Tau-Function Theory of Chaotic Quantum Transport with β = 1, 2, 4

We study the cumulants and their generating functions of the probability distributions of the conductance, shot noise and Wigner delay time in ballistic quantum dots. Our approach is based on the integrable theory of certain matrix integrals and applies to all the symmetry classes ${\beta \in \{1, 2, 4\}}$ of Random Matrix Theory. We compute the weak localization corrections to the mixed cumulants of the conductance and shot noise for β = 1, 4, thus proving a number of conjectures of Khoruzhenko et al. (in Phys Rev B 80:(12)125301, 2009). We derive differential equations that characterize the cumulant generating functions for all ${\beta \in \{1, 2, 4 \} }$ . Furthermore, when β = 2 we show that the cumulant generating function of the Wigner delay time can be expressed in terms of the Painlevé III′ transcendant. This allows us to study properties of the cumulants of the Wigner delay time in the asymptotic limit ${n \to \infty}$ . Finally, for all the symmetry classes and for any number of open channels, we derive a set of recurrence relations that are very efficient for computing cumulants at all orders.     

Self-adjoint Operators as Functions I

Observables of a quantum system, described by self-adjoint operators in a von Neumann algebra or affiliated with it in the unbounded case, form a conditionally complete lattice when equipped with the spectral order. Using this order-theoretic structure, we develop a new perspective on quantum observables. In this first paper (of two), we show that self-adjoint operators affiliated with a von Neumann algebra ${\mathcal{N}}$ can equivalently be described as certain real-valued functions on the projection lattice ${\mathcal{P}(\mathcal{N}})$ of the algebra, which we call q-observable functions. Bounded self-adjoint operators correspond to q-observable functions with compact image on non-zero projections. These functions, originally defined in a similar form by de Groote (Observables II: quantum observables, 2005), are most naturally seen as adjoints (in the categorical sense) of spectral families. We show how they relate to the daseinisation mapping from the topos approach to quantum theory (Döring and Isham , New Structures for Physics, Springer, Heidelberg, 2011). Moreover, the q-observable functions form a conditionally complete lattice which is shown to be order-isomorphic to the lattice of self-adjoint operators with respect to the spectral order. In a subsequent paper (Döring and Dewitt, 2012, preprint), we will give an interpretation of q-observable functions in terms of quantum probability theory, and using results from the topos approach to quantum theory, we will provide a joint sample space for all quantum observables.     

Weak Multiplicativity for Random Quantum Channels

It is known that random quantum channels exhibit significant violations of multiplicativity of maximum output p-norms for any p > 1. In this work, we show that a weaker variant of multiplicativity nevertheless holds for these channels. For any constant p > 1, given a random quantum channel ${\mathcal{N}}$ (i.e. a channel whose Stinespring representation corresponds to a random subspace S), we show that with high probability the maximum output p-norm of ${\mathcal{N}^{\otimes n}}$ decays exponentially with n. The proof is based on relaxing the maximum output ∞-norm of ${\mathcal{N}}$ to the operator norm of the partial transpose of the projector onto S, then calculating upper bounds on this quantity using ideas from random matrix theory.     

Quantum Mechanical Algorithm for Solving Quadratic Residue Equation

We propose a quantum mechanical algorithm for solving quadratic residue equation z ²=b (mod M) based on Grover quantum search. The quantum algorithm will take O( ?M\sqrt{M} ) steps for finding the solutions to the equation by exploiting the properties of quantum superposition and interference effect, while classical algorithm to the same problem will take O(M) steps. The success probability of the algorithm approaches to unity and the cost of the algorithm mainly depends on the calculations of quadratic residue modulo M and the number of iterations. Furthermore, we show that the algorithm can be used to solve the prime factorization problem, and the computing complexity is O( ?N\sqrt{N} ).     

Recurrence for Discrete Time Unitary Evolutions

We consider quantum dynamical systems specified by a unitary operator U and an initial state vector ${\phi}$ . In each step the unitary is followed by a projective measurement checking whether the system has returned to the initial state. We call the system recurrent if this eventually happens with probability one. We show that recurrence is equivalent to the absence of an absolutely continuous part from the spectral measure of U with respect to ${\phi}$ . We also show that in the recurrent case the expected first return time is an integer or infinite, for which we give a topological interpretation. A key role in our theory is played by the first arrival amplitudes, which turn out to be the (complex conjugated) Taylor coefficients of the Schur function of the spectral measure. On the one hand, this provides a direct dynamical interpretation of these coefficients; on the other hand it links our definition of first return times to a large body of mathematical literature.     

Relativistic New Yukawa-Like Potential and Tensor Coupling

We approximately solve the Dirac equation for a new suggested generalized inversely quadratic Yukawa potential including a Coulomb-like tensor interaction with arbitrary spin-orbit coupling quantum number ${\kappa}$ . In the framework of the spin and pseudospin (p-spin) symmetry, we obtain the energy eigenvalue equation and the corresponding eigenfunctions, in closed form, by using the parametric Nikiforov–Uvarov method. The numerical results show that the Coulomb-like tensor interaction, ?T/r, removes degeneracies between spin and p-spin state doublets. The Dirac solutions in the presence of exact spin symmetry are reduced to Schr?dinger solutions for Yukawa and inversely quadratic Yukawa potentials.     

An updated description of heavy-hadron interactions in Geant-4

Exotic stable massive particles (SMP) are proposed in a number of scenarios of physics beyond the Standard Model. LHC experiments are expected to be able both to detect and extract the quantum numbers of any SMP with masses around the TeV scale. An understanding of the interactions of SMPs in matter is required to optimise the detection methods and calculate acceptances in an SMP search. In this paper a regge-based model of R-hadron scattering is extended and implemented in Geant-4. In addition, the implications of R-hadron scattering for collider searches are discussed.     

Grothendieck–Teichmüller and Batalin–Vilkovisky

It is proven that, for any affine supermanifold M equipped with a constant odd symplectic structure, there is a universal action (up to homotopy) of the Grothendieck–Teichmüller Lie algebra ${\mathfrak{grt}_1}$ on the set of quantum BV structures (i.e. solutions of the quantum master equation) on M.     

A Necessary Condition for Quantum Adiabaticity Applied to the Adiabatic Grover Search

Numerous sufficient conditions for adiabaticity of the evolution of a driven quantum system have been known for quite a long time. In contrast, necessary adiabatic conditions are scarce. Recently a practicable necessary condition well suited for many-body systems has been proved. Here we tailor this condition for estimating run times of adiabatic quantum algorithms. As an illustration, the condition is applied to the adiabatic algorithm for searching in an unstructured database (adiabatic Grover search algorithm). We find that the thus obtained lower bound on the run time of this algorithm reproduces \( \sqrt{N} \) scaling (with N being the number of database entries) of the explicitly known optimum run time. This is in contrast to the poor performance of the known sufficient adiabatic conditions, which guarantee adiabaticity only for a run time on the order of O(N), which does not constitute any speedup over the classical database search. This observation highlights the merits of the new adiabatic condition and its potential relevance to adiabatic quantum computing.     

Non-Almost Periodicity of Parallel Transports for Homogeneous Connections

Let ${\cal A}$ be the affine space of all connections in an SU(2) principal fibre bundle over ?³. The set of homogeneous isotropic connections forms a line l in ${\cal A}$ . We prove that the parallel transports for general, non-straight paths in the base manifold do not depend almost periodically on l. Consequently, the embedding $l \hookrightarrow {\cal A}$ does not continuously extend to an embedding $\overline{l} \hookrightarrow \overline{\cal A}$ of the respective compactifications. Here, the Bohr compactification $\overline{l}$ corresponds to the configuration space of homogeneous isotropic loop quantum cosmology and $\overline{\cal A}$ to that of loop quantum gravity. Analogous results are given for the anisotropic case.     

Bounds of the Pinsker and Fannes Types on the Tsallis Relative Entropy

Pinsker’s and Fannes’ type bounds on the Tsallis relative entropy are derived. The monotonicity property of the quantum f -divergence is used fot its estimation from below. For order $\alpha \in (0,1)$ , a family of lower bounds of Pinsker type is obtained. For $\alpha >1$ and the commutative case, upper continuity bounds on the relative entropy in terms of the minimal probability in its second argument are derived. Both the lower and upper bounds presented are reformulated for the case of Rényi’s entropies. The Fano inequality is extended to Tsallis’ entropies for all $\alpha >0$ . The deduced bounds on the Tsallis conditional entropy are used to obtain inequalities of Fannes’ type.     

Robust Quantum Search with Uncertain Number of Target States

The quantum search algorithm is one of the milestones of quantum algorithms. Compared with classical algorithms, it shows quadratic speed-up when searching marked states in an unsorted database. However, the success rates of quantum search algorithms are sensitive to the number of marked states. In this paper, we study the relation between the success rate and the number of iterations in a quantum search algorithm of given λ=M/N, where M is the number of marked state and N is the number of items in the dataset. We develop a robust quantum search algorithm based on Grover–Long algorithm with some uncertainty in the number of marked states. The proposed algorithm has the same query complexity ON as the Grover’s algorithm, and shows high tolerance of the uncertainty in the ratio M/N. In particular, for a database with an uncertainty in the ratio M±MN, our algorithm will find the target states with a success rate no less than 96%.     

Dirac multimode ket-bra operators’ \mathfrak{Q}-ordered and \mathfrak{P}-ordered integration theory and general squeezing operator

We develop quantum mechanical Dirac ket-bra operator’s integration theory in $\mathfrak{Q}$ -ordering or $\mathfrak{P}$ -ordering to multimode case, where $\mathfrak{Q}$ -ordering means all Qs are to the left of all Ps and $\mathfrak{P}$ -ordering means all Ps are to the left of all Qs. As their applications, we derive $\mathfrak{Q}$ -ordered and $\mathfrak{P}$ -ordered expansion formulas of multimode exponential operator $e^{ - iP_l \Lambda _{lk} Q_k } $ . Application of the new formula in finding new general squeezing operators is demonstrated. The general exponential operator for coordinate representation transformation $\left| {\left. {\left( {_{q_2 }^{q_1 } } \right)} \right\rangle \to } \right|\left. {\left( {_{CD}^{AB} } \right)\left( {_{q_2 }^{q_1 } } \right)} \right\rangle $ is also derived. In this way, much more correpondence relations between classical coordinate transformations and their quantum mechanical images can be revealed.     

Reichenbachian Common Cause Systems

A partition C_{i i∈ I} of a Boolean algebra $\mathcal{S}$ in a probability measure space $(\mathcal{S},p)$ is called a Reichenbachian common cause system for the correlated pair A,B of events in $\mathcal{S}$ if any two elements in the partition behave like a Reichenbachian common cause and its complement, the cardinality of the index set I is called the size of the common cause system. It is shown that given any correlation in $(\mathcal{S},p)$ , and given any finite size n>2, the probability space $(\mathcal{S},p)$ can be embedded into a larger probability space in such a manner that the larger space contains a Reichenbachian common cause system of size n for the correlation. It also is shown that every totally ordered subset in the partially ordered set of all partitions of $\mathcal{S}$ contains only one Reichenbachian common cause system. Some open problems concerning Reichenbachian common cause systems are formulated.     

Differential calculus on the quantum superspace and deformation of phase space

We investigate non-commutative differential calculus on the supersymmetric version of quantum space in which quantum supergroups are realized. Multiparametric quantum deformation of the general linear super-group,GL _q(m|n), is studied and the explicit form for the \(\hat R - matrix\) is presented. We apply these results to the quantum phase-space construction ofOSp _q(2n|2m) and calculate their \(\hat R - matrices\) .     

Quantum billiards in multidimensional models with branes

A gravitational $D$ -dimensional model with $l$ scalar fields and several forms is considered. When a cosmological-type diagonal metric is chosen, an electromagnetic composite brane ansatz is adopted and certain restrictions on the branes are imposed; the conformally covariant Wheeler–DeWitt (WDW) equation for the model is studied. Under certain restrictions asymptotic solutions to WDW equation are found in the limit of the formation of the billiard walls which reduce the problem to the so-called quantum billiard on the $(D+ l -2)$ -dimensional Lobachevsky space. Two examples of quantum billiards are considered. The first one deals with $9$ -dimensional quantum billiard for $D = 11$ model with $330$ four-forms which mimic space-like $M2$ - and $M5$ -branes of $D=11$ supergravity. The second one deals with the $9$ -dimensional quantum billiard for $D =10$ gravitational model with one scalar field, $210$ four-forms and $120$ three-forms which mimic space-like $D2$ -, $D4$ -, $FS1$ - and $NS5$ -branes in $D = 10$ $II A$ supergravity. It is shown that in both examples wave functions vanish in the limit of the formation of the billiard walls (i.e. we get a quantum resolution of the singularity for $11D$ model) but magnetic branes could not be neglected in calculations of quantum asymptotic solutions while they are irrelevant for classical oscillating behavior when all $120$ electric branes are present.     

The effects of non-Kolmogorov turbulence on the orbital angular momentum of a photon-beam propagation in a slant channel

We analyze the effects of non-Kolmogorov turbulence on the orbital angular momentum of a photon-beam propagation through atmosphere. The probability models of the orbital angular momentum crosstalk for single photons propagation in the channel with the non-Kolmogorov turbulence aberration have been established. It is found that the crosstalk among orbits increases as the orbital angular momentum quantum number of launch beam rises, the ground turbulence strength ${C_n^{2} \left( 0 \right)}$ enhances or the non-Kolmogorov parameter α of turbulence-channel increases. As non-Kolmogorov parameter α approaches 4, the crosstalk probabilities among neighbor orbits are approximately the same.     

Dephasing in the semiclassical limit is system-dependent

Dephasing in open quantum chaotic systems has been investigated in the limit of large system sizes to the Fermi wavelength ratio, L/λ_F 〉 1. The weak localization correction g ^wl to the conductance for a quantum dot coupled to (i) an external closed dot and (ii) a dephasing voltage probe is calculated in the semiclassical approximation. In addition to the universal algebraic suppression g ^wl ∝ (1 + τ_D/τ_?)^?1 with the dwell time τ_D through the cavity and the dephasing rate τ _? ^?1 , we find an exponential suppression of weak localization by a factor of ∝ exp[? $\tilde \tau $ /τ_?], where $\tilde \tau $ is the system-dependent parameter. In the dephasing probe model, $\tilde \tau $ coincides with the Ehrenfest time, $\tilde \tau $ ∝ ln[L/λ_F], for both perfectly and partially transparent dot-lead couplings. In contrast, when dephasing occurs due to the coupling to an external dot, $\tilde \tau $ ∝ ln[L/ξ] depends on the correlation length ξ of the coupling potential instead of λ_F.     

Some Properties of Generalized Quantum Operations

Let $\mathcal{B}(\mathcal{H})$ be the set of all bounded linear operators on the separable Hilbert space  $\mathcal{H}$ . A (generalized) quantum operation is a bounded linear operator defined on  $\mathcal{B}(\mathcal{H})$ , which has the form $\varPhi_{\mathcal{A}}(X)=\sum_{i=1}^{\infty}A_{i}XA_{i}^{*}$ , where $A_{i}\in\mathcal{B}(\mathcal{H})$ (i=1,2,…) satisfy $\sum_{i=1}^{\infty}A_{i}A_{i}^{*}\leq \nobreak I$ in the strong operator topology. In this paper, we establish the relationship between the (generalized) quantum operation $\varPhi_{\mathcal{A}}$ and its dual $\varPhi_{\mathcal {A}}^{\dag}$ with respect to the set of fixed points and the noiseless subspace. In particular, we also partially characterize the extreme points of the set of all (generalized) quantum operations and give some equivalent conditions for the correctable quantum channel.     

CP violation and the widthZ→b \bar b

We discuss the effect of CP-violatingZb $\bar b$ Zb $\bar b$ G andZb $\bar b$ γ couplings on the width Γ(Z→b $\bar b$ X). The presence of such couplings leads in a natural way to an increase of this width relative to the prediction of the standard model. Various strategies of a direct search for such CP-violating couplings by using CP-odd observables are outlined. The number ofZ bosons required to obtain significant information on the couplings in this way is well within the reach of present LEP experiments.