MATH

Complex Numbers, Linear Algebra:

Historically, a ‘larger’ set of numbers (field extension) was needed to solve equations like \(x^2=-1\).

Definition: \(\mathbb C =\{a+ib|a,b \in \mathbb R\}\) is called the set of complex numbers where \(\mathbb R\) is the set of real numbers and \(i=\sqrt{-1}\).
Taking \(b=0\), one gets \(\mathbb{R} \subset \mathbb{C}\).

A natural one-to-one correspondence between \(\mathbb{C}\) and \(\mathbb{R}^2\) would be:\(a+ib\in \mathbb{C} \rightarrow (a,b) \in \mathbb{R}^2\). Therefore, elements of \(\mathbb{C}\) can be represented on the plane.

For now, picture vector spaces as real or complex vector spaces like a plane (\(\mathbb{C}\) or \(\mathbb{R}^2\)), or a 3 dimensional Euclidean space \(\mathbb{R}^3\) (the usual space).
For more details, look up the definitions of a vector space, a basis of a vector space, and try to prove thatR2is indeed a 2 dimensional vector space (over R).

So, if we take two lines, one numbered by real numbers (the real line) and one numbered by the pure imaginary numbers (multiples of i), and put them so that they intersect each other orthogonally, we get a basis {1, i} for the plane (that is equivalent to C).

Also, we know that there is a polar representation for any point z on a plane:\((a,b) \in \mathbb{R}^2\) \(\rightarrow\) \( r\cdot e^{i\phi}\) such that \(r^2=a^2+b^2\) and \(\phi \in [0,2\pi)\). Here, \(r \in [0,\infty)\) is the (Euclidean) length of the vector \(z\) denoted by \(|z|\). Pictorially, it means that a point on the plane can be determined by a rectangle going through it and centered on the origin or a circle going through it and centered on the origin.

Complex conjugate of a complex number is obtained by inserting -i instead of i. Namely, the complex conjugate of \(z=a+ib\) is \(\bar z =a-ib\).

Example: Show that \(|z|^2 = z\cdot \bar z\).
Solution: Let \(z=a+ib\), then \(|z|^2= a^2+b^2= (a^2 + b^2+ iab-iab)= (a+ib)\cdot(a-ib)\) = \(z\cdot \bar z\).

Bracket notation: Since now on, vectors are represented by \(|\rangle\) called “ket” and dual vectors are represented by \(\langle|\) “bra”.

Dual space is the space of all linear functionals on a vector space. Linear functionals are linear maps taking vectors of the space to its underlying field. So, they have field values. Therefore, applying the bra on the ket, one gets a value in the field (real or complex here). For quantum computing, we usually work with finite dimensional vector spaces. In this case (and not for infinite dimensions in general), the vector space and its dual have the same dimension (not obvious). So, they are isomorphic (there is a bijective bi-linear map between the vector space and its dual space). This map can be defined to be a map that takes a vector to its complex conjugate. In other words, if …. (fill it)

Eigenvalues and Eigenvectors:

The vectors invariant (up to a scaling) under a linear map are called eigenvectors. In other words, for a matrix \(A\), the non-zero vectors \(v\) that satisfy \(Av=\lambda \cdot v\) for some (complex) values \(\lambda\) are called eigenvectors. The values \(\lambda\) are called eigenvalues.

Therefore, we need to find vectors that satisfy \(Av=\lambda \cdot v\) or equivalently, \((A-\lambda I)\cdot v=0\). This is possible only if the matrix \((A-\lambda I)\) is not invertible, i.e. only if \(det\,(A-\lambda I)=0\).

Example (Y Pauli matrix): \(Y=\begin{bmatrix} 0&-i\\i&0\end{bmatrix}\). Then, the eigenvalues are found by solving \(det\,(Y-\lambda I)=det\,\begin{bmatrix} -\lambda&-i\\i&-\lambda\end{bmatrix}=0\). Namely, \(\lambda^2-1=0 \Rightarrow \lambda = \pm 1\). Now that we have the eigenvalues, finding the eigenvectors is just solving the defintion equation. …

Qubits and the Bloch Sphere:

It is postulated in quantum mechanics (the state-postulate) that states of a quantum particle |ψ> are unit vectors in a Hilbert space of dimension (?).

Entanglement and Multiplication spaces:
Concurrence test: E:=2|ad-bc|
If E=0, the two qubits are not entangled.

Reading Assignment:
1)Artur Ekert’s public lecture at the Royal Institute.
2) Artur Ekert’s lecture notes 1