Classical capacity per unit cost for quantum channels

In most communication scenarios, sending a symbol encoded in a quantum state requires spending resources such as energy, which can be quantified by a cost of communication. A standard approach in this context is to quantify the performance of communication protocol by classical capacity, quantifying the maximal amount of information that can be transmitted through a quantum channel per single use of the channel. However, different figures of merit are also possible and a particularly well-suited one is the classical capacity per unit cost, which quantifies the maximal amount of information that can be transmitted per unit cost. I generalize this concept to account for the quantum nature of the information carriers and communication channels and show that if there exists a state with cost equal to zero, e.g., a vacuum state, the capacity per unit cost can be expressed by a simple formula containing maximization of the relative entropy between two quantum states. This enables me to analyze the behavior of photon information efficiency for general communication tasks and show simple bounds on the capacity per unit cost in terms of quantities familiar from quantum estimation theory. I calculate also the capacity per unit cost for general Gaussian quantum channels.

Figure 1

Basic scheme of classical communication through a quantum channel. Input symbols $x$, distributed according to the prior probability distribution $p\left(x\right)$, are encoded into quantum states ${\rho }_x$ of information carriers by a modulation operation, represented by a quantum channel ${\varepsilon }_x$. The noisy communication link is modeled by some quantum channel $\mathrm{\Lambda }$, which outputs states $\mathrm{\Lambda }\left[{\rho }_x\right]$, which are then measured by a positive-operator-valued measure ${\mathrm{\Pi }}_y$, resulting in an output symbol $y$ used to reconstruct the input message.

Figure 2

Photon information efficiency as a function of the average number of photons $\overline{n}$ for a lossy (red solid line) and thermal channel (black solid line) for $\eta =0.9$ and $\tilde{N}=1$. The dashed line indicates the capacity per unit cost for the thermal channel.