Orientation Entanglement - Formal Details

Formal Details

In three dimensions, the problem illustrated above corresponds to the fact that the Lie group SO(3) is not simply connected. Mathematically, one can tackle this problem by exhibiting the special unitary group SU(2), which is also the spin group in three Euclidean dimensions, as a double cover of SO(3). If X = (x₁,x₂,x₃) is a vector in R3, then we identify X with the 2 × 2 matrix with complex entries

Note that −det(X) gives the square of the Euclidean length of X regarded as a vector, and that X is a trace-free, or better, trace-zero Hermitian matrix.

The unitary group acts on X via

where M ∈ SU(2). Note that, since M is unitary,

, and

is trace-zero Hermitian.

Hence SU(2) acts via rotation on the vectors X. Conversely, since any change of basis which sends trace-zero Hermitian matrices to trace-zero Hermitian matrices must be unitary, it follows that every rotation also lifts to SU(2). However, each rotation is obtained from a pair of elements M and −M of SU(2). Hence SU(2) is a double-cover of SO(3). Furthermore, SU(2) is easily seen to be itself simply connected by realizing it as the group of unit quaternions, a space homeomorphic to the 3-sphere.

A unit quaternion has the cosine of half the rotation angle as its scalar part and the sine of half the rotation angle multiplying a unit vector along some rotation axis (here assumed fixed) as its pseudovector (or axial vector) part. If the initial orientation of a rigid body (with unentangled connections to its fixed surroundings) is identified with a unit quaternion having a zero pseudovector part and +1 for the scalar part, then after one complete rotation (2pi rad) the pseudovector part returns to zero and the scalar part has become -1 (entangled). After two complete rotations (4pi rad) the pseudovector part again returns to zero and the scalar part returns to +1 (unentangled), completing the cycle.