Landauer's erasure principle vividly shows the significance of information in physics, implying the "duality" of entropy: information theoretic and thermodynamic ones. This concept has been widely accepted and taken for granted in many situations in quantum information theory as well. In thermodynamics, on the other hand, the second law is the most fundamental rule that describes the behaviour of entropy. A natural question then arises: "Does the second law imply any quantum information theoretic result through the duality of entropy?" Taking accessible information in a system as an example, we show that thermodynamics implies a weaker bound on it than the quantum mechanical one (the Holevo bound). This implies a discrepancy between the erasure principle and the second law, which are commonly believed to give always the same results. It also suggests that if any post-quantum physics should allow more information storage it could still be under the umbrella of thermodynamics.