Learning Linear Algebra through Diagrams #4: The Hilbert Space Formed by Matrices

List of articles in this series:
#1 Basics of Diagrams and Linear Algebra
#2 Spectral Decomposition and Singular Value Decomposition
#3 Tensor Products, Traces, Transposes, and Inner Products
#4 The Hilbert Space Formed by Matrices
Extra Edition: Differentiation with Column Vectors and Matrices
Extra Edition Part 2: Vector Analysis

Last time, we introduced cup column vectors and cap row vectors, and showed that using these allows for a natural representation of traces and other operations. This time, we will state that matrices can be converted into column vectors using cup column vectors (and the inverse conversion is also possible), allowing the set of all $${m \times n}$$ matrices, $${ \Complex^{m \times n} }$, to be viewed as an $${ mn }$-dimensional complex Hilbert space. We will also state that the set of all $${ n }$-th order Hermitian matrices (denoted as $${ \mathbf{Her}_n }$) can be viewed as an $${ n^2 }$-dimensional real Hilbert space, and that the set of all $${ n }$-th order positive semi-definite matrices (denoted as $${ \mathbf{Pos}_n }$) possesses a structure called a self-dual cone. These structures help in understanding the mathematical framework of quantum theory.

When a cup column vector is applied to a matrix $${ X }$, it becomes a column vector. This operation is called lowering. Lowering is an invertible mapping, and its inverse mapping is called raising. In diagrams, these operations can be performed simply by "connecting bent strings."

Note that an operation known as the vec operation is generally used to represent a matrix as a column vector. This operation can be defined as $${ \mathrm{vec} X \coloneqq (I_n \otimes X) \ket{\cup_n} }$. Lowering and the vec operation differ only in the ordering of components and have no essential difference. To be specific about the ordering of components, for a $${ 3 \times 2 }$$ matrix $${ X }$:

$$
X =
\begin{bmatrix}
X_{1,1} & X_{1,2} \\
X_{2,1} & X_{2,2} \\
X_{3,1} & X_{3,2} \\
\end{bmatrix}, \quad
\ket{X_\cup} =
\begin{bmatrix}
X_{1,1} \\
X_{1,2} \\
X_{2,1} \\
X_{2,2} \\
X_{3,1} \\
X_{3,2} \\
\end{bmatrix}, \quad
\mathrm{vec} X =
\begin{bmatrix}
X_{1,1} \\
X_{2,1} \\
X_{3,1} \\
X_{1,2} \\
X_{2,2} \\
X_{3,2} \\
\end{bmatrix}
$$

where $${ X_{i,j} }$$ is the component in the $${ i }$-th row and $${ j }$-th column of $${ X }$

In lowering, we can consider the input space to be replaced by the output space. Conversely, in raising, we can consider the output space to be replaced by the input space. Thus, it can be said that lowering and raising are operations that mutually convert inputs and outputs.

The decomposition represented by this proposition is called the Schmidt decomposition. The Schmidt decomposition can be considered the "lowering version of singular value decomposition."

It is immediately clear from the diagram that lowering $${ XYZ }$$ is equivalent to multiplying the lowering of $${ Y }$$ by $${ X \otimes Z^\mathrm{T} }$. Equations of this form often appear when dealing with CP (completely positive) maps in quantum theory. Note that for the vec operation, $${ \mathrm{vec} (XYZ) = (Z^\mathrm{T} \otimes X) \mathrm{vec} Y }$$ holds as well.

Since matrices can be freely converted into column vectors via lowering, it is immediately clear that the set of all $${ m \times n }$$ matrices, $${ \Complex^{m \times n} }$, can be viewed as a complex vector space, and its dimension is $${ mn }$. In fact, by considering the standard scalar multiplication and addition of matrices, it is easy to confirm that the definition of a vector space is satisfied. Furthermore, by introducing a natural inner product called the "Hilbert-Schmidt inner product," $${ \Complex^{m \times n} }$$ can be viewed as a complex Hilbert space. Note that this inner product appears frequently in quantum theory.

These are the basic properties regarding the Hilbert-Schmidt inner product of two positive semi-definite matrices.

These are the basic properties regarding the Hilbert-Schmidt inner product with a positive definite matrix.

$${ \mathbf{Her}_n }$$ can be viewed as an $${ n^2 }$-dimensional real Hilbert space (not a complex Hilbert space). In quantum theory, it is often more straightforward to consider this real Hilbert space rather than the complex Hilbert space $${ \Complex^{n \times n} }$.

$${ \mathbf{Pos}_n }$$ is a space with relatively clean properties called a self-dual cone (it is also a symmetric cone).

This is a geometric representation of $${ \mathbf{Pos}_n }$. In the case of $${ n=2 }$, it has the shape of a 4-dimensional version of a cone.

Positive semi-definite matrices and non-negative column vectors have similar properties. Note that states in quantum theory are represented by positive semi-definite matrices, while states in classical theory (classical probability theory) are represented by non-negative column vectors; therefore, these properties can be interpreted as being common to both quantum and classical theory.

This is an inequality that holds between the trace of a positive semi-definite matrix and the Hilbert-Schmidt inner product.

For a given set of projection matrices, the sum being a projection matrix is equivalent to them being mutually orthogonal.

For a positive semi-definite matrix $${ S }$$ and a projection matrix $${ P }$, $${ \mathrm{Tr}(PSP) ≤ \mathrm{Tr}(S) }$$ always holds.

Over the course of four articles, we have explained the basics of linear algebra useful for learning quantum theory. This concludes the series for now (though there are also extra editions). Thank you for reading.

In the series "Learning Quantum Theory Through Diagrams", we explain quantum theory using the diagrams introduced in this series, so please take a look at that as well if you are interested.

This article introduces a portion of the content from the book "Quantum Theory via Diagrams and Operational Probabilistic Theories" (there are also parts where this article provides a more detailed explanation).