Quantum Zeno effect as a topological phase transition in full counting statistics and spin noise spectroscopy

When the interaction of a quantum system with a detector is changing from weak to strong coupling limits, the system experiences a transition from the regime with quantum mechanical coherent oscillations to the regime with a frozen dynamics. In addition to this quantum Zeno transition, we show that the full counting statistics of detector signal events experiences a topological phase transition at the boundary between two phases at intermediate coupling of a quantum system to the detector. We demonstrate that this transition belongs to the class of topological phase transitions that can be classified by elements of the braid group. We predict that this transition can be explored experimentally by means of the optical spin noise spectroscopy.Comment: 5 pages, 2 figure

Topologically protected Grover's oracle for the partition problem

The Number Partitioning Problem (NPP) is one of the NP-complete computational problems. Its definite exact solution generally requires a check of all $N$ solution candidates, which is exponentially large. Here we describe a path to the fast solution of this problem in $\sqrt{N}$ quasi-adiabatic quantum annealing steps. We argue that the errors due to the finite duration of the quantum annealing can be suppressed if the annealing time scales with $N$ only logarithmically. Moreover, our adiabatic oracle is topologically protected, in the sense that it is robust against small uncertainty and slow time-dependence of the physical parameters or the choice of the annealing protocol.Comment: v2: final version; to appear in Physical Review

Randomized channel-state duality

Channel-state duality is a central result in quantum information science. It refers to the correspondence between a dynamical process (quantum channel) and a static quantum state in an enlarged Hilbert space. Since the corresponding dual state is generally mixed, it is described by a Hermitian matrix. In this article, we present a randomized channel-state duality. In other words, a quantum channel is represented by a collection of $N$ pure quantum states that are produced from a random source. The accuracy of this randomized duality relation is given by $1/N$, with regard to an appropriate distance measure. For large systems, $N$ is much smaller than the dimension of the exact dual matrix of the quantum channel. This provides a highly accurate low-rank approximation of any quantum channel, and, as a consequence of the duality relation, an efficient data compression scheme for mixed quantum states. We demonstrate these two immediate applications of the randomized channel-state duality with a chaotic $1$-dimensional spin system