Corrigendum: Quantum state estimation with nuisance parameters (2020 J. Phys. A: Math. Theor.53 453001)

We have found the two theorems (theorems 5.2 and 5.3) in section 5 appeared incorrectly. In particular, equations (112) and (114) were not correct.

Firstly, theorem 5.2 should state:

Theorem 5.2. Given a d-parameter regular model ${\cal M}$, for each d-dimensional (column) vector ${\boldsymbol{v}}\in{\mathbb{R}}^d$, the infimum of the MSE matrix in the direction of v is

$$\begin{equation} \inf_{\hat{\Pi}\mathrm{\,:l.u.at\,}{\boldsymbol{\theta}} } {\boldsymbol{v}}^{\mathrm{T}} V_{\boldsymbol{\theta}}\left[\hat{\Pi}\right] {\boldsymbol{v}} = {\boldsymbol{v}}^{\mathrm{T}} \left(J_{\boldsymbol{\theta}}^{\mathrm{S}}\right)^{-1}{\boldsymbol{v}}, \end{equation} \tag{ 112 }$$

where $J_{\boldsymbol{\theta}}^{\mathrm{S}}$ is the SLD quantum Fisher information matrix. An optimal measurement is given by a projection measurement about the linear combination of the SLD operators:

$$\begin{equation} L^{\mathrm{S}}_{{\boldsymbol{\theta}};{\boldsymbol{v}}} = \sum_{i,j = 1}^d v_i J^{\mathrm{S};i,j}_{{\boldsymbol{\theta}}} L_{{\boldsymbol{\theta}};j}^{\mathrm{S}} = \sum_{i = 1}^d v_i L_{{\boldsymbol{\theta}}}^{\mathrm{S};i} . \end{equation} \tag{ 113 }$$

Secondly, theorem 5.3 should state:

Theorem 5.3. Given a d-parameter regular model ${\cal M}$, suppose that we are interested in estimating the parameter θ₁ in the presence of the nuisance parameters $\boldsymbol{\theta}_{\mathrm{N}} = (\theta_2,\dots,\theta_d)$. The achievable lower bound for the MSE about the parameter of interest $V_{\boldsymbol{\theta};\mathrm{I}}[\hat{\Pi}_{\mathrm{I}}]$ is given by

$$\begin{equation} C_{\boldsymbol{\theta};\mathrm{I}}\left[W_\mathrm{I} = 1,{\cal M}\right]: = \min_{\hat{\Pi}_{\mathrm{I}}\mathrm{\,:l.u.\,at\,}\boldsymbol{\theta} \mathrm{\, for \,} \boldsymbol{\theta}_{\mathrm{I}}} V_{\boldsymbol{\theta};\mathrm{I}}\left[\hat{\Pi}_{\mathrm{I}}\right] = J_{\boldsymbol{\theta}}^{\mathrm{S};1,1} = \left( J^{\mathrm{S}}_{\boldsymbol{\theta}}\left({\mathrm{I}}|{\mathrm{N}}\right)\right)^{-1}, \end{equation} \tag{ 114 }$$

where the minimization is taken over all locally unbiased estimators $\hat{\Pi}_{\mathrm{I}}$ for the parameter of interest at θ , $J_{\boldsymbol{\theta}}^{\mathrm{S};1,1}$ is the (1, 1)th element of the inverse SLD matrix, and $J^{\mathrm{S}}_{\boldsymbol{\theta}}({\mathrm{I}}|{\mathrm{N}})$ is the partial SLD Fisher information (85). An optimal measurement is given by a projection measurement about the operator:

$$\begin{equation} L_{{\boldsymbol{\theta}}}^{\mathrm{S};1} = \sum_{j = 1}^d J_{\boldsymbol{\theta}}^{\mathrm{S};j,1} L_{{\boldsymbol{\theta}};j}^{\mathrm{S}}. \end{equation} \tag{ 115 }$$