Incompatibility measures in multiparameter quantum estimation under hierarchical quantum measurements

The incompatibility of the measurements constrains the achievable precisions in multiparameter quantum estimation. Understanding the tradeoff induced by such incompatibility is a central topic in quantum metrology. Here we provide an approach to study the incompatibility under general $p$-local measurements, which are the measurements that can be performed collectively on at most $p$ copies of quantum states. We demonstrate the power of the approach by presenting a hierarchy of analytical bounds on the tradeoff among the precisions of different parameters. These bounds lead to a necessary condition for the saturation of the quantum Cramér-Rao bound under $p$-local measurements, which recovers the partial commutative condition at $p=1$ and the weak commutative condition at $p=\infty$. As a further demonstration of the power of the framework, we present another set of tradeoff relations with the right logarithmic operators.

Figure 1

Upper bounds on ${\mathrm{\Gamma }}_p$ obtained with $C_p$ and $T_p$, together with the QCRB and Holevo bounds at the case $\delta =0$.

Figure 2

Comparison of different bounds of ${\mathrm{\Gamma }}_p$ for the estimation of ${\rho }_x=\frac{1}{2}(I+\delta {\sigma }_3+x_1{\sigma }_1+x_2{\sigma }_2+x_3{\sigma }_3)$ at $x_1=x_2=x_3=0$.

Figure 3

Precision bounds ${\mathrm{\Gamma }}_p$ for $p$-local measurements and the Holevo bound when $n=3$.

Figure 4

Comparison of different bounds of ${\mathrm{\Gamma }}_p$ for the estimation of ${\rho }_x=\frac{1}{3}I+\delta G_3+x_1{G}_1+x_2{G}_2+x_5{G}_5$ at $x_1=x_2=x_5=0$.

Figure 5

Eigenvalues and multiplicities of ${\sum }_{r=1}^p[G_j^{\left(r\right)},G_k^{\left(r\right)}]$.

Figure 6

Upper bound on ${\mathrm{\Gamma }}_p$ and the QCRB and Holevo bounds with the number of parameters equal to $8,4,3,2$, respectively.