### Testing Nonlocality for Continuous Variables

Nonlocality, one of the most astonishing properties of quantum mechanics, has been a renewed hot topic since the advent of the Bell inequality in the 1960s. A great number of experimental tests have been made for its demonstration, all to date for discrete variables, like atomic spins or photon polarizations. We recently proposed how to demonstrate nonlocality for continuous variables, precisely speaking, for the quadrature amplitudes of a light field, which correspond to the position and the momentum of a massive particle. The proposed method is based on conditional homodyne detection and may be currently accessible in experiments. [H. Nha and H. J. Carmichael, Physical Review Letters 93, 020401 (2004)]

### Entanglement in open quantum systems

In most cases, quantum systems are inevitably coupled to environment, such that the state of the system becomes mixed. It is thus very important to characterize or quantify entanglement in the case of mixed states for many applications, especially in the quantum information field. Some physical properties of mixed states, including entanglement, are, however, intrinsically ambiguous, due to the arbitrariness in decomposing a given mixed state into a mixture of pure states. Despite this ambiguity, researchers in the quantum information field tried to set up a universal measure of entanglement for mixed states. The entanglement of formation, for example, among several reasonable measures proposed so far, regards the minimum average value of the von Neumann entropy over all possible decompositions as the degree of entanglement for the given mixed state.

In a recent work we suggested a new perspective in which the multiplicity of mixed-state decompositions, and that of the corresponding degrees of entanglement, can be understood in view of the complementary principle. In many cases the mixedness of a quantum system arises from a well-defined dynamics of the system-environment interaction. One can then unravel it in the context of a quantum measurement made on the environment. We illustrated, in a cavity QED example, how such an unraveling can be performed on the basis of the quantum trajectory theory, and particularly, what such an approach can tell us to understand entanglement for a mixed state. [H. Nha and H. J. Carmichael, Physical Review Letters 93, 120408 (2004)]

### Decoherence of a two-state atom driven by coherent light

Some quantum algorithms and quantum logic gates have been experimentally realized; for example, in an NMR system and in trapped ion systems. The use of laser fields to manipulate two-state atomic transitions is crucial to these systems, since such manipulations realize the one- and two-qubit operations that form the basic building blocks of a quantum computation. When modeling these manipulations, the laser fields are mostly taken to be classical; thus it is implicitly assumed that the interaction of the atom with the laser field contributes nothing to its decoherence. At a fundamental level, however, the laser should be treated as a quantum field, in which case its interaction with the atom would generally lead to entanglement of the two. It is natural then to ask whether the resolution of this entanglement (tracing out the laser field) leads to additional decoherence, i.e., in addition to the decoherence due to spontaneous emission. This issue has been discussed for the past years by some authors, with some disagreement in the conclusion. We recently addressed this problem based on the cascaded quantum systems approach, and showed that there is no additional decoherence by the quantum nature of the laser light as long as the laser field is in a coherent state. [H. Nha and H. J. Carmichael, Physical Review A 71, 013805 (2005)]

### Distinguishing two Gaussian states by homodyne detection

Quantum fidelity, as a measure of the closeness of two quantum states, is operationally equivalent to the minimal overlap of the probability distributions of the two states over all possible POVMs; the POVM realizing the minimum is optimal. We have considered the ability of homodyne detection to distinguish two single-mode Gaussian states, and investigate to what extent it is optimal in this information-theoretic sense. Specifically, we completely identify the conditions under which homodyne detection makes an optimal distinction between two single-mode Gaussian states of the same mean. [H. Nha and H. J. Carmichael, Physical Review A 71, 032336 (2005)]