Entangled Inputs Cannot Make Imperfect Quantum Channels Perfect

Entangled inputs can enhance the capacity of quantum channels, this being one of the consequences of the celebrated result showing the nonadditivity of several quantities relevant for quantum information science. In this work, we answer the converse question (whether entangled inputs can ever render noisy quantum channels to have maximum capacity) to the negative: No sophisticated entangled input of any quantum channel can ever enhance the capacity to the maximum possible value, a result that holds true for all channels both for the classical as well as the quantum capacity. This result can hence be seen as a bound as to how “nonadditive quantum information can be.” As a main result, we find first practical and remarkably simple computable single-shot bounds to capacities, related to entanglement measures. As examples, we discuss the qubit amplitude damping and identify the first meaningful bound for its classical capacity.

Figure 1

Upper bound to the classical information capacity in terms of entanglement measures between the output and its environment in a dilation of the quantum channel.

Figure 2

Upper bound to the classical information capacity of the qubit amplitude-damping channel as a function of $p\in [0,1/2]$. The chosen Kraus operators delivering good bounds for $k=2$ are given by $P_0=|0⟩⟨0|/2$, $P_1=\sum _{i,j=0,1}|i⟩⟨j|/x$, $P_2=(\mathbb{1}-P_0^{†}{P}_0-P_1^{†}{P}_1{)}^{1/2}$ for $x=4$ (dashed line) and $x=3$ (solid line). Any choice for Kraus operators yields a bound, and with the given choice (determined via numerical explorations) one obtains a particularly tight bound.