Embeddings of spaces of quregisters into special linear groups

TL;DR

This paper investigates how the structure of symmetry groups of quregisters can be embedded into linear groups, analyzing the case of qubits and higher-dimensional systems, with implications for entanglement measures.

Contribution

It introduces embeddings of spaces of quregisters into linear groups and explores their algebraic and entanglement-related properties for higher dimensions.

Findings

For n=1, the sphere of qubits is identified with SU(2).

Embeddings for n=2 do not establish a bijection with SU(4).

Embeddings relate to entanglement measures consistent with von Neumann entropy.

Abstract

We study embeddings of the unit sphere of complex Hilbert spaces of dimension a power $2^n$ into the corresponding groups of non-singular linear transformations. For the case of $n=1$, the sphere $S_2$ of qubits is identified with $\mbox{SU}(2)$ and the algebraic structure of this last group is carried into $S_2$. Hence it is natural to analyse whether is it possible, for $n\geq 2$, to carry the structure of the symmetry group $\mbox{SU}(2^n)$ into the unit sphere $S_{2^n}$. For $n=2$ the embeddings of $S_{2^2}$ into $\mbox{GL}(2^2)$, obtained as tensor products of the above embedding, fails to determine a bijection between $S_{2^2}$ and $\mbox{SU}(2^2)$, but they determine entanglement measures consistent with von Neumann entropy.