§         Dynamical maps and master equations for open quantum systems with entanglement: Unitary evolution of a bipartite system that is initially in an entangled state induces, in general, positive and not completely positive maps on the density matrix of each of the subsystems. The accepted wisdom in the theory of open quantum systems is that the reduced dynamics must be completely positive (CP). Not completely positive (NCP) evolution has been studied, albeit briefly, in the past but in my view it has not received the attention it deserves. A simple example of two entangled qubits evolving unitarily according to a fixed Hamiltonian that I studied shows that the reduced dynamics is NCP. I have also developed a parameterized form for NCP maps [1]. The maps acting on certain states can lead to unphysical states with negative density matrices. I find that in the example I looked at the action of the ncp map on all the states of a single qubit that are consistent with the specification of the initial entangled two qubit state is such that it is positivity preserving. This suggests that a coherent interpretation of ncp maps as physical evolution of an open quantum system is possible. At present I am working on the general properties of such maps including how the action of ncp maps affect the distribution of information between the sub-systems that are involved. Formulation of master equations of the Kossakowski-Lindblad type corresponding to NCP maps, if possible without using the Markov approximation is the direction of research that I want to pursue in the near future.

§         Measuring quantum systems – the quantum Zeno and anti-Zeno effects: The quantum Zeno (QZE) and anti-Zeno (AZE) effects can be used as much as a tool for understanding the nature of quantum measurements as it can be used as an example of an essentially quantum phenomenon. It encapsulates the effects of classical interventions on the evolution of a quantum system. I have constructed and analyzed a model of indirect pre-measurements on an unstable quantum state and showed under what conditions QZE and AZE can occur if the pre-measurements lead on to full, non-unitary measurements [2]. The relationship between the nature of the pre-measurements and the appearance of the effects was the focus of this study. Pre-measurements represent the first and simplest step in modeling a realistic measurement but they are unitary processes resulting in no irreversible transfer of information between the measured and measuring systems. Replacing the pre-measurement in the model by a non-unitary process like the action of a dynamical map on the unstable state is the next step I plan to consider. Another interesting possiblility is to model the measurement as an interaction with a classical system treated as a quantum system with additional super-selection rules.

§         The spin-statistics connection and other manifestations of quantum properties at the classical level: My interest in the spin-statistics connection is directed at non-relativistic proofs of the connection and exploring the limits of applicability of such proofs. The consequences of the spin-statistics connection are implicitly used in our understanding and ability to manipulate many manifestly non-relativistic aggregations of quantum systems starting from solid-state systems to ions in a trap. A clear understanding of the connection without using the language of relativistic quantum field theory could provide further insights into how the consequences of spin and statistics can be used to control such systems in novel ways. For instance, my preliminary investigations suggest that it may be possible to include particles or quasi-particles that obey para-statistics within the scope of a consistent non-relativistic elucidation of the connection between spin and statistics. The result I have at present is a particular way of showing the spin-statistics connection without using relativistic arguments [3]. Extending it para-Bose, para-Fermi and maybe even anyonic excitations is what I am currently working on.

§         Constructing quantum information processing devices using quantum waveguide networks: Along with my co-workers, I have shown that it is possible to generate and use the quantum resources of superposition and entanglement in a network of quantum electron-waveguides even in the presence of errors and reflections at the gate elements in the network [4]. This serves as a proof of principle that a quantum information processor could be implemented using such a network. The expected fidelity of the output of the network can be computed numerically starting from first principles. This provides a direct handle on understanding the nature of superposition of states and entanglement present in the network and the effects of imperfections and external influences on them. Currently we are trying to establish collaboration with an experimental group to try out some of our ideas and verify the results we have obtained. Possible extensions of the idea include devising networks that can implement more complicated quantum algorithms.