Combinatorial Approaches in Quantum Information Theory

Quantum entanglement is one of the most remarkable aspects of quantum physics. If two particles are in a entangled state, then, even if the particles are physically separated by a great distance, they behave in some repects as a single entity. Entanglement is a key resource for quantum information processing and spatially separated entangled pairs of particles have been used for numerous purposes such as teleportation, superdense coding and cryptography based on Bell’s theorem. Just as two distant particles could be entangled, it is also possible to entangle three or more separated particles. A well-known application of multipartite entanglement is in testing nonlocality from different directions. Recently, it has also been used for many multi-party computation and communication tasks and multi-party cryptography. One of the major issues in dealing with multi-partite entangled states is of purification. Distilling pure maximally entangled state in this case may not be as simple as that of bipartite case. But if it is possible to create multi-partite entangled states from the bipartite ones then we can first distill pure maximal bipartite states and can then prepare the multi-partite ones. To this end, we consider the problem of creating maximally entangled multi-partite states out of Bell pairs distributed in a communication network from a physical as well as from a combinatorial perspective. We investigate the minimal combinatorics of Bell pairs distribution required for this purpose and discuss how this combinatorics gives rise to resource minimization for practical implementations. We present two protocols for this purpose. The first protocol enables to prepare a GHZ state using two Bell pairs shared amongst the three users with help of two cbits of communication and local operations. The protocol involes all the three users dynamically and thus can find applications in cryptographic tasks. Second protocol entails the use of O(n) cbits of communication and local operations to prepare an n partite maximally entangled state in a distributed network of bell pairs along a spanning tree of EPR graph of the n users. We show that this spanning tree structure is the minimal combinatorial requirement. We also characterize the minimal combinatorics of agents in the creation of pure maximal multi-partite entanglement amongst the set N of n agents in a network using apriori multi-partite entanglement states amongst subsets of N . Another major and interesting issue is of quantifying multi-partite entangled states. Multi-partite entangled states, unlike the bipartite ones, lack convenient mathematical properties like Schmidt decompostion and therefore it becomes difficult to characterize them. Some approches, essentially using the generalization of Schmidt decomposition, have been taken in this direction ; however a general formulation in this case is still an outstanding unresolved problem. State transformations under local operations and classical communication (LOCC) are very important while quantifying entanglement because LOCC can at the best increase classical correlations and therefore a good measure of entanglement is not supposed to increase under LOCC. All the current approaches to study the state transformation under LOCC are based on entropic criterion. We present an entirely different approach based on nice combinatorial properties of graphs and set systems. We introduce a technique called bicolored merging and obtain several results about such transformations. We demostrate a partial ordering of multi-partite states and various classes of incomparable multi-partite states. We utilize these results to establish the impossibility of doing selective teleportation in a case where the apriori entanglement is in the form of a GHZ state. We also discuss the minimum number of copies of a state required to prepare another state by LOCC and present bounds on this number in terms of quantum distance between the two states. The ideas developed in this work continues the combinatorial setting mentioned above and can been extended to incorporate other new kinds of multi-partite states. Moreover, the idea of bicolored merging may also be appropriate to some other areas of information sciences. Key distribution is a fundamental problem in secure communication and quantum key distribution (QKD) protocols for key distribution between two parties on the account of quantum uncertainty and no-cloning principles was realized two decades ago, however the more rigorous and comprehensive proofs of this task, taking into consideration source, device and channel noise as well as an arbitrarily powerful eavesdropper, have been only recently studied by various authors. We consider QKD between two parties extended to that between n trustful parties, that is, how the n parties may share an identical secret key among themselves. We propose a protocol for this purpose and prove its unconditional security. The protocol is simple in the