Using the Hadamard and related transforms for simplifying the spectrum of the quantum baker's map

We rationalize the somewhat surprising efficacy of the Hadamard transform in simplifying the eigenstates of the quantum baker's map, a paradigmatic model of quantum chaos. This allows us to construct closely related, but new, transforms that do significantly better, thus nearly solving for many states of the quantum baker's map. These new transforms, which combine the standard Fourier and Hadamard transforms in an interesting manner, are constructed from eigenvectors of the shift permutation operator that are also simultaneous eigenvectors of bit-flip (parity) and possess bit-reversal (time-reversal) symmetry.Comment: Version to appear in J. Phys. A. Added discussions; modified title; corrected minor error

Entanglement transitions in random definite particle states

Entanglement within qubits are studied for the subspace of definite particle states or definite number of up spins. A transition from an algebraic decay of entanglement within two qubits with the total number $N$ of qubits, to an exponential one when the number of particles is increased from two to three is studied in detail. In particular the probability that the concurrence is non-zero is calculated using statistical methods and shown to agree with numerical simulations. Further entanglement within a block of $m$ qubits is studied using the log-negativity measure which indicates that a transition from algebraic to exponential decay occurs when the number of particles exceeds $m$. Several algebraic exponents for the decay of the log-negativity are analytically calculated. The transition is shown to be possibly connected with the changes in the density of states of the reduced density matrix, which has a divergence at the zero eigenvalue when the entanglement decays algebraically.Comment: Substantially added content (now 24 pages, 5 figures) with a discussion of the possible mechanism for the transition. One additional author in this version that is accepted for publication in Phys. Rev.

Classical bifurcations and entanglement in smooth Hamiltonian system

We study entanglement in two coupled quartic oscillators. It is shown that the entanglement, as measured by the von Neumann entropy, increases with the classical chaos parameter for generic chaotic eigenstates. We consider certain isolated periodic orbits whose bifurcation sequence affects a class of quantum eigenstates, called the channel localized states. For these states, the entanglement is a local minima in the vicinity of a pitchfork bifurcation but is a local maxima near a anti-pitchfork bifurcation. We place these results in the context of the close connections that may exist between entanglement measures and conventional measures of localization that have been much studied in quantum chaos and elsewhere. We also point to an interesting near-degeneracy that arises in the spectrum of reduced density matrices of certain states as an interplay of localization and symmetry.Comment: 7 pages, 6 figure

Local Identities Involving Jacobi Elliptic Functions

We derive a number of local identities of arbitrary rank involving Jacobi elliptic functions and use them to obtain several new results. First, we present an alternative, simpler derivation of the cyclic identities discovered by us recently, along with an extension to several new cyclic identities of arbitrary rank. Second, we obtain a generalization to cyclic identities in which successive terms have a multiplicative phase factor exp(2i\pi/s), where s is any integer. Third, we systematize the local identities by deriving four local ``master identities'' analogous to the master identities for the cyclic sums discussed by us previously. Fourth, we point out that many of the local identities can be thought of as exact discretizations of standard nonlinear differential equations satisfied by the Jacobian elliptic functions. Finally, we obtain explicit answers for a number of definite integrals and simpler forms for several indefinite integrals involving Jacobi elliptic functions.Comment: 47 page

Small but versatile: the extraordinary functional and structural diversity of the β-grasp fold

<p>Abstract</p> <p>Background</p> <p>The β-grasp fold (β-GF), prototyped by ubiquitin (UB), has been recruited for a strikingly diverse range of biochemical functions. These functions include providing a scaffold for different enzymatic active sites (e.g. NUDIX phosphohydrolases) and iron-sulfur clusters, RNA-soluble-ligand and co-factor-binding, sulfur transfer, adaptor functions in signaling, assembly of macromolecular complexes and post-translational protein modification. To understand the basis for the functional versatility of this small fold we undertook a comprehensive sequence-structure analysis of the fold and developed a natural classification for its members.</p> <p>Results</p> <p>As a result we were able to define the core distinguishing features of the fold and numerous elaborations, including several previously unrecognized variants. Systematic analysis of all known interactions of the fold showed that its manifold functional abilities arise primarily from the prominent β-sheet, which provides an exposed surface for diverse interactions or additionally, by forming open barrel-like structures. We show that in the β-GF both enzymatic activities and the binding of diverse co-factors (e.g. molybdopterin) have independently evolved on at least three occasions each, and iron-sulfur-cluster-binding on at least two independent occasions. Our analysis identified multiple previously unknown large monophyletic assemblages within the β-GF, including one which unifies versions found in the fasciclin-1 superfamily, the ribosomal protein L25, the phosphoribosyl AMP cyclohydrolase (HisI) and glutamine synthetase. We also uncovered several new groups of β-GF domains including a domain found in bacterial flagellar and fimbrial assembly components, and 5 new UB-like domains in the eukaryotes.</p> <p>Conclusion</p> <p>Evolutionary reconstruction indicates that the β-GF had differentiated into at least 7 distinct lineages by the time of the last universal common ancestor of all extant organisms, encompassing much of the structural diversity observed in extant versions of the fold. The earliest β-GF members were probably involved in RNA metabolism and subsequently radiated into various functional niches. Most of the structural diversification occurred in the prokaryotes, whereas the eukaryotic phase was mainly marked by a specific expansion of the ubiquitin-like β-GF members. The eukaryotic UB superfamily diversified into at least 67 distinct families, of which at least 19–20 families were already present in the eukaryotic common ancestor, including several protein and one lipid conjugated forms. Another key aspect of the eukaryotic phase of evolution of the β-GF was the dramatic increase in domain architectural complexity of proteins related to the expansion of UB-like domains in numerous adaptor roles.</p> <p>Reviewers</p> <p>This article was reviewed by Igor Zhulin, Arcady Mushegian and Frank Eisenhaber.</p

Reconstructing the ubiquitin network - cross-talk with other systems and identification of novel functions

A computational model of the yeast Ubiquitin system highlights interesting biological features including functional interactions between components and interplay with other regulatory mechanisms