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arXiv:1711.10471 · Bulletin of the American Physical Society

Complexity and geometry of quantum state manifolds

Zhoushen Huang, Alexander V. Balatsky

Published Nov 28, 2017• 1 citation
arXivHTMLPDF

Abstract

We show that the Hilbert space spanned by a continuously parametrized wavefunction family---i.e., a quantum state manifold---is dominated by a subspace, onto which all member states have close to unity projection weight. Its characteristic dimensionality $D_P$ is much smaller than the full Hilbert space dimension, and is equivalent to a statistical complexity measure $e^{S_2}$, where $S_2$ is the $2^{nd}$ Renyi entropy of the manifold. In the thermodynamic limit, $D_P$ closely approximates the quantum geometric volume of the manifold under the Fubini-Study metric, revealing an intriguing connection between information and geometry. This connection persists in compact manifolds such as a twisted boundary phase, where the corresponding geometric circumference is lower bounded by a term proportional to its topological index, reminiscent of entanglement entropy.

DOI
10.48550/arxiv.1711.10471

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A fundamental starting point in quantum information theory is the consideration of the von Neumann entropy and its generalization to relative quantum entropies. A particular feature of quantum relative entropies is their relation via their Hessian to monotonic Riemannian metrics on the dense set of invertible mixed quantum states (Lesniewski and Ruskai 1999). These metrics are also known as quantum Fisher information metrics and provide a direct link to quantum estimation theory (Helstrom 1969). Quantum Fisher information metrics which are extendable to pure states coincide all with the Fubini Study metric of the projective Hilbert space of complex rays. This theses outlines possible advantages of an inverse approach to quantum information theory, by starting with the Fubini Study metric rather then with the von Neumann entropy. This is done in a first step by associating to the Fubini Study metric a covariant and a contra-variant structure on the punctured Hilbert space as being available in the geometric formulation of quantum mechanics. While the contra-variant structure leads to a quantum version of the Cramér-Rao inequality for general 1-dimensional submanifolds of pure states, the covariant structure provides alternative entanglement monotones by identifying an inner product on the pullback tensor fields on local unitary group orbits of quantum states. It is shown in the case of two qubits that these monotones yield a more efficient estimation of entanglement than standard measures from the literature as those associated with the linearization of the von Neumann entropy.
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