different settings. This provides a measure for assessing the performance of illumination under these settings.

Quantifiers of performance
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We consider two quantifiers of performance of illumination: the accessible information and Holevo information.

Let $\mathcal{M} = {\{E_k\}}$ be a positive operator-valued measure (POVM) that mathematically represents a measurement. The POVM elements $E_k$ are non-negative, self-adjoint operators satisfying $\sum_k E_k = 1$, where the subscript $k$ labels the outcome of the measurement. The probability of the measurement outcome $k$ on a state $\rho^{(x)}$ is then given by $q_k^{(x)}=\Tr (\rho^{(x)} E_k)$. Let this be governed by random variable $K_{\mathcal{M}}$. In the communication setting described in the last section, the amount of information obtained by Bob after measurement of the state $\rho^{(x)}$ is given by the mutual information, $$\label{eqn_mutual_int}
I_{\rm mut}(X,K_{\mathcal{M}})
 = \sum_k \sum_{x=0}^1 p_x q^{(x)}_k \log_2 (\frac{q^{(x)}_k}{q_k}),$$ where $q_k=\sum_{x=0}^1 p_x q^{(x)}_k$. The accessible information is the maximization of the mutual information over all POVMs, $$\label{eqn_accessible}
A\left(\rho^{(0)},\rho^{(1)}\right) = \max_{\mathcal{M}}I_{\rm mut}\left(X,K_{\mathcal{M}}\right).$$ The accessible information quantifies Bob’s knowledge when each $\rho^{(x)}$ from $N$ trials is measured separately using an optimal POVM. In the context of communication, illumination can be regarded as classical information exchange over a noisy channel. By the Shannon’s noisy-channel coding theorem [@6773024], 