Quantum Gate And Entanglement

In quantum information science, there are two basic conceptions: quantum gate operation and quantum state. Quantum gates are unitary operations used to describe the evolution of quantum states. Quantum computers are believed to be more powerful than classical computers in computation and simulation, and quantum computation can be regarded as a process that a series of basic quatum gates apply to a standard initial state and then a standard measurement follows. Entangled state is the most important quantum state. It can not only be used to examine the basis of quantum mechanics but also has many interesting applications, such as the well-known quantum teleportation, dense coding and key distribution. Quantum gate operation and entangled state have a close relation: on the one hand, quantum gate can generate or increase entanglement, on the other hand, entangled state can be used to construct quantum gate operation.In this dissertation we mainly focus on the construction of quantum gates, the conversion between quantum gates and entanglement, and the application of entangled states. The work is listed as follows:1. The construction of quantum gates.We introduce two new conceptions on two-qubit gate construction: mirror gate and super-controlled gate. Mirror gate gives a relation between two gates. A theorem on mirror gate is presented through which we can prove that CNOT and DCNOT gates have the same gate construction ability and that only the special B gate has the ability that two applications can construct any two-qubit gate when single-qubit gates are available. Super-controlled gate is a kind of special gates that three applications can construct any two-qubit gate.We give a necessary condition on two-qubit gate construction. When single-qubit gates are available, we want to know whether it is possible to construct the gate U₃ with only one application of the given gates U₁ and U₂. The question is not resolved but we presented a condition that should be satisfied when the construction is possible.We also discuss the gate construction in a solid state quantum computation protocol which uses controllable Heisenberg exchange interactions and a global magnetic field. Compared to the known protocols, our protocol has some advantages.2. Entanglement generation power of quantum gates.Entanglement may be increased when a quantum gate applies to an initial state. When single-qubit gates are available, we calculate the achievable maximal entanglement when a two-qubit gate is applied to an initial state with entanglement C₀. Our result can be used to discuss how to maximize the difference between the finial and the initial entanglement and how to generated entanglement optimally using a given two-qubit interaction.3. Gate construction using quantum entanglement.Entanglement can be used to construct non-local and remote quantum gates. Using entanglement to construct non-local quantum gate is crucial for distributed quantum computation. Implementation of controlled rotations using entanglement is considered. We use a general two-qubit pure state to implement controlled rotations on two separate qubits. The probability of the success we achieve is higher than the known results in some parameter regions. Qualitatively the probability of the success we achieve is an increasing function of the entanglement and a decreasing function of the controlled-rotation angle, and the probability of success will approach the unit when the entangled state approaches a Bell state or the controlled-rotation angle approaches zero.4. Faithful remote state preparation using entanglement.Remote state preparation can be regarded as a kind of state transmission. We present many ensembles of states that can be remotely prepared by using minimum classical bits from Alice to Bob and their previously shared entangled state and prove that we have found all the ensembles in two-dimensional case. Furthermore we show that any pure quantum state can be remotely and faithfully prepared by using finite classical bits from Alice to Bob and their previously shared nonmaximally entangled state though no faithful quantum teleportation protocols can be achieved by using a nonmaximally entangled state.