Hypothesis testing and entropies of quantum channels

Hypothesis testing is an important task in mathematics and physics. Hypothesis testing of two random variables is related to the Kullback-Leibler divergence of the two corresponding distributions. Similarly, quantum hypothesis testing of two quantum states is characterized by the quantum relative entropy. While a quantum state can be abstracted as a device that only has outputs but no inputs, the most general quantum device is a quantum channel that also has inputs. In this work, we extend hypothesis testing to general quantum channels. In both the one-shot and asymptotic scenario, we study several quantifiers for hypothesis testing under different assumptions of how the channels are used. As the quantifiers are analogs to the quantum relative entropy of states, we call them the quantum relative entropy of channels. Then, we define the entropy of channels based on relative entropies from the target channel to the completely depolarizing channel. We investigate the properties that the quantum relative entropy of channels should satisfy, and we study its interplay with entanglement. With the broad applications of the quantum relative entropy of states, our results can be useful for understanding general properties of quantum channels.

Figure 1

Quantum states and channels. (a) A quantum state $\rho$ can be regarded as a device that has null input and only outputs ${\rho }_{\text{out}}=\rho$. (b) A quantum channel is a generalized device that inputs ${\rho }_{\text{in}}$ and outputs ${\rho }_{\text{out}}$. When ${\rho }_{\text{in}}$ has dimension zero or ${\rho }_{\text{out}}$ is a classical state, a quantum channel can be regarded as a state preparation or a demolition measurement, respectively.

Figure 2

Hypothesis testing of two channels. (a) For each use of the quantum channel, no extra ancilla is allowed. (b) One party of the maximally entangled state ${\mathrm{\Phi }}^{+}$ is input to the channel. (c) One party of a joint state is input to the channel.