An alternative quantum fidelity for mixed states of qudits

We give an alternative definition of quantum fidelity for two density operators on qudits in terms of the Hilbert-Schmidt inner product between them and their purity. It can be regarded as the well-defined operator fidelity for the two operators and satisfies all Jozsa's four axioms up to a normalization factor. One desire property is that it is not computationally demanding.Comment: 3 pages, no fig

A Memristor Model with Piecewise Window Function

In this paper, we present a memristor model with piecewise window function, which is continuously differentiable and consists of three nonlinear pieces. By introducing two parameters, the shape of this window function can be flexibly adjusted to model different types of memristors. Using this model, one can easily obtain an expression of memristance depending on charge, from which the numerical value of memristance can be readily calculated for any given charge, and eliminate the error occurring in the simulation of some existing window function models

Entangling Power in the Deterministic Quantum Computation with One Qubit

The deterministic quantum computing with one qubit (DQC1) is a mixed-state quantum computation algorithm that evaluates the normalized trace of a unitary matrix and is more powerful than the classical counterpart. We find that the normalized trace of the unitary matrix can be directly described by the entangling power of the quantum circuit of the DQC1, so the nontrivial DQC1 is always accompanied with the non-vanishing entangling power. In addition, it is shown that the entangling power also determines the intrinsic complexity of this quantum computation algorithm, i.e., the larger entangling power corresponds to higher complexity. Besides, it is also shown that the non-vanishing entangling power does always exist in other similar tasks of DQC1.Comment: 6 pages and 1 figur

Modeling the Flux-Charge Relation of Memristor with Neural Network of Smooth Hinge Functions

The memristor was proposed to characterize the flux-charge relation. We propose the generalized flux-charge relation model of memristor with neural network of smooth hinge functions. There is effective identification algorithm for the neural network of smooth hinge functions. The representation capability of this model is theoretically guaranteed. Any functional flux-charge relation of a memristor can be approximated by the model. We also give application examples to show that the given model can approximate the flux-charge relation of existing piecewise linear memristor model, window function memristor model, and a physical memristor device

Refinements of two identities on (n,m)(n,m)-Dyck paths

For integers $n, m$ with $n \geq 1$ and $0 \leq m \leq n$, an $(n,m)$-Dyck path is a lattice path in the integer lattice $\mathbb{Z} \times \mathbb{Z}$ using up steps $(0,1)$ and down steps $(1,0)$ that goes from the origin $(0,0)$ to the point $(n,n)$ and contains exactly $m$ up steps below the line $y=x$. The classical Chung-Feller theorem says that the total number of $(n,m)$-Dyck path is independent of $m$ and is equal to the $n$-th Catalan number $C_n=\frac{1}{n+1}{2n \choose n}$. For any integer $k$ with $1 \leq k \leq n$, let $p_{n,m,k}$ be the total number of $(n,m)$-Dyck paths with $k$ peaks. Ma and Yeh proved that $p_{n,m,k}$=$p_{n,n-m,n-k}$ for $0 \leq m \leq n$, and $p_{n,m,k}+p_{n,m,n-k}=p_{n,m+1,k}+p_{n,m+1,n-k}$ for $1 \leq m \leq n-2$. In this paper we give bijective proofs of these two results. Using our bijections, we also get refined enumeration results on the numbers $p_{n,m,k}$ and $p_{n,m,k}+p_{n,m,n-k}$ according to the starting and ending steps.Comment: 9 pages, with 2 figure