  We investigate dense coding by imposing various locality restrictions to our
decoder by employing the resource theory of asymmetry framework. In this task,
the sender Alice and the receiver Bob share an entangled state. She encodes the
classical information into it using a symmetric preserving encoder and sends
the encoded state to Bob through a noiseless quantum channel. The decoder is
limited to a measurement to satisfy a certain locality condition on the
bipartite system composed of the receiving system and the preshared
entanglement half. Our contributions are summarized as follows: First, we
derive an achievable transmission rate for this task dependently of conditions
of encoder and decoder. Surprisingly, we show that the obtained rate cannot be
improved even when the decoder is relaxed to local measurements, two-way LOCCs,
separable measurements, or partial transpose positive (PPT) measurements for
the bipartite system. Moreover, depending on the class of allowed measurements
with a locality condition, we relax the class of encoding operations to
super-quantum encoders in the framework of general probability theory (GPT).
That is, when our decoder is restricted to a separable measurement,
theoretically, a positive operation is allowed as an encoding operation.
Surprisingly, even under this type of super-quantum relaxation, the
transmission rate cannot be improved. This fact highlights the universal
validity of our analysis beyond quantum theory.
