Direct Measurement Of Quantum Entanglement

Quantum information is an emerging interdiscipline of information science and quantum mechanics, including quantum commutation, quantum computation and quantum manipulation. Quantum entanglement is a key source in the current quantum information processing. One can even say that, quantum information cannot exit without quantum entanglement.Usually, to fulfill quantum information processing, the quantum channel must be maximally entangled states, or partially entangled states with known degree of entanglement. Consequently, some fundamental problems of quantum entanglement have ignited lots of research interests:how to know whether a quantum state is an entangled state or not, how to quantify the entanglement of a quantum state, and how to directly measure quantum entanglement in an experiment. In this dissertation, we carried out our research work on how to directly measuring quantum entanglement via measuring the concurrence of the entangled states (including two-qubit general pure states and mixed states) in optical and cavity QED systems. The dissertation contains the following several works:1. Based on the definition of the concurrence for a two-qubit state, we derived the analytical expression of the concurrence of some quantum states. A general scheme for directly measuring the concurrence of two-qubit arbitrary pure state was proposed. If there are two copies of the two-qubit pure state, we can directly obtain the concurrence of them based on the probability for two single-state-based measurements on the two rearranged two-qubit pairs from two copies of the state.2. Based on the detection of cavity decay, a direct measurement scheme for two-atom entangled pure state is designed. In this protocol, each Λ-type atom have an excited state and two ground ones, and the cavity mode and the classical laser field are lager detuned from their respective transitions, so the upper level can be decoupled from the evolution. We also select the spontaneous decay rate is far litter than the cavity decay rate. Then, the effect Hamiltonian of the three-level atom-cavity can be described with a two-level atom interacting with a cavity. To directly measure the concurrence of the atomic entangled state, we first let four atoms in two copies of the two-atomic entangled state are trapped in four cavities. Four identical classical laser pulses will be applied on the four atoms simultaneously. If no photons are emitted from the cavities, the state of each atom-cavity system is governed by a non-Hermit interaction Hamiltonian. In this interaction stage, the states of the atoms will be swapped to the cavities. In the detection stage, by single-photon detectors, we detect continuously the photons leaking from the cavities for a finite time interval. We found that it is possible to encode the concurrence of two-atom entangled states in the photon detection results on the cavity decays.3. The above protocol can be used to directly measure the concurrence of the two-atomic Collins-Gisin mixed state. Comparing the interaction time between the cavities and the atoms, the disentanglement time for an atomic entangled state, the time for the entanglement transfer, the effective cavity decay time, and the detecting time, we found that the time required to complete the protocol can be satisfied. Furthermore, the atomic state is used as stationary qubit, photonic state as flying qubit, which is more feasible and more practical because of the inclusion of cavity decay into consideration. Thus, we believe that this scheme may be implementable within the current technology.4. We proposed a protocol for directly measuring entanglement of the two-photon polarization general pure state via Bell-state analyzer. The concurrence of the state can be obtained by singlet-state measurement on the two copies of the state to be measured. The singlet state can be fully discriminated from the other three Bell states by a modified Bell-state analyzer, which can be constructed with half-wave plates, polarization beam splitters and single-photon detectors.5. A protocol for directly measuring the concurrence of a two-photon polarization entangled general pure state was presented, where the prior quantum state tomography was not needed. By parity-check measurements and simple operations on two copies of the two-photon polarization general pure state, the concurrence is encoded in the total probability of picking up the odd parity states from the signal states when they passed through three parity-check gates. This parity-check measurement makes use of highly efficient homodyne measurement, polarization beam splitters, half wave plates and cross-Kerr nonlinearity, and our protocol could be feasible in the near future with the help of the weak cross-Kerr nonlinearity.6. We present a protocol for directly measuring the concurrence of two-photon polarization-entangled pure state via polarization-independent beam splitter and single-photon detectors. In each detection round, two signal photon pairs in the same polarization-entangled states are needed. Let one signal photon of each pairs pass through a polarization-independent beam splitter simultaneously. We then detect the photon number in the two output modes of the beam splitter with single-photon detectors. The concurrence of the photon pairs will be encoded in the total probability of picking up the balanced states. In this protocol, we only detect two signal photons of four photons via a general beam splitter with single-photon detectors. Compared with the previous schemes, it is simpler and more feasible within the current experimental technology.7. With the help of the weak cross-Kerr nonlinearity and the polarization-independent beam splitter, a scheme for direct measurement of the concurrence of a two-photon polarization mixed state (Collins-Gisin state) was proposed, where two copies of the mixed state were also needed. Let two rearranged signal photon pairs pass through two polarization-independent beam splitter simultaneously. The concurrence of the Collins-Gisin mixed state will be encoded in the total probability of picking up the balanced states in two output modes of each beam splitters. The detection of balanced states can be implemented with weak cross-Kerr nonlinearities and homodyne measurements. Furthermore, this scheme can be used to directly measuring the concurrence of two-photon polarization-entangled pure state with high efficiency and high fidelity. Based on the discussions about the case where the two copies of the state are not perfect, this scheme is insensitive to the copy error of the state to be measured.