Volume of the quantum mechanical state space

TL;DR

This paper calculates the volume of quantum state spaces across different Hilbert spaces and metrics, providing explicit formulas and characterizing conditions for infinite volume, which enhances understanding of quantum state space geometry.

Contribution

It introduces explicit volume formulas for quantum state spaces under various metrics and characterizes when these volumes become infinite, advancing geometric understanding in quantum theory.

Findings

Explicit volume formulas for quantum state spaces in real, complex, and quaternionic Hilbert spaces.

Identification of conditions under which the volume of qubit state spaces is infinite.

Characterization of monotone metrics that lead to infinite volume.

Abstract

The volume of the quantum mechanical state space over $n$-dimensional real, complex and quaternionic Hilbert-spaces with respect to the canonical Euclidean measure is computed, and explicit formulas are presented for the expected value of the determinant in the general setting too. The case when the state space is endowed with a monotone metric or a pull-back metric is considered too, we give formulas to compute the volume of the state space with respect to the given Riemannian metric. We present the volume of the space of qubits with respect to various monotone metrics. It turns out that the volume of the space of qubits can be infinite too. We characterize those monotone metrics which generates infinite volume.