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Optimal finite measurements and Gauss quadratures

Sofyan Iblisdir, Jérémie Roland

Abstract

We exhibit measurements for optimal state estimation which have a finite number of outcomes. This is achieved by a connection between finite optimal measurements and Gauss quadratures. The example we consider to illustrate this connection is that of state estimation on $N$ qubits, all in a same pure state. Extensions to state estimation of mixed states are also discussed.

Optimal finite measurements and Gauss quadratures

Abstract

We exhibit measurements for optimal state estimation which have a finite number of outcomes. This is achieved by a connection between finite optimal measurements and Gauss quadratures. The example we consider to illustrate this connection is that of state estimation on qubits, all in a same pure state. Extensions to state estimation of mixed states are also discussed.

Paper Structure

This paper contains 1 theorem, 40 equations, 2 figures, 1 table.

Key Result

Theorem 1

Let $[a,b] \subset \mathbf{R}$, and let $\{p_n\}$ denote a complete set of orthogonal polynomials on $\mathbf{L}^2([a,b])$. If $x_1 < \ldots < x_n$ denote the zeros of $\{p_n(x)\}$, there exist real numbers $\lambda_1, \ldots, \lambda_n$ such that whenever $\rho(x)$ is an arbitrary polynomial of degree at most $2n-1$. Moreover, the distribution $d\alpha(x)$ and the integer $n$ uniquely determine

Figures (2)

  • Figure 1: Lebedev quadrature of order $N=59$. Distribution of $1800$ points for Lebedev quadrature of order $N=59$ (taken from lebe94).
  • Figure 2: Spherical design with $n=60$ points defining a Gauss quadrature on the sphere of order $N=10$hard96.

Theorems & Definitions (1)

  • Theorem 1