Lossless quantum data compression and secure direct communication

CONTENTS 5 7 Summary and Outlook 107 Acknowledgements V 6 CONTENTS Introduction Quantum information theory is the combination of quantum mechanics and information theory. The profit is on both sides: quantum mechanics gains valuable aspects concerning the physical interpretation of the theory, and information theory gains enhanced capabilities of information processing and communication. Classically, it is the logical yes/no decision, the bit, which forms the elementary unit of information. A sequence of bits forms the basic object of information theory, the message. A message can be composed and read out by addressing each bit individually. In quantum information theory the elementary unit of information is the qubit, which represents the linear superposition of yes and no. A sequence of qubits forms the quantum message and here another phenomenon shows up: entanglement. A quantum message consisting of entangled qubits contains " non-local " information, which means that the information cannot be stored and read out by addressing each qubit individually, but only by performing a joint operation on all qubits. There is no analogon for this in classical information theory. The extension of information theory to quantum information theory enables us to search for new algorithms and communication protocols. Shor's factoring algorithm [1] shows that the capabilities of a quantum computer exceed those of a classical computer. The Shor algorithm represents the only known efficient algorithm for prime number factoriza-tion. Here, " efficient " means that the time needed for the computation is a polynomial function of the input length. Any superpolynomial relation between input length and computation time implies that the algorithm is " inefficient ". The origin for the gap of efficieny between classical and quantum computers is the fact that it is impossible to efficiently simulate a quantum computer on a classical computer. The input of a quantum computer is a sequence of qubits, say of length N. The corresponding Hilbert space is of dimension 2 N , where each dimension represents one degree of freedom. A classical computer has to carry out calculations by addressing any of these 2 N degrees of freedom. In contrast to that, the quantum computer performs his algorithms by addressing the N qubits only. The quantum processor does not have to know about linear algebra: the laws of quantum mechanics do the job for free. If we would be able to build a quantum computer with enough resources, then by using the …