Corrections to “Saturability of the Quantum Cramér-Rao Bound in Multiparameter Quantum Estimation at the Single-Copy Level”
Hendra I. Nurdin · IEEE Control Systems Letters · 2024
The corrigendum uses the notation in [1]. As with [2], the work [1] was concerned with saturability of the QCRB for a finite-dimensional rank-deficient density operator $\rho _{\theta }$ in the sense that there exists a POVM with a corresponding probability distribution (on the measurement outcomes) with a classical Fisher information matrix that equals the quantum Fisher information (QFI) matrix of $\rho _{\theta }$ . It does not address the existence of an unbiased estimator that achieves the lowest mean square estimation error for $C\theta $ under this POVM for any real row vector C of the same length as $\theta $ . Reference [1, Th. 2] claims the following necessary and sufficient conditions: 1) $[L_{\theta _{l},++},L_{\theta _{m},++}]=0$ for $l,m=1,\ldots,p$ .2)For each $\theta $ there exists a unitary $U_{\theta } \in \mathbb {C}^{r_{+} \times r_{+}}$ such that $U_{\theta }^{\dagger }(\partial _{l} U_{\theta } - U_{\theta } V_{\theta }^{\dagger } \partial _{l} V_{\theta })\rho _{\theta,++} + \rho _{\theta,++} (\partial _{l} U_{\theta } - U_{\theta } V_{\theta }^{\dagger } \partial _{l} V_{\theta })^{\dagger } U_{\theta }=0$ for $l=1,\ldots,p$ , where $\partial _{l} = \partial /\partial \theta _{l}$ , $V_{\theta } =\left [{{\begin{array}{ccc} |\psi _{1,\theta } \rangle & ~ \ldots & ~ |\psi _{r_{+},\theta } \rangle \end{array}}}\right ]$ , and $\rho _{\theta,++}$ is represented in the basis $\mathcal {B}_{+,\theta }$ . However, an implicit assumption in the proof of [1, Th. 2] that is not valid in general (that $E_{k,00}$ sums to $I_{r_{0}}$ over all k corresponding to null POVM operators) invalidates these conditions for general POVMs. A counterexample to the necessity of Condition 1 can be found for $\rho _{\theta }$ belonging to the quantum exponential family of parameterized density operators [3] of the form \begin{equation*} \rho _{\theta } = e^{\frac {1}{2}\sum _{j=1}^{p} \left ({{\theta _{j} F_{j} + \omega \left ({{\theta }}\right)I }}\right)} \rho _{0} e^{\frac {1}{2}\sum _{j=1}^{p} \left ({{\theta _{j} F_{j} + \omega \left ({{\theta }}\right)I }}\right)},\end{equation*} with $F_{j}=1,\ldots,p$ taken to be invertible and mutually commuting observables, $\rho _{0}$ is some fixed rank-deficient density operator (of the same dimension as the $F_{j}$ ’s), and $\omega (\theta)$ is a scalar normalizing factor such that $\mathrm {tr}(\rho _{\theta }) = 1$ . It is easily seen that the SLDs can be chosen to be $L_{\theta _{j}} = F_{j} + \partial _{j} \omega (\theta) I$ and they are mutually commuting $[L_{\theta _{j}},L_{\theta _{k}}]=0$ for all $j,k$ . Thus the QCRB is saturated by a POVM that consists of the common spectral projectors of $\{L_{\theta _{j}}\}_{j=1,\ldots,p}$ . However, the SLDs will not satisfy Condition 1 in general. It is satisfied, for instance, in the case where $\rho _{0}$ also commutes with all the SLDs, which is not generic. There is also an error in Condition 2 since there can exist a unitary solution under this condition that does not necessarily correspond to saturation of the QCRB. For example, when $r_{+}=1$ or when $\rho _{\theta,++}=(1/r_{+})I_{r_{+}}$ then $U_{\theta }=I_{r_{+}}$ satisfies Condition 2 (since $V_{\theta }$ is an isometry, $V_{\theta }^{\dagger }V_{\theta }=I_{r_{+}}$ , and therefore $\partial _{l} V_{\theta }^{\dagger }V_{\theta }$ is skew-hermitian) without imposing any constraints on $V_{\theta }$ as would be expected.