Eavesdropping on quantum secure direct communication in quantum channels with arbitrarily low loss rate

Quantum attacks that provide an undetectable eavesdropping of the ping-pong protocol operating over lossy quantum channels have already been demonstrated by Wójcik (Phys Rev Lett 90(15):157901, 2003) and Zhang et al. (Phys Lett A 333(12):46–50, 2004). These attacks provide a maximum information gain of 0.311 bits per protocol cycle as long as the induced loss rate remains acceptable. Otherwise, the skipping of some protocol cycles is advised to stay within an accepted loss limit. Such policy leads to a reduction in information gain proportional to the number of skipped cycles. The attack transformation parametrized by the induced loss ratio is proposed. It provides smaller reduction in information gain when the losses accepted by the communicating parties are too low to mount the most effective attack. Other properties of the attack remain the same.