The geometry of state space

Abstract

The geometry of the state space of a finite-dimensional quantum mechanical system, with particular reference to four dimensions, is studied. Many novel features, not evident in the two-dimensional space of a single spin, are found. Although the state space is a convex set, it is not a ball, and its boundary contains mixed states in addition to the pure states, which form a low-dimensional submanifold. The appropriate language to describe the role of the observer is that of flag manifolds.The geometry of the state space of a single spin, E(C2), is so well known as to be a standard textbook topic, but there is surprisingly little known about the geometry of the state space of larger systems, apart from general statements such as it is a convex region which is generated by the pure states which are extremal points. Even for E(C4), which is the state space for a pair of spins and hence the manifold appropriate to the EinsteinPodolsky-Rosen paradox, little appears to be known.It is the purpose of this article to describe the structure of E(C4), and, as it turns out, many major features of E(C n) for arbitrary finite n. The principal motive for this inquiry arose out of the work of the first named authors ~1) on intuitionist quantum mechanics. In this formulation a special neighborhood basis was introduced, and it was thought desirable to test the properties of this basis on some special examples. The case of E(C 2) is too special to be of much use, and so the case of E(C 4) was selected. It turns out that this is sufficient to illustrate the general situation.The structure of this paper is as follows. In Section 1, a brief review of E ( C 2) is given, following the very nice article of Urbantke (2J drawing atten-

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Geometry of mixed states for a q-bit and the quantum Fisher information tensor

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Geometry of sets of quantum maps: A generic positive map acting on a high-dimensional system is not completely positive

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