The efficiency of repeaters based on the Einstein-Podolsky-Rosen effect for quantum cryptography in a damping channel

Based on the fundamental Holevo inequality and on the direct calculations, it is argued that the number of commitments required per one bit in a key in a damping channel increases exponentially with channel length. It is shown that the conclusion drawn recently by Duan $et~al.$ [4] that the exponential increase in resources for quantum cryptography in a damping channel can be reduced to the polynomial law by generating a through Einstein-Podolsky-Rosen pair is erroneous. Therefore, the results of [4] do not solve the fundamental problem restricting practical application of quantum cryptography at distances larger than the damping length.