We prove lower bounds for the error of optimal cubature formulas for $d$ -variate functions from Besov spaces of mixed smoothness $B^{\alpha}_{p,\theta}({\mathbb G}^d)$ in the case $0 1/p$ , and upper bounds for QMC methods on the Fibonacci lattice for two-variate periodic functions from $B^{\alpha}_{p,\theta}({\mathbb G}^2)$ in the case $1\leq p \leq \infty$ , $0 1/p$ , where ${\mathbb G}^d$ is either the $d$ -dimensional torus ${\mathbb T}^d$ or the $d$ -dimensional cube $[0,1]^d$ . In combination these results yield the correct asymptotic of the error of optimal cubature formulas for functions from $B^{\alpha}_{p,\theta}({\mathbb G}^2)$ and indicate that a corresponding results is most likely also true in case $d>2$ . This is compared to the obtained correct asymptotic of optimal cubature formulas on Smolyak grids which results in the observation that any cubature formula on Smolyak grids is never optimal for the general setting.We prove lower bounds for the error of optimal cubature formulas for $d$-variate functions from Besov spaces of mixed smoothness $B^{\alpha}_{p,\theta}({\mathbb G}^d)$ in the case $0 1/p$, and upper bounds for QMC methods on the Fibonacci lattice for two-variate periodic functions from $B^{\alpha}_{p,\theta}({\mathbb G}^2)$ in the case $1\leq p \leq \infty$, $0 1/p$, where ${\mathbb G}^d$ is either the $d$-dimensional torus ${\mathbb T}^d$ or the $d$-dimensional cube $[0,1]^d$. In combination these results yield the correct asymptotic of the error of optimal cubature formulas for functions from $B^{\alpha}_{p,\theta}({\mathbb G}^2)$ and indicate that a corresponding results is most likely also true in case $d>2$. This is compared to the obtained correct asymptotic of optimal cubature formulas on Smolyak grids which results in the observation that any cubature formula on Smolyak grids is never optimal for the general setting.

arXiv:1311.1563Nov 7, 20130 citations

Coherent state phase estimation based on digital nonlinear phase-locked loop

Xu Han, Shuxin Chen, Hao Wu +2 more

The navigation mechanism of obtaining phase parameters based on quantum theory can break through the limitation of classical physical limit to navigation accuracy. In order to achieve the accurate estimation of the coherent state phase, it is usually assumed that the local oscillator phase must be orthogonal to the coherent state phase in the method of quantum homodyne detection. However, the coherent state phase is unknown and the hypothesis cannot be guaranteed to be correct in practice. In this paper we design a nonlinear phase-locked loop to solve the problem. Firstly, in order to obtain the Wigner distribution for a coherent state, we start with the Wigner distribution for the vacuum state and analyze the noise characteristics according to Wigner distribution of coherent state, then the output of homodyne detection is derived. Secondly, in order to avoid introducing errors in theory, caused by linearization and cope with the limiting requirement between local oscillator phase and coherent state phase in the phase tracking of coherent state, we design an orthogonal simplex cubature Kalman filter (OSCKF) algorithm to achieve the function of the nonlinear phase-locked loop. The algorithm converges by updating the state of the local oscillator phase multiple times, and then, the accurate coherent phase is obtained. Finally, according to the design of the phase-locked loop, we observe the data of homodyne detection and then verify the correctness of the OSCKF algorithm. The simulation results show that the OSCKF can converge to the real phase after observing 200 sampled data and the accuracy is higher than extended Kalmn filter (EKF) and cubature Kalman filter (CKF), and the real phase can be obtained under different local oscillator phase. In conclusion, the nonlinear phase-locked loop based on OSCKF algorithm breaks the limitation of traditional way in which the initial local oscillator phase is required to be orthogonal to the coherent state phase, and effectively avoid the linearization error and improve the anti-nonlinear ability. It is very significant in theory and application .The navigation mechanism of obtaining phase parameters based on quantum theory can break through the limitation of classical physical limit to navigation accuracy. In order to achieve the accurate estimation of the coherent state phase, it is usually assumed that the local oscillator phase must be orthogonal to the coherent state phase in the method of quantum homodyne detection. However, the coherent state phase is unknown and the hypothesis cannot be guaranteed to be correct in practice. In this paper we design a nonlinear phase-locked loop to solve the problem. Firstly, in order to obtain the Wigner distribution for a coherent state, we start with the Wigner distribution for the vacuum state and analyze the noise characteristics according to Wigner distribution of coherent state, then the output of homodyne detection is derived. Secondly, in order to avoid introducing errors in theory, caused by linearization and cope with the limiting requirement between local oscillator phase and coherent state phase in the phase tracking of coherent state, we design an orthogonal simplex cubature Kalman filter (OSCKF) algorithm to achieve the function of the nonlinear phase-locked loop. The algorithm converges by updating the state of the local oscillator phase multiple times, and then, the accurate coherent phase is obtained. Finally, according to the design of the phase-locked loop, we observe the data of homodyne detection and then verify the correctness of the OSCKF algorithm. The simulation results show that the OSCKF can converge to the real phase after observing 200 sampled data and the accuracy is higher than extended Kalmn filter (EKF) and cubature Kalman filter (CKF), and the real phase can be obtained under different local oscillator phase. In conclusion, the nonlinear phase-locked loop based on OSCKF algorithm breaks the limitation of traditional way in which the initial local oscillator phase is required to be orthogonal to the coherent state phase, and effectively avoid the linearization error and improve the anti-nonlinear ability. It is very significant in theory and application .

Acta Physica SinicaJan 1, 20192 citations

Digital Feedback Control

Diego Ristè, L. DiCarlo

Quantum science and technologyJan 1, 20166 citations

Equiangular lines, projective symmetries and nice error frames

Tuan-Yow Chien

The existence of maximal sets of equiangular lines (SIC-POVMs) is of interest to the mathematics and physics communities due to their connection to quantum information theory, quantum cryptography, spherical 2-designs and cubature rules. This thesis looks at a way to recover analytic SIC-POVMs from numerical SIC-POVMs, including the discovery of a new analytic SIC-POVM in 17 dimensions. It also develops a way to comprehensively find nice error bases (used for finding SIC-POVMs). Finally, a general theory to classify sequences of vectors up to projective unitary equivalence is developed and applied to study the projective symmetry groups of sequences of vectors.The existence of maximal sets of equiangular lines (SIC-POVMs) is of interest to the mathematics and physics communities due to their connection to quantum information theory, quantum cryptography, spherical 2-designs and cubature rules. This thesis looks at a way to recover analytic SIC-POVMs from numerical SIC-POVMs, including the discovery of a new analytic SIC-POVM in 17 dimensions. It also develops a way to comprehensively find nice error bases (used for finding SIC-POVMs). Finally, a general theory to classify sequences of vectors up to projective unitary equivalence is developed and applied to study the projective symmetry groups of sequences of vectors.

Jan 1, 20154 citations

We prove lower bounds for the error of optimal cubature formulas for

d

-variate functions from Besov spaces of mixed smoothness

B^{\alpha}_{p,\theta}({\mathbb G}^d)

in the case

0 1/p

, and upper bounds for QMC methods on the Fibonacci lattice for two-variate periodic functions from

B^{\alpha}_{p,\theta}({\mathbb G}^2)

in the case

1\leq p \leq \infty

0 1/p

, where

{\mathbb G}^d

is either the

d

-dimensional torus

{\mathbb T}^d

or the

d

-dimensional cube

[0,1]^d

. In combination these results yield the correct asymptotic of the error of optimal cubature formulas for functions from

B^{\alpha}_{p,\theta}({\mathbb G}^2)

and indicate that a corresponding results is most likely also true in case

d>2

. This is compared to the obtained correct asymptotic of optimal cubature formulas on Smolyak grids which results in the observation that any cubature formula on Smolyak grids is never optimal for the general setting.