Accuracy matrix in a generalized simultaneous measurement of a qubit system

We formulate the accuracy of a quantum measurement for a qubit (spin-1/2) system in terms of a $3\times 3$ matrix. This matrix, which we refer to as the accuracy matrix, can be calculated from a positive operator-valued measure (POVM) corresponding to the quantum measurement. Based on the accuracy matrix, we derive trade-off relations between the measurement accuracy of two or three noncommuting observables of a qubit system. These trade-off relations offer a quantitative information-theoretic representation of Bohr’s principle of complementarity. They can be interpreted as the uncertainty relations between measurement errors in simultaneous measurements and also as the trade-off relations between the measurement error and back-action of the measurement. A no-cloning inequality is derived from the trade-off relations. Furthermore, our formulation and the results obtained can be applied to analyze quantum-state tomography. We also show that the accuracy matrix is closely related to the maximum-likelihood estimation and the Fisher information matrix for a finite number of samples; the accuracy matrix tells us how accurately we can estimate the probability distributions of observables of an unknown state by a finite number of quantum measurements.

Figure 1

Trade-off relation for the accuracy of noncommuting observables. P indicates the regimes satisfying the inequality for the case of $\theta =\pi /2$, the union of P and Q indicates the regimes satisfying inequality (63) for the case of $\theta =\pi /6$, and the union of P, Q, and R indicates the regimes satisfying the inequality for the case of $\theta =0$. We can only access regime Q through simultaneous measurement for the case of $\theta =\pi /6$.

Figure 2

(Color online) Maximum-likelihood estimators $p{\left(+;{\mathit{n}}_A\right)}^{\ast }$ (red curves) and $p{\left(+;{\mathit{n}}_B\right)}^{\ast }$ (blue curves) for the case of $\theta =\pi /6$, ${\chi }_A=0.10$, and ${\chi }_B\simeq 0.97$. The abscissa indicates sample number $N$ and the ordinate indicates the value of the estimators. The number of simulations is 20.