scholarly journals Crystallizing highly-likely subspaces that contain an unknown quantum state of light

2016 ◽  
Vol 6 (1) ◽  
Author(s):  
Yong Siah Teo ◽  
Dmitri Mogilevtsev ◽  
Alexander Mikhalychev ◽  
Jaroslav Řeháček ◽  
Zdeněk Hradil
Keyword(s):  
Maximum Likelihood ◽  
Quantum State ◽  
Measurement Data ◽  
Dimensional Subspace ◽  
Continuous Variable ◽  
Finite Dimensional Subspace ◽  
Model Subspaces ◽  
Finite Dimensional ◽  
Unknown Quantum State ◽  
Principle Of Maximum

Abstract In continuous-variable tomography, with finite data and limited computation resources, reconstruction of a quantum state of light is performed on a finite-dimensional subspace. In principle, the data themselves encode all information about the relevant subspace that physically contains the state. We provide a straightforward and numerically feasible procedure to uniquely determine the appropriate reconstruction subspace by extracting this information directly from the data for any given unknown quantum state of light and measurement scheme. This procedure makes use of the celebrated statistical principle of maximum likelihood, along with other validation tools, to grow an appropriate seed subspace into the optimal reconstruction subspace, much like the nucleation of a seed into a crystal. Apart from using the available measurement data, no other assumptions about the source or preconceived parametric model subspaces are invoked. This ensures that no spurious reconstruction artifacts are present in state reconstruction as a result of inappropriate choices of the reconstruction subspace. The procedure can be understood as the maximum-likelihood reconstruction for quantum subspaces, which is an analog to, and fully compatible with that for quantum states.

scholarly journals Deterministic quantum teleportation through fiber channels

2018 ◽  
Vol 4 (10) ◽  
pp. eaas9401 ◽  
Author(s):  
Meiru Huo ◽  
Jiliang Qin ◽  
Jialin Cheng ◽  
Zhihui Yan ◽  
Zhongzhong Qin ◽  
...  
Keyword(s):  
Communication Networks ◽  
Quantum State ◽  
Optical Fibers ◽  
Quantum Teleportation ◽  
Continuous Variable ◽  
Entangled State ◽  
Optical Modes ◽  
Unknown Quantum State ◽  
Einstein Podolsky Rosen ◽  
Fiber Channels

Quantum teleportation, which is the transfer of an unknown quantum state from one station to another over a certain distance with the help of nonlocal entanglement shared by a sender and a receiver, has been widely used as a fundamental element in quantum communication and quantum computation. Optical fibers are crucial information channels, but teleportation of continuous variable optical modes through fibers has not been realized so far. Here, we experimentally demonstrate deterministic quantum teleportation of an optical coherent state through fiber channels. Two sub-modes of an Einstein-Podolsky-Rosen entangled state are distributed to a sender and a receiver through a 3.0-km fiber, which acts as a quantum resource. The deterministic teleportation of optical modes over a fiber channel of 6.0 km is realized. A fidelity of 0.62 ± 0.03 is achieved for the retrieved quantum state, which breaks through the classical limit of1/2. Our work provides a feasible scheme to implement deterministic quantum teleportation in communication networks.

scholarly journals Functional ARCH and GARCH Models: A Yule-Walker Approach

2020 ◽  
Author(s):  
Sebastian Kühnert
Keyword(s):  
Financial Time Series ◽  
Sufficient Conditions ◽  
Dimensional Subspace ◽  
Stationary Solutions ◽  
Finite Dimensional Subspace ◽  
Estimation Errors ◽  
Linear Processes ◽  
Finite Dimensional ◽  
Garch Processes ◽  
Asymptotic Upper Bound

Conditional heteroskedastic financial time series are commonly modelled by ARCH and GARCH. ARCH(1) and GARCH processes were recently extended to the function spaces C[0,1] and L2[0,1], their probabilistic features were studied and their parameters were estimated. The projections of the operators on finite-dimensional subspace were estimated, as were the complete operators in GARCH(1,1). An explicit asymptotic upper bound of the estimation errors was stated in ARCH(1). This article provides sufficient conditions for the existence of strictly stationary solutions, weak dependence and finite moments of ARCH and GARCH processes in various Lp[0,1] spaces, C[0,1] and other spaces. In L2[0,1] we deduce explicit asymptotic upper bounds of the estimation errors for the shift term and the complete operators in ARCH and GARCH and for the projections of the operators on a finite-dimensional subspace in ARCH. The operator estimaton is based on Yule-Walker equations. The estimation of the GARCH operators also involves a result concerning the estimation of the operators in invertible, linear processes which is valid beyond the scope of ARCH and GARCH. Through minor modifications, all results in this article regarding functional ARCH and GARCH can be transferred to functional ARMA.

THE REGULAR PART OF A SEMIGROUP OF LINEAR TRANSFORMATIONS WITH RESTRICTED RANGE

2017 ◽  
Vol 103 (3) ◽  
pp. 402-419 ◽  
Author(s):  
WORACHEAD SOMMANEE ◽  
KRITSADA SANGKHANAN
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Dimensional Subspace ◽  
Finite Dimensional Subspace ◽  
Restricted Range ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Finite Dimensional ◽  
Idempotent Rank ◽  
Image Position

Let$V$be a vector space and let$T(V)$denote the semigroup (under composition) of all linear transformations from$V$into$V$. For a fixed subspace$W$of$V$, let$T(V,W)$be the semigroup consisting of all linear transformations from$V$into$W$. In 2008, Sullivan [‘Semigroups of linear transformations with restricted range’,Bull. Aust. Math. Soc.77(3) (2008), 441–453] proved that$$\begin{eqnarray}\displaystyle Q=\{\unicode[STIX]{x1D6FC}\in T(V,W):V\unicode[STIX]{x1D6FC}\subseteq W\unicode[STIX]{x1D6FC}\} & & \displaystyle \nonumber\end{eqnarray}$$is the largest regular subsemigroup of$T(V,W)$and characterized Green’s relations on$T(V,W)$. In this paper, we determine all the maximal regular subsemigroups of$Q$when$W$is a finite-dimensional subspace of$V$over a finite field. Moreover, we compute the rank and idempotent rank of$Q$when$W$is an$n$-dimensional subspace of an$m$-dimensional vector space$V$over a finite field$F$.

Implementing Galerkin Finite Element Methods for Semilinear Elliptic Differential Inclusions

2013 ◽  
Vol 13 (1) ◽  
pp. 95-118 ◽  
Author(s):  
Janosch Rieger
Keyword(s):  
Finite Element ◽  
Differential Inclusions ◽  
Dimensional Subspace ◽  
Finite Dimensional Subspace ◽  
Basic Algorithm ◽  
Galerkin Finite Element ◽  
Finite Dimensional ◽  
Partial Differential Inclusions ◽  
Galerkin Finite Element Methods ◽  
Element Approach

Abstract. This paper presents the first feasible method for the approximation of solution sets of semi-linear elliptic partial differential inclusions. It is based on a new Galerkin Finite Element approach that projects the original differential inclusion to a finite-dimensional subspace of . The problem that remains is to discretize the unknown solution set of the resulting finite-dimensional algebraic inclusion in such a way that efficient algorithms for its computation can be designed and error estimates can be proved. One such discretization and the corresponding basic algorithm are presented along with several enhancements, and the algorithm is applied to two model problems.

scholarly journals Quasi-reliable estimates of effective sample size

2020 ◽  
Author(s):  
Youhan Fang ◽  
Yudong Cao ◽  
Robert D Skeel
Keyword(s):  
Sample Size ◽  
Dimensional Subspace ◽  
Monte Carlo Algorithm ◽  
Finite Dimensional Subspace ◽  
Effective Sample Size ◽  
Finite Dimensional ◽  
Adequate Sampling ◽  
Autocorrelation Time ◽  
The Mean ◽  
The Cost

Abstract The efficiency of a Markov chain Monte Carlo algorithm for estimating the mean of a function of interest might be measured by the cost of generating one independent sample, or equivalently, the total cost divided by the effective sample size, defined in terms of the integrated autocorrelation time. To ensure the reliability of such an estimate, it is suggested that there be an adequate sampling of state space— to the extent that this can be determined from the available samples. A sufficient condition for adequate sampling is derived in terms of the supremum of all possible integrated autocorrelation times, which leads to a more stringent condition for adequate sampling than that simply obtained from integrated autocorrelation times for functions of interest. A method for estimating the supremum of all integrated autocorrelation times, based on approximation in a finite-dimensional subspace, is derived and evaluated empirically.

scholarly journals Characterizations of Operator Birkhoff–James Orthogonality

2017 ◽  
Vol 60 (4) ◽  
pp. 816-829 ◽  
Author(s):  
Mohammad Sal Moslehian ◽  
Ali Zamani
Keyword(s):  
Unit Vector ◽  
Dimensional Subspace ◽  
Linear Operators ◽  
Inner Product ◽  
Finite Dimensional Subspace ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
James Orthogonality ◽  
New Type ◽  
Norm Attaining

AbstractIn this paper, we obtain some characterizations of the (strong) Birkhoff–James orthogonality for elements of Hilbert C*-modules and certain elements of . Moreover, we obtain a kind of Pythagorean relation for bounded linear operators. In addition, for we prove that if the norm attaining set is a unit sphere of some finite dimensional subspace of and , then for every , T is the strong Birkhoff–James orthogonal to S if and only if there exists a unit vector such that . Finally, we introduce a new type of approximate orthogonality and investigate this notion in the setting of inner product C*-modules.

About Weyl and Wigner Tomography in Finite-Dimensional Hilbert Spaces

2006 ◽  
Vol 13 (04) ◽  
pp. 403-413 ◽  
Author(s):  
Thomas Durt
Keyword(s):  
Quantum State ◽  
Hilbert Spaces ◽  
Higher Dimensions ◽  
Point Of View ◽  
Finite Dimensional ◽  
Complete Knowledge ◽  
Unknown Quantum State ◽  
The One ◽  
First Time

We study different techniques that allow us to gain complete knowledge about an unknown quantum state, e.g. to perform full tomography of this state. In a first time, we focus on two simple cases, full tomography of one- and two-qubit systems. We analyze and compare those techniques according to two criteria. Our first criterion is the minimisation of the redundancy of the data acquired during the tomographic process. In the case of two-qubits tomography, we also analyze this process from the point of view of factorisability, so to say we analyze the possibility to realise the tomographic process through local operations and classical communications between local observers. Finally, we present new results that concern the extension of the one- and two-qubit cases to higher dimensions.

scholarly journals CALDERÓN’S INVERSE PROBLEM WITH A FINITE NUMBER OF MEASUREMENTS

2019 ◽  
Vol 7 ◽  
Author(s):  
GIOVANNI S. ALBERTI ◽  
MATTEO SANTACESARIA
Keyword(s):  
Finite Number ◽  
Reconstruction Algorithm ◽  
Dimensional Subspace ◽  
Finite Dimensional Subspace ◽  
Stability Estimates ◽  
Inverse Conductivity Problem ◽  
Finite Dimensional ◽  
First Results ◽  
Boundary Measurements ◽  
Inverse Boundary Value Problems

We prove that an $L^{\infty }$ potential in the Schrödinger equation in three and higher dimensions can be uniquely determined from a finite number of boundary measurements, provided it belongs to a known finite dimensional subspace ${\mathcal{W}}$ . As a corollary, we obtain a similar result for Calderón’s inverse conductivity problem. Lipschitz stability estimates and a globally convergent nonlinear reconstruction algorithm for both inverse problems are also presented. These are the first results on global uniqueness, stability and reconstruction for nonlinear inverse boundary value problems with finitely many measurements. We also discuss a few relevant examples of finite dimensional subspaces ${\mathcal{W}}$ , including bandlimited and piecewise constant potentials, and explicitly compute the number of required measurements as a function of $\dim {\mathcal{W}}$ .

scholarly journals Galerkin approximation of linear problems in Banach and Hilbert spaces

2020 ◽  
Author(s):  
W Arendt ◽  
I Chalendar ◽  
R Eymard
Keyword(s):  
Hilbert Spaces ◽  
A Priori Estimates ◽  
A Priori ◽  
Galerkin Approximation ◽  
Dimensional Subspace ◽  
Adjoint Problem ◽  
Finite Dimensional Subspace ◽  
Finite Dimensional ◽  
Regularity Properties ◽  
Sesquilinear Form

Abstract In this paper we study the conforming Galerkin approximation of the problem: find $u\in{{\mathcal{U}}}$ such that $a(u,v) = \langle L, v \rangle $ for all $v\in{{\mathcal{V}}}$, where ${{\mathcal{U}}}$ and ${{\mathcal{V}}}$ are Hilbert or Banach spaces, $a$ is a continuous bilinear or sesquilinear form and $L\in{{\mathcal{V}}}^{\prime}$ a given data. The approximate solution is sought in a finite-dimensional subspace of ${{\mathcal{U}}}$, and test functions are taken in a finite-dimensional subspace of ${{\mathcal{V}}}$. We provide a necessary and sufficient condition on the form $a$ for convergence of the Galerkin approximation, which is also equivalent to convergence of the Galerkin approximation for the adjoint problem. We also characterize the fact that ${{\mathcal{U}}}$ has a finite-dimensional Schauder decomposition in terms of properties related to the Galerkin approximation. In the case of Hilbert spaces we prove that the only bilinear or sesquilinear forms for which any Galerkin approximation converges (this property is called the universal Galerkin property) are the essentially coercive forms. In this case a generalization of the Aubin–Nitsche Theorem leads to optimal a priori estimates in terms of regularity properties of the right-hand side $L$, as shown by several applications. Finally, a section entitled ‘Supplement’ provides some consequences of our results for the approximation of saddle point problems.

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