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arXiv · 1802.01162

Information storing yields a point-asymmetry of state space in general probabilistic theories

Keiji Matsumoto·Gen Kimura
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Abstract

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.

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Keiji Matsumoto, Gen Kimura. 2018-02-04. Information storing yields a point-asymmetry of state space in general probabilistic theories. https://arxiv.org/abs/1802.01162

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