Open Systems & Information Dynamics

George Sudarshan was an eclectic thinker, and one of the most profound physicists of the last century. We review here a small part of his oeuvre, focusing on his pioneering contributions to the quantum Zeno effect, quantum channels (the Kraus–Sudarshan representation) and quantum semigroups (the Gorini–Kossakowski–Lindblad–Sudarshan equation). These topics are of crucial importance in the booming field of quantum mechanics and applications.

The role of mixed states in topological quantum matter is less known than that of pure quantum states. Generalisations of topological phases appearing in pure states have received attention in the literature only quite recently. In particular, it is still unclear whether the generalisation of the Aharonov–Anandan phase for mixed states due to Uhlmann plays any physical role in the behaviour of the quantum systems. We analyse, from a general viewpoint, topological phases of mixed states and the robustness of their invariance. In particular, we analyse the role of these phases in the behaviour of systems with periodic symmetry and their evolution under the influence of an environment preserving its crystalline symmetries.

These are my recollections of working with George Sudarshan from 2002 to 2008 when I was a PhD student in his group. During these years I learnt a lot of physics and also was a witness to some remarkable occurrences.

In quantum systems theory one of the fundamental problems boils down to: given an initial state, which final states can be reached by the dynamic system in question. Here we consider infinite-dimensional open quantum dynamical systems following a unital Kossakowski–Lindblad master equation extended by controls. More precisely, their time evolution shall be governed by an inevitable potentially unbounded Hamiltonian drift term H₀, finitely many bounded control Hamiltonians H_j allowing for (at least) piecewise constant control amplitudes $u_j\left(t\right)\in ℝ$ plus a bang-bang (i.e., on-off) switchable noise term Г_V in Kossakowski–Lindblad form. Generalizing standard majorization results from finite to infinite dimensions, we show that such bilinear quantum control systems allow to approximately reach any target state majorized by the initial one as up to now it only has been known in finite dimensional analogues. The proof of the result is currently limited to the bounded control Hamiltonians H_j and for noise terms Г_V with compact normal V.

We study the nonlinear quantum master equation describing a laser under the mean field approximation. The quantum system is formed by a single mode optical cavity and two level atoms, which interact with reservoirs. Namely, we establish the existence and uniqueness of the regular solution to the nonlinear operator equation under consideration, as well as we get a probabilistic representation for this solution in terms of a mean field stochastic Schrödinger equation. To this end, we find a regular solution for the nonautonomous linear quantum master equation in Gorini–Kossakowski–Sudarshan–Lindblad form, and we prove the uniqueness of the solution to the nonautonomous linear adjoint quantum master equation in Gorini–Kossakowski–Sudarshan–Lindblad form. Moreover, we obtain rigorously the Maxwell–Bloch equations from the mean field laser equation.

The superposition of pure quantum states explicitly expressed in terms of a nonlinear addition rule of state density operators is reviewed. The probability representation of density matrices of qudit states is used to formulate the interference of the states as a combination of the probability distributions describing pure states. The formalism of quantizer–dequantizer operators is developed. Examples of spin-1/2 states and f-oscillator systems are considered.