Classical Estimation Theory

Abstract

To formulate error and disturbance in quantum measurement, the estimation process from the measurement outcomes has an essential role. In this chapter, we review classical estimation theory [1][3] and introduce Fisher information, which gives the upper bound of the accuracy of the estimation [3]

Appendix. Generalized Inverse Matrices

If the rank of an $n\times n$ square matrix $A$ is $n$, the inverse of $A$ that satisfies $AA^{-1} = A^{-1}A = I$ is uniquely determined. However, for the case in which $\mathrm{{rank}}A < n$ or $A$ is not a square matrix, the inverse does not exist. By relaxing the condition, we can define an inverse for an arbitrary matrix as follows.

Definition 1

(Moore-Penrose pseudoinverse [4, 5].) Let $A$ be an $n\times m$ matrix, and $P[\mathrm{{supp}}(A)]$ and $P[\mathrm{{img}}(A)]$ be the projections onto the support and the image of $A$, respectively. If an $m \times n$ matrix $B$ satisfies

$$\begin{aligned} BA = P[\mathrm{{supp}}(A)], \qquad AB = P[\mathrm{{img}}(B)], \end{aligned}$$

(3.96)

$$\begin{aligned} \mathrm{{supp}}(B) = \mathrm{{img}}(A), \qquad \mathrm{{img}}(B) = \mathrm{{supp}}(A), \end{aligned}$$

(3.97)

then $B$ is called the Moore-Penrose pseudoinverse of $A$.

In the following, we denote the pseudoinverse by $A^{-1}$. The pseudoinverse exists for an arbitrary matrix, and is uniquely determined. If $A$ is a square matrix and has full rank, its pseudoinverse reduces to the inverse. If all the elements of $A$ are $0$, the pseudoinverse is $A^{{\mathrm {T}}}$. The pseudoinverse of a column vector $\mathbf{{v}}$ is $\mathbf{{v}}^\dagger /|\mathbf{{v}}|^2$.

The pseudoinverse is obtained in terms of the singular value decomposition (SVD). An arbitrary $n \times m$ matrix $A$ can be decomposed as

$$\begin{aligned} A = \sum _i^r s_i \mathbf{{u}}_i \mathbf{{v}}_i^{{\mathrm {T}}}, \end{aligned}$$

(3.98a)

$$\begin{aligned} s_i > 0,\quad \mathbf{{u}}_i \in \mathbb {R}^n,\quad \mathbf{{v}}_i\in \mathbb {R}^m,\quad \mathbf{{u}}_i\cdot \mathbf{{u}}_j = \mathbf{{v}}_i\cdot \mathbf{{v}}_j = \delta _{ij}, \end{aligned}$$

(3.98b)

where $r = \mathrm{{rank}}[A]$, and $s_i$ is called a singular value of $A$. Such the decomposition is called the singular value decomposition. The vectors $\mathbf{{v}}_i$ and $\mathbf{{u}}_i$ satisfy the following relation:

$$\begin{aligned} \sum _i^r \mathbf{{v}}_i\mathbf{{v}}_i^{{\mathrm {T}}}= P[\mathrm{{supp}}(A)], \qquad \sum _i^r \mathbf{{u}}_i\mathbf{{u}}_i^{{\mathrm {T}}}= P[\mathrm{{img}}(A)]. \end{aligned}$$

(3.99)

Therefore, the pseudoinverse of $A$ is given by

$$\begin{aligned} A^{-1} = \sum _i^r s_i^{-1} \mathbf{{v}}_i \mathbf{{u}}_i^{{\mathrm {T}}}. \end{aligned}$$

(3.100)