Entanglement Properties And Quality Characterization Of Noisy Quantum Operations

In recent years,the entanglement properties of quantum states are playing more and more important roles in the field of quantum information processing.A great deal of theoretical work has been devoted to the study of the properties and manipulation of quantum states and has made some fruitful achievements which have been applied to such fields as quantum computation,quantum measurement,quantum communication and so on.Quantum entanglement has gradually become an indispensable physical resource which often needs to be generated by means of nonlocal quantum operations.So both the study on entanglement properties associated with quantum operations and the study on quantum entangled states are equally important.Entanglement properties of quantum operations have been studied widely and deeply and some analytical descriptions of entanglement properties of quantum operations are given.But these studies are only for ideal unitary quantum operations without involving the influence of environmental noise on the entanglement properties of quantum operations.That is to say,the influence mechanism of environmental noise on quantum operations is not clear.What's more,the quantum system inevitably interacts with its environment,and the implementation quality of quantum operations realized by virtue of the interaction between quantum systems will also be influeced by the environment,so the quality characterization of quantum operation is also a concern of the community.In order to characterize the implementation quality of quantum operations,the concepts of average purity and average fidelity which are calculated by averaging over all possible input states of quantum operations have been defined.However,quantum operation is a relatively independent concept,so the quality of quantum operations should be accurately described without the aid of quantum states.In addition,it is often required to change the entanglement of quantum states using nonlocal quantum operations in quantum information processing,and the entanglement properties of quantum operations may be related to the efficiency of quantum information processing,so it is necessary to study the role of quantum operations with different entanglement properties in quantum information applications,and thus the optimal quantum operation can be found for quantum information processing.In this dissertation,based on the analysis on the existing measures(entangling power and operator entanglement)of entanglement properties of unitary quantum operations,the study on entangling power and operator entanglement is extended to the general noisy nonunitary quantum operation case.Furthermore,a new method to depict the influence of noise on the implementation quality of quantum operation are given,and the optimal matching relations between the entangling power of joint unitary operation and the quantum channel parameter in standard entanglement purification and entanglement concentration are found.Specificly,the main results of this dissertation are as follows:1.A method for calculating the operator entanglement and entangling power of noisy nonunitary operations is proposed.By decomposing the Kraus operators of noisy evolution as the sum of products of Pauli matrices,we derive the analytical expression of the operator entanglement for a general nonunitary operation.The definition of entangling power is extended from the ideal unitary operation case to the nonunitary case via a Kraus operator representation and the analytical expression of the entangling power for a general nonunitary operation is derived.To demonstrate the effectiveness of the above method,we study the properties of operator entanglement and entangling power of nonunitary operations caused by phase damping noise.Our findings imply that the pure phase damping noise has its own operator entanglement and entangling power,which increase exponentially with time and asymptotically approach their respective upper bounds.In addition,when the phase damping noise is added to an ideal operation,such as an ISWAP operation or a controlled-Z operation,it can make the operation's entangling power grow exponentially with the strength of noise,but leave its operator entanglement invariant.In this sense,we can conclude that,for a general operation,operator entanglement is a more intrinsic property than entangling power.2.State-independent(SI)purity and fidelity of a general quantum operation(evolution)are defined.The purity and fidelity of quantum operations are of great importance in characterizing the quality of quantum operations.The currently available definitions of the purity and fidelity of quantum operations are based on the average over all possible input pure quantum states,i.e.they are state-dependent(SD).Here,without resorting to quantum states,we define the state-independent(SI)purity and fidelity of a general quantum operation(evolution)in virtue of new density matrix formalism for quantum operations,which is extended from the quantum state level to quantum operation level.The SI purity and fidelity gain more intrinsic physical properties of quantum operations than state-dependent ones,such as the purity of a one-qubit amplitude damping channel(with damping rate 1)is 1/2,which is in line with the fact that the channel is still a nonunitary operation described by two Kraus operators rather than a unitary one.But the state-dependent Haar average purity is 1 in this case.So the SI purity and fidelity proposed here can help the experimentalists to exactly quantify the implementation quality of an operation.As a byproduct,a new measure of the operator entanglement is proposed for a quantum evolution(unitary or nonunitary)in terms of the linear entropy of its density matrix on the orthonormal operator bases(OOBs)in Hilbert-Schmidt space.3.The optimal matching relations between the entangling power of the joint unitary operation and the non-maximal entangled channels are found in entanglement concentration and purification.The bilateral controlled NOT(CNOT)operation plays a key role in standard entanglement purification process,but the CNOT operation may not be the optimal joint operation in the sense that the output entanglement is maximized.Here,the CNOT operations in both the Schmidt-projection based entanglement concentration and the entanglement purification schemes are replaced with a general joint unitary operation,and the optimal matching relations between the entangling power of the joint unitary operation and the non-maximal entangled channel are found for optimizing the entanglement increment or the output entanglement.The result is somewhat counter-intuitive for entanglement concentration.The output entanglement is maximized when the entangling power of the joint unitary operation and the quantum channel satisfy certain relation.There exist a variety of joint operations with non-maximal entangling power that can induce a maximal output entanglement,which will greatly broaden the set of the potential joint operations in entanglement concentration.In addition,the entanglement increment in purification process is maximized only by the joint unitary operations(including CNOT)with maximal entangling power.