Abstract
The von Neumann entropy of a quantum state is a central concept in physics and information theory, having a number of compelling physical interpretations. There is a certain perspective that the most fundamental notion in quantum mechanics is that of a quantum channel, as quantum states, unitary evolutions, measurements, and discarding of quantum systems can each be regarded as certain kinds of quantum channels. Thus, an important goal is to define a consistent and meaningful notion of the entropy of a quantum channel. Motivated by the fact that the entropy of a state $\rho$ can be formulated as the difference of the number of physical qubits and the "relative entropy distance" between $\rho$ and the maximally mixed state, here we define the entropy of a channel $\mathcal{N}$ as the difference of the number of physical qubits of the channel output with the "relative entropy distance" between $\mathcal{N}$ and the completely depolarizing channel. We prove that this definition satisfies all of the axioms, recently put forward in [Gour, IEEE Trans. Inf. Theory 65, 5880 (2019)], required for a channel entropy function. The task of quantum channel merging, in which the goal is for the receiver to merge his share of the channel with the environment's share, gives a compelling operational interpretation of the entropy of a channel. The entropy of a channel can be negative for certain channels, but this negativity has an operational interpretation in terms of the channel merging protocol. We define Renyi and min-entropies of a channel and prove that they satisfy the axioms required for a channel entropy function. Among other results, we also prove that a smoothed version of the min-entropy of a channel satisfies the asymptotic equipartition property.
Highlights
In his foundational work on quantum statistical mechanics, von Neumann extended the classical Gibbs entropy concept to the quantum realm [1]
We prove that the entropy of a channel is additive, which is the second axiom proposed in Ref. [18] for a channel entropy function
We have introduced a definition for the entropy of a quantum channel, based on the channel relative entropy between the channel of interest and the completely randomizing channel
Summary
In his foundational work on quantum statistical mechanics, von Neumann extended the classical Gibbs entropy concept to the quantum realm [1]. In analogy to the operational interpretation for D(ρA πA) mentioned above, it is known that D(N R) is equal to the optimal rate at which the channel NA→B can be distinguished from the completely randomizing channel RA→B, by allowing for any possible quantum strategy to distinguish the channels [15]. This statement holds in the Stein setting of quantum hypothesis testing It cannot be considered an entropy function according to the approach of Ref. [18]
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