Unitary 2-designs are random unitaries simulating up to the second order statistical moments of the uniformly distributed random unitaries, often referred to as Haar random unitaries. They are used in a wide variety of theoretical and practical quantum information protocols and also have been used to model the dynamics in complex quantum many-body systems. Here, we show that unitary 2-designs can be approximately implemented by alternately repeating random unitaries diagonal in the Pauli-Z basis and Pauli-X basis. We also provide a converse about the number of repetitions needed to achieve unitary 2-designs. These results imply that the process after l repetitions achieves a Theta(d(-l))-approximate unitary 2-design. Based on the construction, we further provide quantum circuits that efficiently implement approximate unitary 2-designs. Although a more efficient implementation of unitary 2-designs is known, our quantum circuit has its own merit that it is divided into a constant number of commuting parts, which enables us to apply all commuting gates simultaneously and leads to a possible reduction of an actual execution time. We finally interpret the result in terms of the dynamics generated by time-dependent Hamiltonians and provide for the first time a random disordered time-dependent Hamiltonian that generates a unitary 2-design after switching interactions only a few times. Published by AIP Publishing.

%B JOURNAL OF MATHEMATICAL PHYSICS %V 58 %R 10.1063/1.4983266 %0 Journal Article %J Physical Review A %D 2014 %T Unified approach to entanglement criteria using the Cauchy-Schwarz and Hölder inequalities %A Wölk, Sabine %A Huber, Marcus %A Gühne, Otfried %X We present a unified approach to different recent entanglement criteria. Although they were developed in different ways, we show that they are all applications of a more general principle given by the Cauchy-Schwarz inequality. We explain this general principle and show how to use it to derive not only already known entanglement criteria, but also criteria which were unknown so far. We systematically investigate its potential and limitations to detect bipartite and multipartite entanglement. Furthermore, we describe how to apply our generalized entanglement detection scheme to find the measurement directions to verify the entanglement of a given state experimentally. %B Physical Review A %V 90 %8 8/2014 %N 2 %! Phys. Rev. A %R 10.1103/PhysRevA.90.022315 %0 Book %D 2012 %T Ultracold atoms in optical lattices: simulating Quantum Many-Body systems %A Lewenstein Maciej %A Sanpera Anna %A Ahufinger Verònica %I Oxford University Press %@ 978-0199573127 %0 Journal Article %J New Journal of Physics %D 2007 %T Ultracold atomic Bose and Fermi spinor gases in optical lattices %A Eckert, Kai %A Zawitkowski, L. %A Leskinen, M. J. %A Sanpera, Anna %A Lewenstein, Maciej %K optical lattices %B New Journal of Physics %V 9 %P 133 %G eng %U http://iopscience.iop.org/1367-2630/9/5/133/pdf/1367-2630\_9\_5\_133.pdf %0 Journal Article %J Advances in Physics %D 2007 %T Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond %A Lewenstein, Maciej %A Sanpera, Anna %A Ahufinger, Veronica %A Damski, B. %A Sen(De), Aditi %A Sen, Ujjwal %K optical lattices %B Advances in Physics %V 56 %P 243 %G eng %U http://www.informaworld.com/smpp/ftinterface\~content=a778175433\~fulltext=713240930 %0 Journal Article %J Foundations of Physics %D 2006 %T Unitarity as Preservation of Entropy and Entanglement in Quantum Systems %A Hulpke, Florian %A Poulsen, Uffe %A Sanpera, Anna %A Sen(De), Aditi %A Sen, Ujjwal %A Lewenstein, Maciej %X Abstract The logical structure of Quantum Mechanics (QM) and its relation to other fundamental principles of Nature has been for decades a subject of intensive research. In particular, the question whether the dynamical axiom of QM can be derived from other principles has been often considered. In this contribution, we show that unitary evolutions arise as a consequences of demanding preservation of entropy in the evolution of a single pure quantum system, and preservation of entanglement in the evolution of composite quantum systems. 6 %B Foundations of Physics %V 36 %P 477–499 %8 04/2006 %G eng %U http://www.springerlink.com/content/tjg6r668u462848r/fulltext.pdf %R 10.1007/s10701-005-9035-7