  It is known that the high-dimensional quantum state space is notoriously
complicated in contrast with the beautiful Bloch ball of the qubit. We examined
the mechanism behind this fact in the frame work of general probabilistic
theory (GPT), and found rather general quantitative relations between the
geometry of the state space and its information storing capability. The main
result is the information-asymmetry identity, which (up to the constant term)
equates the {\it Minkowski measure of asymmetry} with the {\it information
storability} which, in addition to its own operational meaning, serves as an
upper bound to common information measures such as semi-classical capacity. As
a consequence, the asymmetry measure is lower-bounded by information storing
capability of the state space, so the increase in the latter enhances the
former. Coming back to the quantum systems, the $d$-level state space cannot be
symmetric "because" it can store more than a single bit of information. Also,
the Holevo capacity of any quantum channel with point-symmetric image is at
most a single bit. In the course of the research, we applied Shannon theory to
GPT, producing a couple of new results. Also presented is a new geometrical
proof of known upper bounds to information measures.
