Introduction to quantum computing: Complex vector space.

Sources: “Quantum computing for computer scientists”, N. Yanofsky and M. Mannucci, Cambridge Press, 2008.

This post is part of a (probably) long list of posts regarding quantum computing. In this post, we will delve into the basics of complex vector spaces.

Disclaimer: while the material below might seem too simple, I suggest not skimming over this post - personally, I always enjoy reading something that I believe I own, since επαναληψη μητηρ πασης μαθησεως (repetition is the mother of any knowledge).

The space of $\mathbb{C}^n$

States of quantum systems are described as vectors in the complex space. Thus, it is important, before we delve into the details of what is a quantum state, to have a crystal clear idea of what complex vectors are.

Let $\mathbf{c} \in \mathbb{C}^n$ denote a complex vector in $n$ complex dimensions, such that:

\[\mathbf{c} = \begin{bmatrix} a_1 + i b_1 \\ a_2 + i b_2 \\ \vdots \\ a_n + i b_n \\ \end{bmatrix}.\]

Observe that a complex vector is a vector where each entry is a complex number.

Let $\mathbf{c}_1$ and $\mathbf{c}_2$ be two vectors in such a $n$-dimensional complex space. The following hold:

\[x \cdot \mathbf{c} = (a_x + i b_x) \cdot \begin{bmatrix} a_1 + i b_1 \\ a_2 + i b_2 \\ \vdots \\ a_n + i b_n \\ \end{bmatrix} = \begin{bmatrix} (a_x + i b_x)(a_1 + i b_1) \\ (a_x + i b_x)(a_2 + i b_2) \\ \vdots \\ (a_x + i b_x)(a_n + i b_n) \\ \end{bmatrix}\]

Inner products

An inner product between two complex vectors, $\mathbf{c}_1 \in \mathbb{C}^n$ and $\mathbf{c}_2 \in \mathbb{C}^n$, is a bi-nary operation that takes two complex vectors as an input and give back a –possibly– complex scalar value. In math terms, we denote this operation as:

\[\left\langle \mathbf{c}_1, \mathbf{c}_2 \right \rangle = \mathbf{c}_1^\dagger \cdot \mathbf{c}_2 \in \mathbb{C}.\]

Some properties of the inner product are as follows:

The space of $\mathbb{C}^{m \times n}$

The above rules naturally generalize to matrix complex spaces $\mathbb{C}^{m \times n}$. Beyond the above:

\((\mathbf{C}^\top)_{jk} = (\mathbf{C})_{kj}\).

\((\overline{\mathbf{C}})_{jk} = \overline{(\mathbf{C})_{jk}}\).

\((\mathbf{C}^\dagger)_{jk} = \overline{(\mathbf{C})_{kj}}\).

Some other properties (the following hold also for conjugate operations):

Matrix multiplication is an operation of the form:

\[\mathbb{C}^{m \times n} \times \mathbb{C}^{n \times q} \mapsto \mathbb{C}^{m \times q}\]

where, for matrices $\mathbf{A} \in \mathbb{C}^{m \times n}$ and $\mathbf{B} \in \mathbb{C}^{n \times q}$, we have:

\[\left(\mathbf{A} \cdot \mathbf{B}\right)_{jk} = \sum_{h = 0}^{n-1} (\mathbf{A})_{jh} \cdot (\mathbf{B})_{hk}.\]

Matrix multiplication satisfies (beyond the standard associative, distributive, etc. rules):

Let us consider the square case $\mathbb{C}^{n \times n}$. Then, a matrix $\mathbf{C} \in \mathbb{C}^{n \times n}$ is called Hermitian if $\mathbf{C} = \mathbf{C}^\dagger$; this translates into an entrywise rule that

\((\mathbf{C})_{jk} = \overline{(\mathbf{C})_{kj}}\).

By definition, the entries on the diagonal of a Hermitian matrix are real.

If $\mathbf{C}$ is a Hermitian matrix, then all its eigenvalues are real. To see this, let $(\lambda, \mathbf{v})$ be an eigenvalue-eigenvector pair of $\mathbf{C}$. Then:

\[\lambda \left\langle \mathbf{v}, \mathbf{v} \right \rangle = \left\langle \lambda \mathbf{v}, \mathbf{v} \right \rangle = \left\langle \mathbf{C} \mathbf{v}, \mathbf{v} \right \rangle = \left\langle \mathbf{v}, \mathbf{C} \mathbf{v} \right \rangle = \left\langle \mathbf{v}, \lambda \mathbf{v} \right \rangle = \bar{\lambda}\left\langle \mathbf{v}, \mathbf{v} \right \rangle.\]

Hermitian matrices and their eigendecompositions play a critical role in quantum information and processing. As we will see later in our posts, every physical observable of a quantum system is associated with a Hermitian matrix. When we measure, the resulting state is represented as one of the eigenvectors of the corresponding Hermitian matrix.

One can define inner products beyond vectors: in the case of matrices, the inner product is defined as:

\[\left\langle \mathbf{C}_1, \mathbf{C}_2 \right \rangle = \text{Trace}\left(\mathbf{C}_1^\dagger \cdot \mathbf{C}_2\right) \in \mathbb{C}.\]

Finally, we will introduce the notion of an invertible matrix: a matrix $\mathbf{C} \in \mathbb{C}^{n \times n}$ is invertible, if there is a matrix $\mathbf{C}^{-1} \in \mathbb{C}^{n \times n}$ such that:

\[\mathbf{C} \cdot \mathbf{C}^{-1} = I.\]

An important class of matrices is that of unitary matrices. In that case, the inverse of a matrix is its adjoint. A fundamental property is the following:

Unitary matrices preserve inner products.

I.e., for a unitary matrix $\mathbf{U}$,

\[\left \langle \mathbf{U} \mathbf{C}_1, \mathbf{U} \mathbf{C_2} \right \rangle = \left \langle \mathbf{C}_1, \mathbf{C}_2 \right \rangle.\]

And, because unitary matrices preserve inner products, they also preserve norms.

Linear indepedence and basis of complex vectors

Let us focus on the complex space $\mathbb{C}^n$. A set of vectors $\mathbf{c}_1, \dots \mathbf{c}_n$ in $\mathbb{C}^n$ is linearly indepedent if:

\[w_1 \mathbf{c}_1 + w_2 \mathbf{c}_2 + \dots + w_n \mathbf{c}_n = 0,\]

implies that $w_i = 0, ~\forall i$.

In other words, any vector on that set cannot be described as a linear combination of the other vectors in that set.

The case of a basis of vectors is somewhat different:

A set of vectors $\mathbf{c}_1, \dots \mathbf{c}_n$ in $\mathbb{C}^n$ form a basis, if the vector set is linear independent and, at the same time, any other vector in $\mathbb{C}^n$ can be written as a linear combination of vectors that in that set.

The dimension of the complex vector space, defined by a basis, is equal to the number of vectors that define the basis.

Using the notion of orthogonality, we can extend the notion of a basis as follows:

A basis, comprised of vectors $n$-dimensional vectors $\mathbf{c}_1, \mathbf{c}_2, \dots, \mathbf{c}_n$, is an orthogonal basis in $n$-dimensions, if and only if, $\left \langle \mathbf{c}_i, \mathbf{c}_j \right\rangle, \forall i \neq j$. In the case where, in addition $\left \langle \mathbf{c}_i, \mathbf{c}_i \right \rangle = 1$, the basis is called orthonormal.

Hilbert space

A Hilbert space is a complex inner product space that is complete. I.e., it is accompanied with an inner product operation (e.g., in vector space $\mathbb{C}^n$, $\left \langle \mathbf{c}_1, \mathbf{c}_2 \right \rangle$, in matrix space $\mathbb{C}^{m \times n}$, $\left \langle \mathbf{C}_1, \mathbf{C}_2 \right \rangle$, etc.); it is complete in the sense that any sequence accumulating somewhere, converges to a point.

In our case –that of finite dimensional complex spaces– the inner products we consider so far are automatically complete. Thus, every space we defined above is a Hilbert space, and we don’t have to worry about it.

For the moment, we will skip the definition of tensor product, since it is a crucial notion by its own and I prefer to describe when we have already started discussing about quantum systems. Stay tuned!