Finite registers and their ordered computational bases

For an integer $d\ge 1$, the $d$-level register is the complex inner-product space $Q_d={\mathbb{C}}^d$ with the standard pairing $⟨z,w{⟩}_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}}={\sum }_{j<d}{z}_j\overline{w_j}$, linear in the first argument as in Real and complex inner-product spaces and their induced length and The standard formulas $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ and ${\sum }_{k<n}{x}_k\overline{y_k}$ on ${\mathbb{C}}^n$ are inner products. Its computational basis is the ordered standard basis $(e_0,\dots ,e_{d-1})$ of The standard list $e:n\to F^n$ with $e_i(i)=1_F$ and $e_i(j)=0_F$ for $j\mathrm{\ne }i$ is an ordered basis of $F^n$; hence ${\dim }_F{F}^n=n$, and $F^0$ is the zero space with basis $\mathrm{\varnothing }$ and dimension $0$. The register's alphabet is the ordered set $\{0,\dots ,d-1\}$.

For $n\ge 0$, the $n$-qubit register is the ordered tensor product over $\mathbb{C}$ of $n$ copies of $Q_2$. Here ${\otimes }_{\mathbb{C}}$ is the algebraic module tensor product (The tensor product $M{\otimes }_{R}N$ from the additive group underlying the free $\mathbb{Z}$-module on $M\times N$, elementary tensors, and finite tensor sums), with its complex-module structure and finite iterated/empty-product conventions from Over a commutative ring, $M{\otimes }_{R}N$ is an $R$-module with $r(m\otimes n)=(rm)\otimes n=m\otimes (rn)$ and Finite iterated tensor products represent multilinear maps independently of parenthesization. The tuple of bit positions is ordered lexicographically, with the first wire as the most significant position; the product-basis theorem below establishes the corresponding elementary tensors as a basis. For $n=0$, the empty tensor product is $\mathbb{C}$ with basis $\{1\}$. This one-dimensional trivial register is distinct from the zero vector space ${\mathbb{C}}^0$. By definition a register has positive dimension; a zero-dimensional space may still occur as a factor in the more general tensor-space construction below.