to the early 1980’s, when Richard Feynman observed that certain quantum mechanical effects cannot be simulated efficiently on a classical computer. This observation led to speculation that perhaps com- putation in general could be done more efficiently if it made use of these quantum effects. However no one was sure how to use the quantum effects to speed up computation until 1994, when Peter Shor surprised the world by describing a polynomial time quantum al- gorithm for factoring integers. This discovery prompted a flurry of activity, both among experimentalists trying to build quantum computers and theoreticians trying to find other quantum algorithms. Since then the field of quantum information and computation develops very quickly. It is attracting more and more people from mathematics, physics and computer science to work on this field.

Quantum information and computation theory has many surprising properties that do not exist in the classical case. For instance, the quantum key distribution can be used to transmit private keys on insecure channels. The quantum dense coding uses one quantum bit together with an EPR (maximally entangled pair of quantum bit) to encode and transmit two classical bits. Since EPR pairs can be distributed ahead of time, only one qubit needs to be physically transmitted to communicate two bits of classical information.

As an unknown quantum state can not be cloned perfectly, a unknown quantum state can only be transferred from one place to another, but not copied. Quantum teleportation is the transfer of a quantum state from one place to another through classical channels and entangled quantum states. Moreover, as it is impossible to isolate the qubits sufficiently from the external environment, the greatest problem for building quantum computers is the decoherence. The breakthrough came from the algorithm, through the invention of quantum error correction techniques. Theories in designing quantum error correcting codes that detect certain kinds of errors are also under development. The quantum key distribution and teleportation are already realized experimently for some simple cases. For quantum computation, a number of proposals for building quantum computers using ion traps, nuclear magnetic resonance, optical and solid state techniques are investigated both theoretically and experimentally.

Above all, the quantum entanglement plays very important roles in the quantum in- formation processing. However, the theory about quantum entanglement is still far from satisfactory. Most proposed measures of entanglement involve extremizations which are dif- ficult to handle analytically. It is also an unsolved problem to judge if a quantum state is entangled or separable. Moreover, there are many interesting properties of entangled states such as bounded ones, quantum correlations such as quantum discord, as well as nonlocality. The structures of quantum states, e.g., the geometric orbits, or the equivalence condition for two states under local unitary transformations are also tricky problems.

These problems are mathematically related to operator algebras, matrix analysis on tensor spaces, stochastic processes and probability theory, geometry, nonlinear differential equations etc..

We had organized successfully the conference “Quantum Information and Quantum En- tanglement” at the Max Planck Institute for Mathematics in the Sciences, July 16- 17, 2010. Many outstanding researchers, both physicists and mathematicians, from China, Germany and other countries joined the conference, and discussed the following topics: measures of quantum entanglement, separability criteria, classification of quantum states under local operations, distillation, evolution of quantum entanglement; quantum computation and al- gorithms, quantum simulation, cryptography, error correction, dense coding, teleportation; Bell inequalities related to quantum entanglement and quantum mechanics; quantum qubits and decoherence in cold atoms, trapped ions, cavity QED, Josephson junction, linear and nonlinear optics and quantum dots.

It is now a good time to take stock of what has been achieved in the meantime, as well as to include the young people that have entered the field in the meantime. There are several strong groups in Poland, Germany, and China, and it should be very fruitful to bring them together in Sanya. We can already build upon successful cooperations between some mathematicians from China and Germany, but it is important for us to also bring the different groups together and to involve some talented graduate students and postdoctors. In particular, it is an explicit aim of the workshop to stimulate new collaborations.

Concerning the financial arrangements, we assume that TSIMF will cover the local ex- penses of up to 50 participants. All participants should take care of their own travel expenses (except that we hope that we can nominate two international participants to have their ex- penses covered by the Simons Foundation).