How useful is a quantum dynamical operation for quantum information processing? Motivated by this question, we investigate several strength measures quantifying the resources intrinsic to a quantum operation. We develop a general theory of such strength measures, based on axiomatic considerations independent of state-based resources. The power of this theory is demonstrated with applications to quantum communication complexity, quantum computational complexity, and entanglement generation by unitary operations.

We investigate the entanglement characteristics of two general bimodal Bose-Einstein condensates -a pair of tunnel-coupled Bose-Einstein condensates and the atom-molecule Bose-Einstein condensate. We argue that the entanglement is only physically meaningful if the system is viewed as a bipartite system, where the subsystems are the two modes. The indistinguishibility of the particles in the condensate means that the atomic constituents are physically inaccessible and thus the degree of entanglement between individual particles, unlike the entanglement between the modes, is not experimentally relevant so long as the particles remain in the condensed state. We calculate the entanglement between the two modes for the exact ground state of the two bimodal condensates and consider the dynamics of the entanglement in the tunnel-coupled case.

How does the classical phase-space structure for a composite system relate to the entanglement characteristics of the corresponding quantum system? We demonstrate how the entanglement in nonlinear bipartite systems can be associated with a fixed-point bifurcation in the classical dynamics. Using the example of coupled giant spins we show that when a fixed point undergoes a supercritical pitchfork bifurcation, the corresponding quantum state-the ground state-achieves its maximum amount of entanglement near the critical point. We conjecture that this will be a generic feature of systems whose classical limit exhibits such a bifurcation.

We compare and contrast the entanglement in the ground state of two Jahn-Teller models. The E ␤ system models the coupling of a two-level electronic system, or qubit, to a single-oscillator mode, while the E models the qubit coupled to two independent, degenerate oscillator modes. In the absence of a transverse magnetic field applied to the qubit, both systems exhibit a degenerate ground state. Whereas there always exists a completely separable ground state in the E ␤ system, the ground states of the E model always exhibit entanglement. For the E ␤ case we aim to clarify results from previous work, alluding to a link between the ground-state entanglement characteristics and a bifurcation of a fixed point in the classical analog. In the E case we make use of an ansatz for the ground state. We compare this ansatz to exact numerical calculations and use it to investigate how the entanglement is shared between the three system degrees of freedom.

The monogamous nature of entanglement has been illustrated by the derivation of entanglement-sharing inequalities-bounds on the amount of entanglement that can be shared among the various parts of a multipartite system. Motivated by recent studies of decoherence, we demonstrate an interesting manifestation of this phenomena that arises in system-environment models where there exists interactions between the modes or subsystems of the environment. We investigate this phenomenon in the spin-bath environment, constructing an entanglement-sharing inequality bounding the entanglement between a central spin and the environment in terms of the pairwise entanglement between individual bath spins. The relation of this result to decoherence will be illustrated using simplified system-bath models of decoherence. While entanglement is argued to be the distinguishing feature of quantum computers, responsible for their power ͓1͔, it is also the source of one of the major obstacles in their construction. Decoherence, the process by which a quantum superposition state decays into a classical, statistical mixture of states, is caused by entangling interactions between the system and its environment ͓2͔. Somewhat paradoxically, the quantum entanglement between a system and its environment induces classicality in the system. While it is still a contentious topic as to whether quantum computation will be possible in the face of decoherence, Zurek ͓3͔ has demonstrated that decoherence is necessary to facilitate the measurement of a quantum system. Understanding decoherence lies at the heart of measurement, quantum information processing, and, more fundamentally, the transition from the quantum to the classical world.The road to studying decoherence by explicitly modeling system-environment interactions has led to simple models of the quantum environment. Environments can be modeled as either baths of harmonic oscillators ͓4͔ or spins ͑with spin 1 2 ͒ argued to represent distinct types of environmental modes ͓5͔. The simplest system-environment models consist of a central spin ͑or qubit͒ coupled to the environment-i.e., the spin-boson model ͓4͔-which has applications to the decoherence of qubits for quantum information processing.Decoherence of a spin-1 2 particle at low temperatures may be conveniently modeled by the "central spin" model, which couples a central spin- where H S and H B are the internal Hamiltonians of the central spin and spin bath, respectively, and H SB is the coupling term. Denote the state of the system-environment at time t by SB ͑t͒. Initially at t = 0 we take the central spin S to be in a pure state, uncorrelated with the bath. That is,for some initial state of the bath B ͑0͒. Typically B ͑0͒ is taken to be a thermal state of the Hamiltonian H B or, at low temperatures, the ground state. As the system evolves under H the central spin becomes coupled to the bath, and its reduced density matrix S ͑t͒ at later times is no longer pure. The central spin is said to have decohered, and the amount of decoherence is t...

What is the computational power of a quantum computer? We show that determining the output of a quantum computation is equivalent to counting the number of solutions to an easily computed set of polynomials defined over the finite field Z_2. This connection allows simple proofs to be given for two known relationships between quantum and classical complexity classes, namely BQP/P/\#P and BQP/PP.

We derive and analyze the Born-Markov master equation for a quantum harmonic oscillator interacting with a bath of independent two-level systems. This hitherto virtually unexplored model plays a fundamental role as one of the four "canonical" system-environment models for decoherence and dissipation. To investigate the influence of further couplings of the environmental spins to a dissipative bath, we also derive the master equation for a harmonic oscillator interacting with a single spin coupled to a bosonic bath. Our models are experimentally motivated by quantum-electromechanical systems and micron-scale ion traps. Decoherence and dissipation rates are found to exhibit temperature dependencies significantly different from those in quantum Brownian motion. In particular, the systematic dissipation rate for the central oscillator decreases with increasing temperature and goes to zero at zero temperature, but there also exists a temperature-independent momentum-diffusion ͑heating͒ rate.

The physics of quantum walks on graphs is formulated in Hamiltonian language, both for simple quantum walks and for composite walks, where extra discrete degrees of freedom live at each node of the graph. It is shown how to map between quantum walk Hamiltonians and Hamiltonians for qubit systems and quantum circuits; this is done for both a single-and multi-excitation coding, and for more general mappings. Specific examples of spin chains, as well as static and dynamic systems of qubits, are mapped to quantum walks, and walks on hyperlattices and hypercubes are mapped to various gate systems. We also show how to map a quantum circuit performing the quantum Fourier transform, the key element of Shor's algorithm, to a quantum walk system doing the same. The results herein are an essential preliminary to a Hamiltonian formulation of quantum walks in which coupling to a dynamic quantum environment is included.