The largest possible global symmetry of a 2-dimensional Hilbert space?

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Suppose we have a quantum system of a 2-dimensional Hilbert space $$\mathcal{H}$$ and a Hamiltonian $$\hat H$$.

My puzzle: What is the largest possible global symmetry for the Hilbert space $$\mathcal{H}$$ and Hamiltonian $$\hat H$$?

My attempt: By symmetry of a quantum system, in some sense, we meant to find the quantum symmetry transformation $$\hat S$$ as an operator in terms of matrx such that $$\hat S \hat H \hat S^{-1} =\hat H .$$

Naively, if we have $$\hat H$$ is proportional to an identity matrix $$\mathbb{I}$$ acting on 2-dim state vector in $$|v \rangle \in\mathcal{H}$$ , we have at most a constraint for $$\hat S \hat S^{-1} =\mathbb{I}.$$ This means that the symmetry forms a invertible matrices with complex entries known as the general linear matrix group $$GL(2,\mathbb{C}).$$ However, this does only include the linear symmetry, but not the anti-linear symmetry such as complex conjugation $$\hat K$$. So again, What is the largest possible global symmetry for the Hilbert space $$\mathcal{H}$$ and Hamiltonian $$\hat H$$?

This post imported from StackExchange Physics at 2020-12-04 11:33 (UTC), posted by SE-user annie marie heart
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