  In a little more detail: if \(\{v_{i}^A\}\) is an orthonormal basis for \(H_A\) and \(\{u_{j}^B\}\) is an orthonormal basis for \(H_B\), then the set of pairs \((v_{i}^A, u_{j}^B)\) is taken to form an orthonormal basis for the tensor product space \(H_A \otimes H_B\). The notation \(v_i^A \otimes u_j^B\) is used for the pair \((v_{i}^A,u_{j}^B)\), and inner product on \(H_A \otimes H_B\) is defined as:[6]