Finite operator and ket conventions

A finite operator space in these conventions is a specified nonzero complex coordinate space $H={\mathbb{C}}^d$, $d\ge 1$, equipped with its standard inner product and a specified ordered orthonormal basis. An ordered basis is a finite ordered list of linearly independent vectors spanning the space. An orthonormal basis is a basis whose vectors have inner products $1$ with themselves and $0$ with each other. Coordinates and matrices below are taken in the specified bases. The root inner product is linear in its first argument; on standard coordinates it is $⟨x,y{⟩}_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}}={\sum }_i{x}_i\overline{y_i}$.

Below, $H={\mathbb{C}}^d$ and $K={\mathbb{C}}^e$, with $d,e\ge 1$, are any two finite operator spaces with their specified ordered orthonormal bases $(h_i{)}_{i=1}^d$ and $(k_j{)}_{j=1}^e$. For $x={\sum }_i{x}_i{b}_i$ in a specified ordered basis $(b_i)$, its coordinate column is $(x_i{)}_i$; the same symbol $x$ also denotes this column. For a linear map $A:K\to H$, its matrix, also denoted $A=(a_{ij})$, is determined by $A{k}_j={\sum }_i{a}_{ij}{h}_i$. Products denote rectangular matrix products, $(BC{)}_{ij}={\sum }_k{B}_{ik}{C}_{kj}$, for compatible shapes. Transposition means $(B^T{)}_{ji}=B_{ij}$, and $\bar{B}$ denotes entrywise complex conjugation.

For vectors $u\in H$ and $v\in K$, the ket $\mathrm{\mid }u⟩$ denotes $u$, the bra $⟨v\mathrm{\mid }$ denotes the linear functional $x\mapsto v^{†}x={\sum }_j\overline{v_j}{x}_j$ on $K$, and the outer product $\mathrm{\mid }u⟩⟨v\mathrm{\mid }:K\to H$ maps $x$ to $u(v^{†}x)$. For a linear map $A:K\to H$, $A^{†}:H\to K$ denotes the adjoint, represented by the conjugate transpose ${\bar{A}}^T$. The bracket $⟨u\mathrm{\mid }A\mathrm{\mid }v⟩$ means $u^{†}Av$, equivalently $⟨Av,u{⟩}_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}}$.

An ordered tensor register consists of a finite ordered list of finite-dimensional complex inner-product spaces, each with a specified ordered orthonormal basis: its factors remain in the displayed order and its product basis is ordered lexicographically. The empty tensor product is $\mathbb{C}$. No infinite tensor product or unspecified basis is implicit.

Real and complex inner product spaces, with the inner product linear in the first argument

The standard formulas ⟨ x,y⟩=∑_k<nxₖ yₖ on ℝⁿ and ∑_k<nxₖyₖ on ℂⁿ are inner products

The adjoint T^*:W→ V is characterised by ⟨ Tv,w⟩_W=⟨ v,T^*w⟩_V

Ordered tensor registers, their inner product, and wire permutations

Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis

Linear map between vector spaces over the same field

Coordinate columns [v]_B and matrices [T]_B^C of linear maps relative to ordered bases

Rectangular matrix multiplication and the identity matrix Iₙ, including zero-sized shapes

The transpose A^T of a matrix

In an orthonormal basis, expanding the vectors in their coordinate columns and using linearity in the first argument and conjugate symmetry gives $⟨x,y{⟩}_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}}={\sum }_i{x}_i\overline{y_i}$ in those coordinates, consistent with the standard-coordinate inner product. These are the coordinate columns of Coordinate columns [v]_B and matrices [T]_B^C of linear maps relative to ordered bases, in the ordered bases of Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis, using Real and complex inner product spaces, with the inner product linear in the first argument and The standard formulas ⟨ x,y⟩=∑_k<nxₖ yₖ on ℝⁿ and ∑_k<nxₖyₖ on ℂⁿ are inner products.

Hence $v^{†}x=⟨x,v{⟩}_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}}$ is linear in $x$, so the stated outer product preserves linear combinations and is a well-defined linear map in the sense of Linear map between vector spaces over the same field. The scalar product of the row and column uses Rectangular matrix multiplication and the identity matrix Iₙ, including zero-sized shapes.

Expanding $x={\sum }_j{x}_j{k}_j$ and applying linearity to the defining matrix columns gives $(Ax{)}_i={\sum }_j{a}_{ij}{x}_j$, the matrix action supplied by the coordinate-column convention and rectangular multiplication. This uses Coordinate columns [v]_B and matrices [T]_B^C of linear maps relative to ordered bases, Linear map between vector spaces over the same field and Rectangular matrix multiplication and the identity matrix Iₙ, including zero-sized shapes.

For $x\in K$ and $y\in H$, the finite coordinate sums give $⟨Ax,y{⟩}_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}}={\sum }_{i,j}{a}_{ij}{x}_j\overline{y_i}=⟨x,{\bar{A}}^{T}y{⟩}_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}}$, where $({\bar{A}}^T{)}_{ji}=\overline{a_{ij}}$ by The transpose A^T of a matrix. Thus the conjugate transpose satisfies the defining adjoint identity of The adjoint T^*:W→ V is characterised by ⟨ Tv,w⟩_W=⟨ v,T^*w⟩_V, with the inner product of Real and complex inner product spaces, with the inner product linear in the first argument and the matrix products of Rectangular matrix multiplication and the identity matrix Iₙ, including zero-sized shapes. In particular $u^{†}Av=⟨Av,u{⟩}_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}}$.

The ordered tensor construction and its product basis are supplied by Ordered tensor registers, their inner product, and wire permutations.