Geometry of quantum states

Quantum states are normalized vectors in Hilbert space, whereby states are equivalent up to a complex multiplicative constant. By considering the Hilbert space as an n-dimensional complex vector space, one can think of the Hilbert space as an (2n-1)-unit sphere embedded in (2n)-dimensional real vector space. At the same time, it is possible to projectivize a complex multiplicative constant of quantum states to form an n-dimensional complex projective space. Since the Hilbert space of a multipartite system consists of the tensor product of its constituents' Hilbert spaces, the probability amplitudes of a multipartite state are endowed with tensorial properties.

All of these algebraic/geometric descriptions can be used to study the properties of quantum states. For instance, diagram on the left shows the polytope of largest eigenvalues of the one-body marginals of three qubits, showcasing the possibility of identifying entanglement properties of three qubits through the one-body marginals.