Informational power of quantum measurements

We introduce the informational power of a quantum measurement as the maximum amount of classical information that the measurement can extract from any ensemble of quantum states. We prove the additivity by showing that the informational power corresponds to the classical capacity of a quantum-classical channel. We restate the problem of evaluating the informational power as the maximization of the accessible information of a suitable ensemble. We provide a numerical algorithm to find an optimal ensemble and quantify the informational power.

Figure 1

Informational power $W(\Pi )$ of the $D$-dimensional POVM ${\Pi }^{(\eta )}$ projecting on the orthonormal basis affected by isotropic noise parameterized by $\eta$ [see Eq. (34)], as a function of $\eta$, for dimension $D=2,3,4$.

Figure 2

Real $Z_N$-symmetric POVMs $\Pi =\{\frac{2}{N}|{\pi }_j\rangle \langle {\pi }_j|\}{}_{j=0}^{N-1}$ (blue vectors labeled $|{\pi }_j\rangle$) for $N=3$ (upper left), $N=5$ (upper right), and $N=7$ (lower left and lower right). A maximally informative ensemble (red vectors labeled $|{\psi }_i\rangle$) $V={p_i}_{,}^{|}$ with a minimal number of states is represented for each POVM. For $N=7$, there are two inequivalent maximally informative ensembles (lower left and lower right). The angle between state $|\psi \rangle =(\begin{array}{c}\hfill \cos \theta \hfill \\ \hfill \sin \theta \hfill \end{array})$ and state $|\psi \rangle =(\begin{array}{c}\hfill \cos \theta \hfill \\ \hfill \sin \theta \hfill \end{array})$ is $2({\theta }_1-{\theta }_0)$, as in the Bloch sphere representation. The length of the vector corresponding to state $|{\psi }_i\rangle$ is proportional to $\sqrt{p_i}$.

Figure 3

Informational power of a mirror-symmetric POVM $\Pi =\{n_j|{\pi }_j\rangle \langle {\pi }_j|\}{}_{j=0}^2$, with $|{\pi }_j\rangle$ as in Eq. (37), as a function of $\theta$. The minimum is attained for $\theta =\pi /6$, where $\Pi$ corresponds to the trine POVM and a maximally informative ensemble is the antitrine. The maxima at $\theta =0$ and $\theta =\pi /4$ correspond to the degenerate case of the POVM projecting on an orthonormal basis. The label $M=2$ or $M=3$ denotes the minimum number $M$ of states of a maximally informative ensemble in each of the three regions.