Twospin

Twospin is a simple quantum mechanical model consisting of two spin-1/2 degrees of freedom. The composite system has a four-dimensional Hilbert space H1 ⊗ H2, spanned by basis states |↑↑>, |↑↓>, |↓↑>, |↓↓>. It's central in studies of quantum entanglement, spin coupling, and two-qubit operations.

The total spin operator S = S1 + S2 has eigenvalues S=0 (singlet) and S=1 (triplet). The singlet state |ψ−> = (|↑↓> - |↓↑>)/√2 is antisymmetric and has total spin zero, while the triplet manifold includes |ψ+> = (|↑↓> + |↓↑>)/√2, |↑↑>, and |↓↓> with total spin 1. The exchange interaction J S1·S2 splits these levels, yielding E_singlet = -3J/4 and E_triplet = J/4 (for appropriate convention). In external magnetic field B, Zeeman terms split the triplet further according to Sz.

Entanglement: the singlet and the Bell state |ψ+> are maximally entangled; |↑↑> and |↓↓> are separable. The two-spin system serves as a basic two-qubit register in quantum information, enabling Bell tests, teleportation, and quantum gates through controlled interactions.

Realizations and applications: in solid-state systems such as coupled quantum dots or nitrogen-vacancy centers, in trapped ions and superconducting circuits, twospin models implement two-qubit gates and study many-body physics via pair interactions. It is a foundational building block in studies of entanglement dynamics, quantum phase transitions, and spin-based quantum computing.