Generally a recap of linear algebra basics.

We denote a field $$F$$ (either the real field $$R$$ or complex field $$C$$). A vector space on F is a set V that is closed under addition $$V\times V \rightarrow V$$ and scalar multiplication $$F \times V \rightarrow V$$. This has the zero element 0.

A Hilbert space is a (finite dimensional) vector space with inner product $$\langle{}\cdot{},\cdot{}\rangle{}: V \times F \rightarrow F$$.

• This inner product is linear in the second argument (Physics convention), i.e. $$\langle{}w,\alpha{}v\rangle{} = \alpha{}\langle{}w,v\rangle{} $$
• Has conjugate $$ \langle{}w,v\rangle{} = \overline{\langle{}v,w\rangle{}} $$
• Positive definite: $$\langle{}v,v\rangle{} \ge{} 0$$ with equality iff v=0
• Canonical norm: $$||v|| = \sqrt{\langle{}v,v\rangle{}} $$
• Cauchy-Schwarz inequality: $$|\langle{}v,w\rangle{}| \le ||v||\cdot{}||w||$$
• Triangle inequality: $$||v+w|| \le ||v|| + ||w||$$
• $$v$$ and $$w$$ are orthogonal if $$\langle{}v,w\rangle{} = 0$$
• $$\{v_i\}_i$$ is orthonormal if $$\langle{}v_i,v_j\rangle{} = \delta{}_{ij}$$
• Orthonormal basis (ONB) of V if $$V = \text{span}\{v_i\} = \{v: v=\sum_i a_i v_i, a_i \in F\}$$
• Dimension of V is number of elements in ONB.

One example are pure states $$|\psi{}\rangle{} \in H$$, with the computational basis defined as $$\{|i\rangle{}\}_{i=0}^{d-1}$$. The inner product is defined $$\langle{}\theta{}|\phi{}\rangle{} \equiv{} \langle{}|\theta{}\rangle{},|\phi{}\rangle{}\rangle{} = \sum_i \overline{\theta}_i \cdot{} \phi_i $$. This is analogous to the matrix representation of dot product between column and row vectors.

Linear operators $$L(V,W)$$ contains all linear maps from V to W.

• $$L(v) := L(V,V)$$
• $$L(V,W)$$ is a vector space over same field (for convenience) as V and W, e.g. $$v\in V: (a_1X_1 + a_2X_2)v = a_1X_1v + a_2X_2v \in W$$.
  • Note the action of X is determined by its action on ONB ($$Xv = \sum_i a_i X v_i$$)
• Adjoint of $$ X\in L(V,W)$$ is $$ X^\dagger{} \in L(W,V) $$ satisfying $$\langle{}w,Xv\rangle{} = \langle{}X^\dagger{}w, v\rangle{} $$, where $$ (X^\dagger{})_{ij} = \overline{X_{ji}}$$.
  • Note $$(XY)^\dagger{} = Y^\dagger{} X^\dagger{} $$, and also $$(X^\dagger{})^\dagger{} = X$$.
• Identity operator on V is $$1_V \in L(V)$$, where $$1_V: v\mapsto v$$, or for any general ONB $$1_V: v \mapsto \sum_{i=0}^{d-1} v_i \langle{} v_i,v\rangle{}$$.
  • If $$v_i$$ are kets, we get the more familiar outer product notation, but this becomes a bit limited when representing matrices.
• An isometry $$U\in L(V,W) $$ satisfies $$U^\dagger{}U = 1_V$$.
• A unitary $$U\in L(V)$$ is a stronger isometry where the adjoint is its inverse, i.e. $$U^\dagger{}U = UU^\dagger{}$$.
  • Isometries preserve inner product: $$\langle{}Uv,Uw\rangle = \langle{} v,U^\dagger{}Uw\rangle{} = \langle{} v,w\rangle{} $$.
  • Unitaries are maps that map from one ONB to another, i.e. $$ U: v\mapsto{} \sum_i w_i \langle{} v_i,v\rangle{} $$, for a map from ONB $$\{v_i\}_i$$ to ONB $$\{w_i\}_i$$, or in other words $$Uv_i = w_i \forall{} j$$
• An operator H is self-adjoint or Hermitian if $$H^\dagger{} = H$$, e.g. density operators, states, effects
  • This must be defined on a real field.
• We can also define an inner product on $$L(V,W)$$, where $$\langle{} X,Y\rangle{} = \sum_i \langle{} Xv_i, Yv_i\rangle{}$$ taking any ONB of V.
  • This is also defined as the Hilbert-Schmidt inner product $$\langle{}X,Y\rangle{} \sim \sum_i \langle{} X|i\rangle{} ,Y|i\rangle{} \rangle{} = \sum_i \langle{} i|X^\dagger{}Y|i\rangle{} = \text{tr}(X^\dagger{}Y)$$, in the matrix picture.
    • Doesn't have to be the diagonal in the computation basis per se, can be any arbitrary inner product, which we can see in the lemma $$\langle{} X,Y\rangle{} = \langle{}Y^\dagger{},X^\dagger{}\rangle{}$$ (proof in handwritten lecture notes).
  • Can be shown to be positive definite by considering $$\langle{}X,X\rangle{} \ge{} 0$$ where equality iff $$X=0$$.

We can use the unitary to define further:

\begin{align*} \langle{}X,Y\rangle{} &= \sum_i \langle{}Xw_i,Yw_i\rangle{} \\ &= \sum_i \langle{}XUw_i,YUw_i\rangle{} \\ &= \langle{}XU,YU\rangle{} \\ &= \langle{}U^\dagger{}Y^\dagger{},U^\dagger{}X^\dagger{}\rangle{} \\ &= \langle{}Y^\dagger{},UU^\dagger{}X^\dagger{}\rangle{} \\ &= \langle{}Y^\dagger{},X^\dagger{}\rangle{} \end{align*}

A list of names of students in the class. Scribing for lecture notes and a final exam.