Quantum Information

How should we think about quantum computing? The usual answer to this question is based on ideas inspired by computer science, such as qubits, quantum gates, and quantum circuits. In this talk I will explain an alternate geometric approach to quantum computation. In the geometric approach, an optimal quantum computation corresponds to "free falling" along the minimal geodesics of a certain Riemannian manifold. This reformulation opens up the possibility of using tools from geometry to understand the strengths and weaknesses of quantum computation, and perhaps to understand what makes certain physical operations difficult (or easy) to synthesize.

In order to predict the future state of a quantum system, we generally do not need to know the past state of the entire universe, but only the state of a finite neighborhood of the system. This locality is best expressed as a restriction on how information "flows" between systems. In this talk I will describe some recent work, inspired by quantum cellular automata, about the information strucutre of local quantum dynamics. Issues to be discussed include the definition of "locality", some characterization theorems, connections between classical and quantum locality for reversible maps, the relation between local and global dynamics, and the dissection of CNOT.

In this talk, I will outline a quantum generalization of causal networks that are used to analyze complex probabilistic inference problems involving large numbers of correlated random variables. I will review the framework of classical causal networks and the graph theoretical constructions that are abstracted from them, including entailed conditional independence, d-separation and Markov equivalence. I will show how to generalize the definition of causal networks to the quantum case, such that the same graph theoretic constructions apply, and give an explicit representation of the states supported on the graph as the Gibbs states of certain classes of Hamiltonians.

Adiabatic Quantum Computation is not only a possibly more robust alternative to standard quantum computation. Since it considers a continuous-time evolution of the system, it also provides a natural bridge towards studying the dynamics of interacting many-particle quantum systems, quantum phase transitions and other issues in fundamental physics. After a brief review of adiabatic quantum computation, I will show our recent results on the dynamics of entanglement and fidelity for the search and Deutsch algorithms including several variations and optimization. I will show how these studies led to suggesting an alternative definition of entanglement and compare the results, and discuss possible implications for considering entanglement a resource. I will conclude with an outlook on further applications and extensions of adiabatic quantum computation.

This talk is concerned with the noise-insensitive transmission of quantum information. For this purpose, the sender incorporates redundancy by mapping a given initial quantum state to a messenger state on a larger-dimensional Hilbert space. This encoding scheme allows the receiver to recover part of the initial information if the messenger system is corrupted by interaction with its environment. Our noise model for the transmission leaves a part of the quantum information unchanged, that is, we assume the presence of a noiseless subsystem or of a decoherence-free subspace. We address the case when the noiseless component cannot contain all the quantum information to be transmitted, and investigate how to best spread the information in a quantum state across the noise-susceptible components. (Joint work with David Kribs and Vern Paulsen.)

We will look at the axioms of quantum mechanics as expressed, for example, in the book by M. A. Nielsen and I. L. Chung ("Quantum Computation and Quantum Information"). We then take a critical look at these axioms, raising several questions as we go. In particular, we will look at the possible informational completeness property of the family of operators that we measure. We will propose physical solutions based on the results of quantum mechanics on phase space and the measurement of quantum particles by quantum mechanical means. We illustrate this with both momentum-position measurements and spin measurements.

Inspired by the notion that the differences between quantum theory and classical physics are best expressed in terms of information theory, Hardy (2001) and Clifton, Bub, and Halvorson (2003) have constructed frameworks general enough to embrace both quantum and classical physics, within which one can invoke principles that distinguish the classical from the quantum. Independently of any view that quantum theory is essentially about quantum information, such frameworks provide a useful tool for exploring the differences between classical and quantum physics, and the relations between the various properties of quantum mechanics that distinguish it from the classical. In particular, we can ask: on which features of quantum physics do our familiar possibility/impossibility theorems depend? It turns out that it is possible to extend the no-cloning theorem and other results, such as the Holevo bound on acquisition of information by a single measurement, beyond the quantum setting.

Entanglement entropy is currently of interest in several areas in physics, such as condensed matter, field theory, and quantum information. One of the most interesting properties of the entanglement entropy is its scaling behavior, especially close to phase transitions. It was believed that for dimensions higher than 1 the entropy scales like surface area of the subsystem. We will describe a recent result for free fermions at zero temperature, where the entropy in fact scales faster. The latter problem will be related to a mathematical conjecture due to H. Widom (1982). This is a joint work with I. Klich.

Hamiltonian oracles are the continuum limit of the standard unitary quantum oracles. In addition to being a potentially useful tool in the study of standard oracles, Hamiltonian oracles naturally introduce the concept of fractional queries and are amenable to study using techniques of differential equations and geometry. As an example of these ideas we shall examine the Hamiltonian oracle corresponding to the problem of oracle interrogation. This talk is intended for all those who wish to apply their knowledge of differential geometry without the risk of creating an event horizon.

I will investigate the creation and detection of multipartite entangled states in systems of ultracold neutral atoms trapped in an optical lattice. These setups are scalable, highly versatile and controllable at the quantum level. Thus they provide an ideal test bed for studying the properties of multipartite entangled states. I will first present methods exploiting incoherent dynamics for initializing an atomic quantum register. The immersion of an optical lattice in a Bose-Einstein condensate leads to spontaneous emission of phonons. This process can be used for irreversibly loading and cooling atoms within the lowest Bloch band of the lattice. I will describe loading and cooling schemes based on this mechanism and compare them to conventional loading schemes. I will then show how coherent dynamics in a very strongly interacting 1D optical lattice setup can be used for the efficient generation of arbitrary graph states in the atomic quantum register. This system can be mapped onto an XY spin chain which itself is equivalent to a system of non-interacting fermions. By exploiting the anticommutation relations between these fictitious fermions I will discuss how any graph state can be realized in an efficient and robust way. In the final part of my talk I will present a practical method for detecting and characterizing multipartite entangled states in atomic quantum registers. This scheme is based on measuring violations of entropic inequalities using simple quantum networks involving only two copies of the quantum state under consideration. I will investigate the performance of this method under realistic conditions taking into account the most common sources of experimental errors.