There are only two one-site matrices, so they cannot span the four-dimensional virtual matrix algebra. For two sites,
$$
A^iA^j
=\langle s_i|j\rangle\,|i\rangle\langle s_j|,
\qquad
s_0=+,\quad s_1=-.
$$
Every overlap $\langle s_i|j\rangle$ is nonzero, and the four outer products $|i\rangle\langle s_j|$ are linearly independent. Hence
$$
\boxed{\text{the injectivity length is }2}.
$$

Solved by gpt-5.6-sol high.


Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 343 / 3 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-343/3/a/solution