diagonal operator
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2021 ◽  
Vol 11 (01) ◽  
pp. 97-104
Author(s):  
小航 贺




2019 ◽  
Vol 07 (03) ◽  
pp. 738-745
Author(s):  
Jin Chen ◽  
Wenjing Lu ◽  
Hanyue Xiao ◽  
Yanan Wang ◽  
Xin Tan


2018 ◽  
Vol 9 (3) ◽  
pp. 297-309 ◽  
Author(s):  
Mojtaba Bakherad ◽  
Khalid Shebrawi


2011 ◽  
Vol 42 (2) ◽  
pp. 193-203
Author(s):  
M. Gupta ◽  
L. R. Acharya

In this paper we establish relationships of the approximation numbers of matrix transformations acting between the vector-valued sequence spaces spaces of the type $\lambda(X)$ defined corresponding to a scalar-valued sequence space $\lambda$ and a Banach space $(X,\|.\|)$ as $$\lambda(X)=\{\overline x=\{x_i\}: x_i\in X, \forall~i\in \mathbb{N},~\{\|x_i\|_X\}\in \lambda\};$$ with those of their component operators. This study leads to a characterization of a diagonal operator to be approximable. Further, we compute the approximation numbers of inclusion maps acting between $\ell^p(X)$ spaces for $1\leq p\leq \infty$.







2009 ◽  
Vol 9 (5&6) ◽  
pp. 461-486
Author(s):  
V.V. Shende ◽  
I.L. Markov

The three-input \TOFFOLI\ gate is the workhorse of circuit synthesis for classical logic operations on quantum data, e.g., reversible arithmetic circuits. In physical implementations, however, \TOFFOLI\ gates are decomposed into six \CNOT\ gates and several one-qubit gates. Though this decomposition has been known for at least 10 years, we provide here the first demonstration of its \CNOT-optimality. We study three-qubit circuits which contain less than six \CNOT\ gates and implement a block-diagonal operator, then show that they implicitly describe the cosine-sine decomposition of a related operator. Leveraging the canonical nature of such decompositions to limit one-qubit gates appearing in respective circuits, we prove that the $n$-qubit analogue of the \TOFFOLI\ requires at least $2n$ \CNOT\ gates. Additionally, our results offer a complete classification of three-qubit diagonal operators by their \CNOT -cost, which holds even if ancilla qubits are available.



2008 ◽  
Vol 103 (1) ◽  
pp. 77
Author(s):  
Junsheng Fang ◽  
Don Hadwin ◽  
Xiujuan Ma

We compute spectra and Brown measures of some non self-adjoint operators in $(M_2(\mathsf {C}), \frac{1}{2}\mathrm{Tr})*(M_2(\mathsf{C}), \frac{1}{2}\mathrm{Tr})$, the reduced free product von Neumann algebra of $M_2(\mathsf {C})$ with $M_2(\mathsf {C})$. Examples include $AB$ and $A+B$, where $A$ and $B$ are matrices in $(M_2(\mathsf {C}), \frac{1}{2}\mathrm{Tr})*1$ and $1*(M_2(\mathsf {C}), \frac{1}{2}\mathrm{Tr})$, respectively. We prove that $AB$ is an R-diagonal operator (in the sense of Nica and Speicher [12]) if and only if $\mathrm{Tr}(A)=\mathrm{Tr}(B)=0$. We show that if $X=AB$ or $X=A+B$ and $A,B$ are not scalar matrices, then the Brown measure of $X$ is not concentrated on a single point. By a theorem of Haagerup and Schultz [9], we obtain that if $X=AB$ or $X=A+B$ and $X\neq \lambda 1$, then $X$ has a nontrivial hyperinvariant subspace affiliated with $(M_2(\mathsf{C}), \frac{1}{2}\mathrm{Tr})*(M_2(\mathsf{C}), \frac{1}{2}\mathrm{Tr})$.



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