We generalize Bell's inequalities to biparty systems with continuous quantum variables. This is achieved by introducing the Bell operator in perfect analogy to the usual spin- 1/2 systems. It is then demonstrated that two-mode squeezed vacuum states display quantum nonlocality by using the generalized Bell operator. In particular, the original Einstein-Podolsky-Rosen states, which are the limiting case of the two-mode squeezed vacuum states, can maximally violate Bell's inequality due to Clauser, Horne, Shimony, and Holt. The experimental aspect of our scheme is briefly considered.

The hybrid entangled states generated, e.g., in a trapped-ion or atom-cavity system, have exactly one ebit of entanglement, but are not maximally entangled. We demonstrate this by showing that they violate, but in general do not maximally violate, Bell's inequality due to Clauser, Horne, Shimony and Holt. These states are interesting in that they exhibit the entanglement between two distinct degrees of freedom (one is discrete and another is continuous). We then demonstrate these entangled states as a valuable resource in quantum information processing including quantum teleportation, entanglement swapping and quantum computation with "parity qubits". Our work establishes an interesting link between quantum information protocols of discrete and continuous variables. [19][20][21][22][23]. In particular, violations of the Bell-type inequalities by the "regularized" EPR states produced in a pulsed nondegenerate optical parametric amplifier was experimentally observed by using homodyning with weak coherent fields and photon counting [22].In connection with the applicability of quantum superposition principle on a macroscopic scale, Schrödinger [5] described a gedanken experiment, in which a cat is placed in a quantum superposition of being dead and alive while entangled with a single radioactive atom. The mesoscopic equivalents of the Schrödinger-cat states [called hybrid entangled states (HES) in the subsequent discussion] have been experimentally realized for a 9 Be + ion in traps [24] and atoms in high-Q cavities [25]. Particularly, in the trapped ion experiment [24], the HES were generated by entangling ion's internal states (|↑, ↓ in the terminology of spin-1/2 particles) with discrete spectrum and motional states with continuous spectrum:where the motional states |x 1 and |x 2 of the ion are two distinguishable wave packets of a harmonic oscillator and thus denote quantum states with continuous variables. For the atom-cavity system, the entanglement of the type (1) occurs between a microwave cavity field and an atom [25]. These HES are of great theoretical interest in addressing some fundamental issues, such as decoherence and the quantum/classical boundary [24][25][26]. The trapped-ion system is a strong candidate for quantum computation [1,27,28]. In this paper we demonstrate the HES as a valuable resource in quantum information processing, building an interesting link between quantum information protocols of discrete and continuous variables. Quantum nonlocality of the HES is also analyzed by using the recently developed formulation [23]. For usual two-qubit (qubit-1 and qubit-2) systems, one can introduce the following Bell-basis spanned by the two-qubit states Ψ ± 1,2 = 1 √ 2 (|↑ 1 |↓ 2 ± |↓ 1 |↑ 2 ) ,The pairs of qubits are maximally entangled when they are in these states. An analogous Bell-basis spanned by four HES ψ ± 1,2 (z) = 1 √ 2 (|↑ 1 |z o2 ± |↓ 1 |z e2 ) , 1

Unknown quantum pure states of arbitrary but definite s-level of a particle can be transferred onto a group of remote two-level particles through two-level EPRs as many as the number of those particles in this group. We construct such a kind of teleportation, the realization of which need a nonlocal unitary transformation to the quantum system that is made up of the s-level particle and all the two-level particles at one end of the EPRs, and measurements to all the single particles in this system. The unitary transformation to more than two particles is also written into the product form of two-body unitary transformations.Quantum mechanics offers us the capabilities of transferring information different from the classical case, either for computation or communication. Bennett et.al.[1], developed a quantum method of teleportation, through which, an unknown quantum pure state of a spin-1 2 particle (we call it 'qubit' [2,3] ) is teleported from the sender 'Alice' at the sending terminal onto the qubit at the receiving terminal where the receiver 'Bob' need to perform a unitary transformation on his qubit. At first it is necessary to prepare two spin-1 2 particles in an Einstein-Podolsky-Rosen (EPR) entangled state [4] or so-called a Bell state and send them to the two different places to establish a quantum channel between Alice and Bob. The second step is that Alice performs a Bell operator measurement [5] to the quantum system involving her share of the two entangled particles together with the particle at an unknown state to be transferred. Then through classical channels, for example, by broadcasting, Alice needs to let Bob know which one she gets of the four possible outcomes of the Bell operator measurement. After Bob performs on his share of the two formerly entangled particles one of four unitary transformations determined by those outcomes, this particle

Making use of the transformation relation between the ordinary form and the antinormal product form of boson exponential quadratic operators ͑BEQO's͒, we present an effective method which can be conveniently used to calculate arbitrary matrix elements of BEQO's. By this method, some important matrix elements have been calculated analytically. As a preliminary application, we obtain the exact solution of the density matrix and partition function for the general boson quadratic Hamiltonian without any information for the energy level. As a natural extension, we also obtain the partition function for a general fermion quadratic system.

We consider the transmission of classical information over a quantum channel by two senders. The channel capacity region is shown to be a convex hull bound by the Von Neumann entropy and the conditional Von Neumann entropy. We discuss some possible applications of our result. We also show that our scheme allows a reasonable distribution of channel capacity over two senders.

No abstract

Based on the generalized linear quantum transformation theory, we present a normal ordering evolution operator for onedimensional quant urn oscillator with time-dependent frequency and mass, then give the exact expression of the evolution matrix elements, wave function and expectation value of arbitrary observable.

By means of the transformation relation between the ordinary form of boson exponential quadratic operators (BEQO) and its anti-normal product form, we present an effective method to conveniently calculate arbitrary matrix elements of BEQO. By this method, many important matrix elements can be calculated analytically. As a direct application, we obtain the exact solutions of the density matrix and partition function for general boson quadratic Hamiltonian without any information about the energy level.