Lecture Notes: Linear Algebra for Quantum Computing

Introduction

In this lecture, we embark on a detailed exploration of linear algebra, the fundamental mathematical language for quantum computing. Linear algebra provides the necessary tools to describe quantum systems, their states, and the operations performed on them. We will begin by introducing Dirac notation, also known as bra-ket notation, a highly convenient and widely adopted formalism for representing quantum states and operators. This notation simplifies many quantum mechanical expressions and facilitates calculations.

Following this, we will delve into the representation of quantum states as vectors within complex vector spaces. A cornerstone of quantum mechanics, the principle of superposition, will be examined in detail, highlighting how quantum states can exist as linear combinations of basis states. We will emphasize the crucial role of complex numbers as coefficients in these superpositions, a departure from classical systems described by real numbers.

Essential concepts such as the inner product in complex vector spaces will be thoroughly discussed. The inner product defines notions of orthogonality and normalization, which are critical for understanding probabilities and measurements in quantum mechanics. We will define orthogonality and orthonormal bases, which are essential for constructing and manipulating quantum states.

We will then formally introduce Hilbert spaces, the abstract vector spaces that serve as the mathematical arena for quantum mechanics. We will explore the defining properties of Hilbert spaces, including completeness and separability, and understand why these properties are necessary for a consistent quantum theory.

Furthermore, we will introduce linear operators, which represent physical transformations and measurements in quantum mechanics. Special attention will be given to projection operators, which play a vital role in describing quantum measurements. We will discuss how operators can be represented and composed, leading to the concept of operator multiplication. Finally, we will illustrate the important concept of non-commuting operators with the example of the Pauli operators \(\sigma_Z\) and \(\sigma_X\). The non-commutativity of certain operators is a uniquely quantum phenomenon with profound implications.

This lecture is designed to equip you with a robust foundation in the linear algebra formalism essential for navigating the concepts and techniques of quantum computing. By mastering these mathematical tools, you will be well-prepared to understand the more advanced topics in quantum computation that follow.

Linear Algebra for Quantum Computing

Dirac Notation: Bra-Ket Notation

In quantum mechanics, we employ a specialized and highly effective notation known as Dirac notation, or bra-ket notation, to represent quantum states and linear operators. Introduced by Paul Dirac, this notation offers significant advantages over traditional vector and matrix representations, particularly in simplifying complex expressions and highlighting the linear algebraic structure of quantum theory. Instead of denoting vectors with arrows (\(\vec{\psi}\)) or using standard column vector notation (\(\psi\)), Dirac notation uses kets to represent state vectors.

Definition 1 (Ket Notation). A ket, denoted as \(\left|{\psi}\right\rangle\), represents a column vector in a complex vector space, specifically a Hilbert space, and describes a quantum state \(\psi\). The label inside the ket, \(\psi\) in this case, is arbitrary and serves to identify the state. For instance, the computational basis states of a qubit are conventionally represented as \(\left|{0}\right\rangle\) and \(\left|{1}\right\rangle\).

The use of \(\left|{\psi}\right\rangle\) instead of \(\vec{\psi}\) or \(\psi\) might seem like a mere change in symbols, but it is profoundly more. Dirac notation is designed to work seamlessly with the mathematical operations of quantum mechanics, especially inner products and linear operators. The notation itself guides the mathematical manipulations and often makes complex quantum mechanical calculations more intuitive and less error-prone. We will see the full power of this notation as we progress.

Quantum States as Vectors: Superposition

A fundamental principle of quantum mechanics is superposition. Unlike classical systems that exist in definite states, quantum systems can exist in a combination of multiple states simultaneously. This is mathematically described by representing quantum states as vectors in a linear vector space, where a state can be a linear combination of basis states.

Definition 2 (Superposition). A quantum state \(\left|{\psi}\right\rangle\) can be expressed as a linear superposition of basis states. For a qubit, using the computational basis states \(\left|{0}\right\rangle\) and \(\left|{1}\right\rangle\), a general quantum state is written as: \[\left|{\psi}\right\rangle = \alpha \left|{0}\right\rangle + \beta \left|{1}\right\rangle\] where \(\alpha\) and \(\beta\) are complex numbers known as coefficients or amplitudes* of the superposition. These coefficients determine the contribution of each basis state to the overall quantum state.*

This concept of superposition is a departure from classical intuition. Classically, a bit is either 0 or 1. A qubit, however, can be in a state that is simultaneously \(\left|{0}\right\rangle\) and \(\left|{1}\right\rangle\), in a probabilistic sense dictated by the coefficients \(\alpha\) and \(\beta\). This superposition is not just a probabilistic mixture in the classical sense; it’s a coherent linear combination that allows for quantum interference and other uniquely quantum phenomena.

For a qubit, the state \(\left|{\psi}\right\rangle = \alpha \left|{0}\right\rangle + \beta \left|{1}\right\rangle\) is represented as a vector in a two-dimensional complex vector space. The coefficients \(\alpha\) and \(\beta\) are complex numbers, which are crucial for capturing the full richness of quantum mechanics, including phenomena like phase and interference.

Complex Vector Spaces

The coefficients in the linear superposition of quantum states are, in general, complex numbers. This necessitates the use of complex vector spaces as the natural setting for quantum mechanics. Working with complex numbers is not just a mathematical convenience; it is essential for describing physical reality at the quantum level.

Definition 3 (Complex Vector Space). A complex vector space* is a vector space where the scalars are complex numbers. In contrast to real vector spaces where scalars are real numbers, complex vector spaces allow for scalar multiplication by complex numbers. Quantum states reside in complex vector spaces, denoted as \(\mathbb{C}^n\) for \(n\)-dimensional spaces. For instance, a qubit state space is \(\mathbb{C}^2\).*

Consider a qubit state \(\left|{\psi}\right\rangle = \alpha \left|{0}\right\rangle + \beta \left|{1}\right\rangle\). Since \(\alpha\) and \(\beta\) are complex numbers, each has a real and an imaginary part. Thus, to fully specify this state, we need four real numbers: \(\alpha = a_1 + i a_2\) and \(\beta = b_1 + i b_2\), where \(a_1, a_2, b_1, b_2\) are real numbers and \(i = \sqrt{-1}\) is the imaginary unit.

We can represent the state \(\left|{\psi}\right\rangle\) as a column vector: \[\left|{\psi}\right\rangle = \begin{pmatrix} \alpha \\ \beta \end{pmatrix} = \begin{pmatrix} a_1 + i a_2 \\ b_1 + i b_2 \end{pmatrix}\] In general, a vector in an \(n\)-dimensional complex vector space \(\mathbb{C}^n\) is represented as a column vector with \(n\) complex numbers as entries. The operations of vector addition and scalar multiplication in \(\mathbb{C}^n\) are defined analogously to real vector spaces, but with complex scalars.

Inner Product in Complex Vector Spaces

To define notions of length, angle, and orthogonality in complex vector spaces, we need the concept of an inner product. The inner product in complex vector spaces is a generalization of the scalar product (or dot product) in real vector spaces, but it incorporates complex conjugation to ensure desirable properties, such as a positive definite norm.

Definition 4 (Inner Product in \(\mathbb{C}^n\)). For two vectors \(x = \begin{pmatrix} x_1 \\ \vdots \\ x_n \end{pmatrix}\) and \(y = \begin{pmatrix} y_1 \\ \vdots \\ y_n \end{pmatrix}\) in \(\mathbb{C}^n\), their inner product, denoted as \(\left\langle{x}\middle|{y}\right\rangle\), is defined as: \[\left\langle{x}\middle|{y}\right\rangle = \sum_{i=1}^{n} x_i^* y_i = x^\dagger y\] where \(x_i^*\) is the complex conjugate of \(x_i\), and \(x^\dagger\) denotes the conjugate transpose* (or Hermitian transpose) of the column vector \(x\). The conjugate transpose is obtained by taking the transpose of the vector and then taking the complex conjugate of each entry.*

The Hermitian transpose operation, denoted by \(\dagger\) or \(\dagger\), is crucial in defining the inner product in complex vector spaces. It combines transposition and complex conjugation. In Dirac notation, the bra \(\left\langle{x}\right|\) is defined to correspond to the Hermitian transpose of the ket \(\left|{x}\right\rangle\).

Definition 5 (Bra Notation). The bra, denoted as \(\left\langle{\phi}\right|\), represents the conjugate transpose of the ket \(\left|{\phi}\right\rangle\). Mathematically, \(\left\langle{\phi}\right| = \left|{\phi}\right\rangle^\dagger\). If \(\left|{\phi}\right\rangle = \begin{pmatrix} \phi_1 \\ \vdots \\ \phi_n \end{pmatrix}\), then the corresponding bra is the row vector \(\left\langle{\phi}\right| = \begin{pmatrix} \phi_1^* & \cdots & \phi_n^* \end{pmatrix}\).

Using bra-ket notation, the inner product of two kets \(\left|{x}\right\rangle\) and \(\left|{y}\right\rangle\) is compactly written as \(\left\langle{x}\middle|{y}\right\rangle = \left\langle{x}\right| \left|{y}\right\rangle\). This notation emphasizes that the inner product is formed by taking a bra (conjugate transpose of the first vector) and acting it on a ket (the second vector), resulting in a complex number.

Remark. Remark 1 (Non-Commutativity of Inner Product). Due to the complex conjugation in the definition, the inner product in complex vector spaces is not commutative, but rather exhibits conjugate symmetry: \[\left\langle{x}\middle|{y}\right\rangle = \left\langle{y}\middle|{x}\right\rangle^*\] This means that swapping the order of vectors in the inner product results in the complex conjugate of the original inner product. In the case of real vector spaces, the inner product is commutative, i.e., \(\left\langle{x}\middle|{y}\right\rangle = \left\langle{y}\middle|{x}\right\rangle\), because the complex conjugation has no effect on real numbers.

Orthogonality and Orthonormal Basis

The inner product allows us to define geometric concepts such as orthogonality and normalization in complex vector spaces, which are essential for describing quantum states and measurements.

Definition 6 (Orthogonality). Two vectors \(\left|{x}\right\rangle\) and \(\left|{y}\right\rangle\) in a complex vector space are said to be orthogonal* if their inner product is zero: \[\left\langle{x}\middle|{y}\right\rangle = 0\] Orthogonality is a generalization of the perpendicularity concept from Euclidean space. In quantum mechanics, orthogonal states are distinguishable and represent mutually exclusive outcomes of a measurement in some basis.*

Definition 7 (Orthonormal Basis). A set of vectors \(\{\left|{v_1}\right\rangle, \left|{v_2}\right\rangle, \dots, \left|{v_n}\right\rangle\}\) forms an orthonormal basis* for \(\mathbb{C}^n\) if it satisfies two conditions:*

1. Orthogonality: The vectors are mutually orthogonal, meaning the inner product of any two distinct vectors in the set is zero: \(\left\langle{v_i}\middle|{v_j}\right\rangle = 0\) for all \(i \neq j\).

2. Normalization: Each vector in the set is normalized, meaning the inner product of each vector with itself is one (unit length): \(\left\langle{v_i}\middle|{v_i}\right\rangle = 1\) for all \(i\).

These two conditions can be compactly combined using the Kronecker delta* symbol \(\delta_{ij}\): \[\left\langle{v_i}\middle|{v_j}\right\rangle = \delta_{ij} = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases}\] where \(\delta_{ij}\) is 1 if \(i=j\) and 0 if \(i \neq j\).*

Orthonormal bases are extremely useful because they provide a convenient coordinate system for vector spaces. Any vector in the space can be easily expressed as a linear combination of orthonormal basis vectors, and the coefficients in this combination are readily obtained using inner products.

Example 1 (Computational Basis). The computational basis \(\{\left|{0}\right\rangle, \left|{1}\right\rangle\}\) for a qubit, where \(\left|{0}\right\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}\) and \(\left|{1}\right\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}\), is a fundamental example of an orthonormal basis for the qubit space \(\mathbb{C}^2\). Let’s verify its orthonormality:

• Normalization: \[\left\langle{0}\middle|{0}\right\rangle = \begin{pmatrix} 1 & 0 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = 1 \cdot 1 + 0 \cdot 0 = 1\] \[\left\langle{1}\middle|{1}\right\rangle = \begin{pmatrix} 0 & 1 \end{pmatrix} \begin{pmatrix} 0 \\ 1 \end{pmatrix} = 0 \cdot 0 + 1 \cdot 1 = 1\]

• Orthogonality: \[\left\langle{0}\middle|{1}\right\rangle = \begin{pmatrix} 1 & 0 \end{pmatrix} \begin{pmatrix} 0 \\ 1 \end{pmatrix} = 1 \cdot 0 + 0 \cdot 1 = 0\] \[\left\langle{1}\middle|{0}\right\rangle = \begin{pmatrix} 0 & 1 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = 0 \cdot 1 + 1 \cdot 0 = 0\]

Thus, the computational basis \(\{\left|{0}\right\rangle, \left|{1}\right\rangle\}\) is indeed an orthonormal basis for \(\mathbb{C}^2\).

Linear Combinations and Linear Manifolds

As we have seen, superposition states are formed through linear combinations of basis states. Understanding linear combinations and the spaces they span, known as linear manifolds or subspaces, is crucial for working with quantum states.

Definition 8 (Linear Combination). A linear combination* of a set of vectors \(\{\left|{v_1}\right\rangle, \left|{v_2}\right\rangle, \dots, \left|{v_k}\right\rangle\}\) is a vector \(\left|{v}\right\rangle\) formed by summing scalar multiples of these vectors: \[\left|{v}\right\rangle = \sum_{i=1}^{k} c_i \left|{v_i}\right\rangle = c_1 \left|{v_1}\right\rangle + c_2 \left|{v_2}\right\rangle + \dots + c_k \left|{v_k}\right\rangle\] where \(c_1, c_2, \dots, c_k\) are complex coefficients.*

Linear combinations are fundamental to the vector space structure. They allow us to construct new vectors from a given set of vectors. In quantum mechanics, superposition states are precisely linear combinations of basis states.

Definition 9 (Linear Manifold). The linear manifold* (or linear subspace) spanned by a set of vectors \(S = \{\left|{v_1}\right\rangle, \left|{v_2}\right\rangle, \dots, \left|{v_k}\right\rangle\}\) is the set of all possible linear combinations of vectors in \(S\). In other words, it is the set of all vectors that can be expressed in the form \(\sum_{i=1}^{k} c_i \left|{v_i}\right\rangle\) for any choice of complex coefficients \(c_1, c_2, \dots, c_k\).*

A linear manifold is a subspace within the larger vector space that is "closed" under linear combinations. If you take any two vectors in a linear manifold and form a linear combination of them, the resulting vector will also be within the same linear manifold. For example, the set of all superposition states of a qubit is the linear manifold spanned by the basis states \(\left|{0}\right\rangle\) and \(\left|{1}\right\rangle\), which is the entire qubit space \(\mathbb{C}^2\).

This section has provided a review of essential linear algebra concepts, tailored for their application in quantum computing. We have emphasized the use of complex numbers, Dirac notation, inner products, orthogonality, orthonormal bases, and linear combinations. These concepts form the bedrock for understanding Hilbert spaces and linear operators, which we will explore in the subsequent sections.

Hilbert Space: The Space of Quantum Systems

In quantum mechanics, the state of any quantum system is represented by a vector in a complex vector space called a Hilbert space. Hilbert space provides the mathematical framework for describing quantum states, observables, and their evolution. It generalizes the familiar Euclidean space to spaces that can be finite or infinite-dimensional and are equipped with an inner product that allows for the definition of geometric concepts like distance and angle.

Definition of Hilbert Space

Definition 12 (Hilbert Space). A Hilbert space* \(\mathcal{H}\) is a complex vector space equipped with an inner product \(\left\langle{\cdot}\middle|{\cdot}\right\rangle\) that satisfies a specific set of axioms, making it a complete inner product space. Formally, a Hilbert space must satisfy the following properties:*

1. Linear Space over \(\mathbb{C}\): \(\mathcal{H}\) is a vector space over the field of complex numbers \(\mathbb{C}\). This means that for any two vectors \(\left|{\psi}\right\rangle, \left|{\phi}\right\rangle \in \mathcal{H}\) and any complex numbers \(a, b \in \mathbb{C}\), their linear combination \(a\left|{\psi}\right\rangle + b\left|{\phi}\right\rangle\) is also a vector in \(\mathcal{H}\). This closure under linear combinations is fundamental to the principle of superposition in quantum mechanics.

2. Inner Product: \(\mathcal{H}\) is equipped with an inner product, denoted by \(\left\langle{\cdot}\middle|{\cdot}\right\rangle : \mathcal{H} \times \mathcal{H} \to \mathbb{C}\), which maps pairs of vectors to complex numbers and satisfies the following axioms for all vectors \(\left|{\psi}\right\rangle, \left|{\phi}\right\rangle, \left|{\chi}\right\rangle \in \mathcal{H}\) and scalars \(a, b \in \mathbb{C}\):

   1. Conjugate Symmetry (Hermitian symmetry): \(\left\langle{\psi}\middle|{\phi}\right\rangle = \left\langle{\phi}\middle|{\psi}\right\rangle^*\). The inner product of \(\left|{\psi}\right\rangle\) with \(\left|{\phi}\right\rangle\) is the complex conjugate of the inner product of \(\left|{\phi}\right\rangle\) with \(\left|{\psi}\right\rangle\). This is a generalization of symmetry for real vector spaces.

   2. Positive-Definiteness: \(\left\langle{\psi}\middle|{\psi}\right\rangle \geq 0\), and \(\left\langle{\psi}\middle|{\psi}\right\rangle = 0\) if and only if \(\left|{\psi}\right\rangle = \left|{0}\right\rangle\), where \(\left|{0}\right\rangle\) is the zero vector in \(\mathcal{H}\). The inner product of a vector with itself is a non-negative real number, and it is zero only for the zero vector. This property ensures that the norm (length) of any non-zero vector is positive.

   3. Linearity in the second argument: \(\left\langle{\psi}\middle|{a\left|{\phi}\right\rangle + b\left|{\chi}\right\rangle}\right\rangle = a\left\langle{\psi}\middle|{\phi}\right\rangle + b\left\langle{\psi}\middle|{\chi}\right\rangle\). The inner product is linear with respect to scalar multiplication and vector addition in the second argument (ket). Combined with conjugate symmetry, this implies anti-linearity in the first argument (bra): \(\left\langle{a\left|{\phi}\right\rangle + b\left|{\chi}\right\rangle}\middle|{\psi}\right\rangle = a^*\left\langle{\phi}\middle|{\psi}\right\rangle + b^*\left\langle{\chi}\middle|{\psi}\right\rangle\).

3. Completeness: \(\mathcal{H}\) is complete* with respect to the norm induced by the inner product. This is a more technical condition related to convergence. It means that every Cauchy sequence of vectors in \(\mathcal{H}\) converges to a limit that is also within \(\mathcal{H}\).*

   Definition 10 (Cauchy Sequence). A sequence of vectors \(\{\left|{\psi_n}\right\rangle\}_{n=1}^\infty\) in \(\mathcal{H}\) is a Cauchy sequence if for every real number \(\epsilon > 0\), there exists an integer \(M\) such that for all integers \(i, j > M\), the distance between \(\left|{\psi_i}\right\rangle\) and \(\left|{\psi_j}\right\rangle\) is less than \(\epsilon\), i.e., \(\|\left|{\psi_i}\right\rangle - \left|{\psi_j}\right\rangle\| < \epsilon\), where the norm \(\|\left|{\psi}\right\rangle\| = \sqrt{\left\langle{\psi}\middle|{\psi}\right\rangle}\) is derived from the inner product.**

   Completeness ensures that the Hilbert space is "not missing any points" in a certain sense. For practical purposes, it guarantees that if you have a sequence of quantum states that are getting closer and closer to each other, the state they are approaching is also a valid quantum state within the Hilbert space.

4. Separability: \(\mathcal{H}\) is separable, meaning it contains a countable dense subset.

   Definition 11 (Dense Subset). A subset \(D \subseteq \mathcal{H}\) is dense in \(\mathcal{H}\) if every vector in \(\mathcal{H}\) can be approximated arbitrarily closely by a vector in \(D\). That is, for every vector \(\left|{\psi}\right\rangle \in \mathcal{H}\) and every \(\epsilon > 0\), there exists a vector \(\left|{\phi}\right\rangle \in D\) such that \(\|\left|{\psi}\right\rangle - \left|{\phi}\right\rangle\| < \epsilon\).**

   Separability implies that there exists a countable set of vectors \(\{\left|{\phi_i}\right\rangle\}_{i \in \mathbb{N}}\) in \(\mathcal{H}\) such that any vector in \(\mathcal{H}\) can be approximated as closely as desired by a vector from this countable set. For finite-dimensional Hilbert spaces, separability is automatically satisfied. For infinite-dimensional spaces, this condition is important for technical reasons, ensuring that we can work with countable bases in some sense. For example, the set of rational numbers is a countable dense subset of the real numbers \(\mathbb{R}\).

Properties of Hilbert Spaces

The defining properties of Hilbert spaces ensure that they are well-behaved and suitable for the mathematical formulation of quantum mechanics. These properties have profound implications for the structure and interpretation of quantum theory.

Linear Space Structure

The requirement that a Hilbert space is a linear space over complex numbers is fundamental because it directly reflects the principle of superposition in quantum mechanics. Quantum states can be added together and multiplied by complex scalars to form new quantum states. This linear structure allows for the representation of quantum phenomena such as interference and entanglement.

Inner Product and Geometric Interpretation

The inner product is what turns a complex vector space into a Hilbert space. It introduces a notion of geometry, allowing us to define:

• Norm (Length): The norm of a vector \(\left|{\psi}\right\rangle\), denoted as \(\|\left|{\psi}\right\rangle\|\), is defined as \(\|\left|{\psi}\right\rangle\| = \sqrt{\left\langle{\psi}\middle|{\psi}\right\rangle}\). It represents the "length" or magnitude of the quantum state vector. In quantum mechanics, for physically realizable states, we often require states to be normalized, i.e., \(\|\left|{\psi}\right\rangle\| = 1\).

• Orthogonality (Angles): The inner product defines the "angle" between two vectors. Orthogonal vectors (inner product is zero) are considered "perpendicular". In quantum mechanics, orthogonal states represent distinguishable states, which is crucial for measurement theory.

• Probability Amplitudes: The inner product \(\left\langle{\phi}\middle|{\psi}\right\rangle\) is often interpreted as a probability amplitude. For example, if \(\{\left|{\phi_i}\right\rangle\}\) is an orthonormal basis, and \(\left|{\psi}\right\rangle = \sum_i c_i \left|{\phi_i}\right\rangle\), then \(c_i = \left\langle{\phi_i}\middle|{\psi}\right\rangle\), and \(|c_i|^2 = |\left\langle{\phi_i}\middle|{\psi}\right\rangle|^2\) represents the probability of finding the system in state \(\left|{\phi_i}\right\rangle\) when the system is in state \(\left|{\psi}\right\rangle\) upon measurement in the basis \(\{\left|{\phi_i}\right\rangle\}\).

Completeness and Convergence

The completeness property is essential for mathematical rigor, especially when dealing with infinite-dimensional Hilbert spaces, which are necessary to describe systems with continuous degrees of freedom (e.g., position and momentum). Completeness ensures that mathematical operations that involve limits, such as infinite sums or integrals, are well-defined within the Hilbert space. For example, when considering the time evolution of quantum states, which can be described by differential equations, completeness guarantees that solutions remain within the Hilbert space.

Separability and Countable Basis

Separability, while being a more technical requirement, has important implications, particularly for infinite-dimensional Hilbert spaces. It ensures that even in infinite-dimensional spaces, there exists at least a countable dense subset, which implies the existence of a countable orthonormal basis (if the Hilbert space is separable and infinite-dimensional). This is often convenient because it allows us to work with sequences and sums rather than dealing with uncountable sets, simplifying many mathematical treatments. For finite-dimensional Hilbert spaces, such as those used to describe systems with a finite number of qubits, separability is automatically guaranteed.

Remark. Remark 2. In the context of quantum computing with a finite number of qubits, we primarily work with finite-dimensional Hilbert spaces, which are isomorphic to \(\mathbb{C}^{2^n}\) for \(n\) qubits. In these finite-dimensional cases, the completeness and separability conditions are always satisfied and are less of a practical concern in computations. However, they are fundamental theoretical requirements that ensure the mathematical consistency of quantum mechanics. The first two properties—being a linear space over \(\mathbb{C}\) and having an inner product—are constantly and directly used in quantum computing and quantum mechanics.

Linear Operators in Quantum Mechanics

In quantum mechanics, linear operators are mathematical entities that act on quantum states, transforming one state into another within the same Hilbert space. These operators are essential for describing physical observables, quantum gates, and the dynamics of quantum systems. They are linear transformations that preserve the vector space structure of the Hilbert space.

Definition of Linear Operators

Definition 13 (Linear Operator). A linear operator* \(A\) on a Hilbert space \(\mathcal{H}\) is a mapping \(A: \mathcal{H} \to \mathcal{H}\) that satisfies the principle of linearity. For any vectors \(\left|{\psi}\right\rangle, \left|{\phi}\right\rangle \in \mathcal{H}\) and any complex scalars \(a, b \in \mathbb{C}\), the action of \(A\) must obey the superposition principle: \[A(a\left|{\psi}\right\rangle + b\left|{\phi}\right\rangle) = a(A\left|{\psi}\right\rangle) + b(A\left|{\phi}\right\rangle)\] This definition ensures that the operator respects linear combinations, a fundamental aspect of quantum mechanics.*

Linear operators are the quantum analogue of matrices in finite-dimensional spaces. In fact, in a finite-dimensional Hilbert space, every linear operator can be represented as a matrix once a basis is chosen. Applying a linear operator to a quantum state is analogous to multiplying a matrix by a vector.

Notable Operators

Several types of linear operators are particularly significant in quantum mechanics and quantum computing. We will introduce a few fundamental ones here.

Identity Operator

Definition 14 (Identity Operator). The identity operator, denoted by \(\mathbb{I}\) (sometimes also by \(I\) or \(1\)), is the operator that leaves every vector in the Hilbert space unchanged. For any quantum state \(\left|{\psi}\right\rangle \in \mathcal{H}\), the identity operator acts as: \[\mathbb{I}\left|{\psi}\right\rangle = \left|{\psi}\right\rangle\] The identity operator is analogous to the number 1 in multiplication; it is the neutral element for operator multiplication.

In matrix form, the identity operator in an \(n\)-dimensional space is represented by the \(n \times n\) identity matrix, which has ones on the main diagonal and zeros elsewhere.

Zero Operator

Definition 15 (Zero Operator). The zero operator, denoted by \(\hat{0}\) (or simply \(0\) when the context is clear), is the operator that maps every vector in the Hilbert space to the zero vector \(\left|{0}\right\rangle\). For any quantum state \(\left|{\psi}\right\rangle \in \mathcal{H}\): \[\hat{0}\left|{\psi}\right\rangle = \left|{0}\right\rangle\] The zero operator is the additive identity in the space of linear operators.

In matrix form, the zero operator is represented by the zero matrix, where all entries are zero.

Projection Operator

Projection operators are of paramount importance in quantum mechanics, especially in the context of quantum measurements. They project a quantum state onto a specific subspace or direction within the Hilbert space.

Definition and Properties of Projection Operators

Definition 16 (Projection Operator). Given a normalized vector \(\left|{u}\right\rangle \in \mathcal{H}\) (i.e., \(\|\left|{u}\right\rangle\| = 1\)), the projection operator \(P_{\left|{u}\right\rangle}\) onto the direction defined by \(\left|{u}\right\rangle\) is defined as: \[P_{\left|{u}\right\rangle} \left|{\psi}\right\rangle = \left\langle{u}\middle|{\psi}\right\rangle \left|{u}\right\rangle\] for any vector \(\left|{\psi}\right\rangle \in \mathcal{H}\). The operator \(P_{\left|{u}\right\rangle}\) takes an arbitrary state \(\left|{\psi}\right\rangle\) and projects it onto the state \(\left|{u}\right\rangle\). The resulting vector is always in the direction of \(\left|{u}\right\rangle\), and its magnitude is given by the scalar coefficient \(\left\langle{u}\middle|{\psi}\right\rangle\), which is the inner product of \(\left|{u}\right\rangle\) and \(\left|{\psi}\right\rangle\).

Geometrically, the projection operator \(P_{\left|{u}\right\rangle}\) finds the component of \(\left|{\psi}\right\rangle\) that lies along the direction of \(\left|{u}\right\rangle\). If \(\left|{\psi}\right\rangle\) already lies along \(\left|{u}\right\rangle\), then \(P_{\left|{u}\right\rangle} \left|{\psi}\right\rangle = \left|{\psi}\right\rangle\). If \(\left|{\psi}\right\rangle\) is orthogonal to \(\left|{u}\right\rangle\), then \(P_{\left|{u}\right\rangle} \left|{\psi}\right\rangle = \left|{0}\right\rangle\).

Proposition 1 (Idempotency of Projection Operators). Projection operators are idempotent, meaning that applying a projection operator twice is the same as applying it once. A key property of projection operators is that they are idempotent. Applying a projection operator twice is equivalent to applying it once: \[P_{\left|{u}\right\rangle}^2 = P_{\left|{u}\right\rangle}\]

Proof. Proof. To prove idempotency, we apply \(P_{\left|{u}\right\rangle}^2\) to an arbitrary vector \(\left|{\psi}\right\rangle\) and show that the result is the same as applying \(P_{\left|{u}\right\rangle} \left|{\psi}\right\rangle\): \[\begin{aligned}P_{\left|{u}\right\rangle}^2 \left|{\psi}\right\rangle &= P_{\left|{u}\right\rangle} (P_{\left|{u}\right\rangle} \left|{\psi}\right\rangle) & \text{By definition of } P_{\left|{u}\right\rangle}^2 \\&= P_{\left|{u}\right\rangle} (\left\langle{u}\middle|{\psi}\right\rangle \left|{u}\right\rangle) & \text{Action of } P_{\left|{u}\right\rangle} \text{ on } \left|{\psi}\right\rangle \\&= \left\langle{u}\middle|{\psi}\right\rangle (P_{\left|{u}\right\rangle} \left|{u}\right\rangle) & \text{Linearity of } P_{\left|{u}\right\rangle} \text{ and scalar } \left\langle{u}\middle|{\psi}\right\rangle \\&= \left\langle{u}\middle|{\psi}\right\rangle (\left\langle{u}\middle|{u}\right\rangle \left|{u}\right\rangle) & \text{Action ofP_{\left|{u}\right\rangle} \text{ on } \left|{u}\right\rangle\end{aligned}\] Since \(\left|{u}\right\rangle\) is normalized, \(\left\langle{u}\middle|{u}\right\rangle = \|\left|{u}\right\rangle\|^2 = 1\). Therefore, \[P_{\left|{u}\right\rangle}^2 \left|{\psi}\right\rangle = \left\langle{u}\middle|{\psi}\right\rangle (1 \cdot \left|{u}\right\rangle) = \left\langle{u}\middle|{\psi}\right\rangle \left|{u}\right\rangle = P_{\left|{u}\right\rangle} \left|{\psi}\right\rangle\] Since \(P_{\left|{u}\right\rangle}^2 \left|{\psi}\right\rangle = P_{\left|{u}\right\rangle} \left|{\psi}\right\rangle\) for all vectors \(\left|{\psi}\right\rangle \in \mathcal{H}\), we conclude that \(P_{\left|{u}\right\rangle}^2 = P_{\left|{u}\right\rangle}\). This property reflects the fact that once a vector is projected onto a subspace, projecting it again onto the same subspace does not change the vector further. ◻

Projection Operator in Bra-Ket Notation

In bra-ket notation, the projection operator \(P_{\left|{u}\right\rangle}\) onto a normalized vector \(\left|{u}\right\rangle\) has a particularly simple and elegant form: \[P_{\left|{u}\right\rangle} = \left|{u}\middle\rangle\middle\langle{u}\right|\] Here, \(\left|{u}\middle\rangle\middle\langle{u}\right|\) is the outer product of the vector \(\left|{u}\right\rangle\) with itself. Applying this operator to a vector \(\left|{\psi}\right\rangle\) yields: \[P_{\left|{u}\right\rangle} \left|{\psi}\right\rangle = \left|{u}\middle\rangle\middle\langle{u}\right| \left|{\psi}\right\rangle = \left|{u}\right\rangle (\left\langle{u}\right| \left|{\psi}\right\rangle) = \left|{u}\right\rangle \left\langle{u}\middle|{\psi}\right\rangle = \left\langle{u}\middle|{\psi}\right\rangle \left|{u}\right\rangle\] This bra-ket notation for projection operators is highly convenient for calculations and theoretical manipulations in quantum mechanics. It directly expresses the projection in terms of the state vector \(\left|{u}\right\rangle\) onto which we are projecting.

Completeness Relation and Resolution of Identity

For any orthonormal basis of a Hilbert space, the sum of the projection operators onto each basis vector yields a fundamental result known as the completeness relation or resolution of the identity. This relation expresses the identity operator in terms of projection operators.

Proposition 2 (Completeness Relation (Resolution of Identity)). The sum of projection operators onto each basis vector of an orthonormal basis equals the identity operator. Let \(\{\left|{v_1}\right\rangle, \left|{v_2}\right\rangle, \dots, \left|{v_n}\right\rangle\}\) be an orthonormal basis for an \(n\)-dimensional Hilbert space \(\mathcal{H}\). Then the sum of the projection operators onto each basis vector is equal to the identity operator on \(\mathcal{H}\): \[\sum_{i=1}^{n} P_{\left|{v_i}\right\rangle} = \sum_{i=1}^{n} \left|{v_i}\middle\rangle\middle\langle{v_i}\right| = \mathbb{I}\] This means that the identity operator can be decomposed into a sum of projections onto the basis states.

Proof. Proof. Let \(\{\left|{v_i}\right\rangle\}_{i=1}^n\) be an orthonormal basis for \(\mathcal{H}\). Any vector \(\left|{\psi}\right\rangle \in \mathcal{H}\) can be uniquely expressed as a linear combination of these basis vectors: \(\left|{\psi}\right\rangle = \sum_{i=1}^{n} c_i \left|{v_i}\right\rangle\), where the coefficients are given by \(c_i = \left\langle{v_i}\middle|{\psi}\right\rangle\). Now, consider the action of the operator \(\sum_{i=1}^{n} \left|{v_i}\middle\rangle\middle\langle{v_i}\right|\) on an arbitrary vector \(\left|{\psi}\right\rangle\): \[\begin{aligned}\left( \sum_{i=1}^{n} \left|{v_i}\middle\rangle\middle\langle{v_i}\right| \right) \left|{\psi}\right\rangle &= \sum_{i=1}^{n} \left|{v_i}\middle\rangle\middle\langle{v_i}\right| \left|{\psi}\right\rangle & \text{Definition of sum of operators} \\&= \sum_{i=1}^{n} \left|{v_i}\middle\rangle\middle\langle{v_i}\right| \left|{\psi}\right\rangle & \text{Definition of } \left|{v_i}\middle\rangle\middle\langle{v_i}\right| \\&= \sum_{i=1}^{n} \left|{v_i}\right\rangle \left\langle{v_i}\middle|{\psi}\right\rangle & \text{Action of } \left|{v_i}\middle\rangle\middle\langle{v_i}\right| \\&= \sum_{i=1}^{n} \left\langle{v_i}\middle|{\psi}\right\rangle \left|{v_i}\right\rangle & \text{Rearranging scalars} \\&= \sum_{i=1}^{n} c_i \left|{v_i}\right\rangle & \text{Since } c_i = \left\langle{v_i}\middle|{\psi}\right\rangle \\&= \left|{\psi}\right\rangle & \text{Expansion of } \left|{\psi}\right\rangle \text{ in basis } \{\left|{v_i}\right\rangle\}\end{aligned}\] Thus, we have shown that for any vector \(\left|{\psi}\right\rangle \in \mathcal{H}\), \(\left( \sum_{i=1}^{n} \left|{v_i}\middle\rangle\middle\langle{v_i}\right| \right) \left|{\psi}\right\rangle = \left|{\psi}\right\rangle = \mathbb{I}\left|{\psi}\right\rangle\). This implies that the operator \(\sum_{i=1}^{n} \left|{v_i}\middle\rangle\middle\langle{v_i}\right|\) acts identically to the identity operator on all vectors in \(\mathcal{H}\), and therefore, \(\sum_{i=1}^{n} \left|{v_i}\middle\rangle\middle\langle{v_i}\right| = \mathbb{I}\). ◻

The completeness relation is a powerful tool. It shows how the identity operator, which does nothing, can be built up from projection operators, which pick out components along specific directions. This relation is frequently used in quantum mechanics to decompose states and operators in terms of basis states and to simplify calculations.

Example: Projection onto a Qubit State

Let’s illustrate projection operators with a concrete example in the context of qubits. We will consider projections onto superposition states of a qubit.

Computational Basis and Superposition States Revisited

Recall the computational basis for a qubit: \(\{\left|{0}\right\rangle, \left|{1}\right\rangle\} = \left\{ \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \end{pmatrix} \right\}\). We also defined the superposition states \(\left|{+}\right\rangle\) and \(\left|{-}\right\rangle\), which are also important basis states for qubits: \[\left|{+}\right\rangle = \frac{1}{\sqrt{2}} (\left|{0}\right\rangle + \left|{1}\right\rangle) = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix}, \quad\left|{-}\right\rangle = \frac{1}{\sqrt{2}} (\left|{0}\right\rangle - \left|{1}\right\rangle) = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -1 \end{pmatrix}\] We have already verified that \(\{\left|{+}\right\rangle, \left|{-}\right\rangle\}\) is also an orthonormal basis for the qubit space \(\mathbb{C}^2\).

Projecting the Basis State \(\left|{0}\right\rangle\) onto the Superposition State \(\left|{+}\right\rangle\)

Let’s calculate the projection of the computational basis state \(\left|{0}\right\rangle\) onto the superposition state \(\left|{+}\right\rangle\). We need to apply the projection operator \(P_{\left|{+}\right\rangle} = \left|{+}\middle\rangle\middle\langle{+}\right|\) to the state \(\left|{0}\right\rangle\): \[\begin{aligned}P_{\left|{+}\right\rangle} \left|{0}\right\rangle &= \left|{+}\middle\rangle\middle\langle{+}\right| \left|{0}\right\rangle = \left|{+}\right\rangle \left\langle{+}\middle|{0}\right\rangle\end{aligned}\] First, we compute the inner product \(\left\langle{+}\middle|{0}\right\rangle\): \[\left\langle{+}\middle|{0}\right\rangle = \left\langle{+}\right| \left|{0}\right\rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \frac{1}{\sqrt{2}} (1 \cdot 1 + 1 \cdot 0) = \frac{1}{\sqrt{2}}\] Now, we can find the projection: \[P_{\left|{+}\right\rangle} \left|{0}\right\rangle = \left\langle{+}\middle|{0}\right\rangle \left|{+}\right\rangle = \frac{1}{\sqrt{2}} \left|{+}\right\rangle = \frac{1}{\sqrt{2}} \left( \frac{1}{\sqrt{2}} (\left|{0}\right\rangle + \left|{1}\right\rangle) \right) = \frac{1}{2} (\left|{0}\right\rangle + \left|{1}\right\rangle) = \frac{1}{2} \left|{0}\right\rangle + \frac{1}{2} \left|{1}\right\rangle\] So, the projection of \(\left|{0}\right\rangle\) onto \(\left|{+}\right\rangle\) is \(\frac{1}{\sqrt{2}}\) times the state \(\left|{+}\right\rangle\), or equivalently, \(\frac{1}{2} (\left|{0}\right\rangle + \left|{1}\right\rangle)\). The scalar coefficient of the projection is \(\left\langle{+}\middle|{0}\right\rangle = \frac{1}{\sqrt{2}}\). The squared magnitude \(|\left\langle{+}\middle|{0}\right\rangle|^2 = \left| \frac{1}{\sqrt{2}} \right|^2 = \frac{1}{2}\) represents the probability of obtaining the state \(\left|{+}\right\rangle\) when measuring a system in state \(\left|{0}\right\rangle\) in the basis \(\{\left|{+}\right\rangle, \left|{-}\right\rangle\}\) (though this interpretation requires more context from quantum measurement theory).

Verifying Orthonormality of \(\left|{+}\right\rangle\) and \(\left|{-}\right\rangle\) States

As a reminder and for completeness, let’s re-verify that the superposition states \(\left|{+}\right\rangle\) and \(\left|{-}\right\rangle\) form an orthonormal set, and thus an orthonormal basis for the qubit space. We need to check normalization and orthogonality.

• Normalization: \[\left\langle{+}\middle|{+}\right\rangle = \left\langle{+}\right| \left|{+}\right\rangle = \left( \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \end{pmatrix} \right) \left( \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix} \right) = \frac{1}{2} (1 \cdot 1 + 1 \cdot 1) = \frac{1}{2} \cdot 2 = 1\] \[\left\langle{-}\middle|{-}\right\rangle = \left\langle{-}\right| \left|{-}\right\rangle = \left( \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & -1 \end{pmatrix} \right) \left( \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -1 \end{pmatrix} \right) = \frac{1}{2} (1 \cdot 1 + (-1) \cdot (-1)) = \frac{1}{2} \cdot 2 = 1\] Both \(\left|{+}\right\rangle\) and \(\left|{-}\right\rangle\) are normalized to 1.

• Orthogonality: \[\begin{aligned} \left\langle{+}\middle|{-}\right\rangle &= \left\langle{+}\right| \left|{-}\right\rangle = \left( \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \end{pmatrix} \right) \left( \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -1 \end{pmatrix} \right) = \frac{1}{2} (1 \cdot 1 + 1 \cdot (-1)) = \frac{1}{2} (1 - 1) = 0 \end{aligned}\] Thus, \(\left|{+}\right\rangle\) and \(\left|{-}\right\rangle\) are orthogonal to each other.

Since \(\left|{+}\right\rangle\) and \(\left|{-}\right\rangle\) are both normalized and orthogonal, they form an orthonormal set. As there are two of them in a 2-dimensional space, they form an orthonormal basis for the qubit space \(\mathbb{C}^2\).

Operator Representation and Composition

In quantum mechanics, linear operators are essential for describing transformations and measurements of quantum states. To perform calculations and understand the effects of these operators, we need ways to represent them mathematically and to understand how they combine.

Representing Operators via Action on Basis Vectors

The behavior of a linear operator on any arbitrary vector is completely determined by its action on a set of basis vectors. If we know how an operator transforms each basis vector, we can deduce its action on any linear combination of these basis vectors, thanks to linearity.

Let \(\{\left|{d_1}\right\rangle, \left|{d_2}\right\rangle, \dots, \left|{d_n}\right\rangle\}\) be an orthonormal basis for an \(n\)-dimensional complex vector space \(\mathbb{C}^n\). Any vector \(\left|{\psi}\right\rangle \in \mathbb{C}^n\) can be uniquely expressed as a linear combination of these basis vectors: \(\left|{\psi}\right\rangle = \sum_{j=1}^{n} c_j \left|{d_j}\right\rangle\), where \(c_j = \left\langle{d_j}\middle|{\psi}\right\rangle\) are the complex coefficients.

The action of a linear operator \(A\) on \(\left|{\psi}\right\rangle\) is then given by: \[A\left|{\psi}\right\rangle = A \left( \sum_{j=1}^{n} c_j \left|{d_j}\right\rangle \right) = \sum_{j=1}^{n} c_j (A\left|{d_j}\right\rangle)\] By linearity, we just need to know the action of \(A\) on each basis vector \(\left|{d_j}\right\rangle\) to determine its action on any vector \(\left|{\psi}\right\rangle\). Since \(A\left|{d_j}\right\rangle\) is also a vector in \(\mathbb{C}^n\), it can itself be expanded in terms of the basis \(\{\left|{d_i}\right\rangle\}\): \[A\left|{d_j}\right\rangle = \sum_{i=1}^{n} A_{ij} \left|{d_i}\right\rangle\] where \(A_{ij}\) are complex numbers. These coefficients \(A_{ij}\) are the matrix elements of the operator \(A\) in the basis \(\{\left|{d_i}\right\rangle\}\).

Matrix Representation of Operators

The set of coefficients \(\{A_{ij}\}\) for \(i, j = 1, \dots, n\) forms an \(n \times n\) matrix \(M_A\), which is the matrix representation of the operator \(A\) in the basis \(\{\left|{d_i}\right\rangle\}\). The entry in the \(i\)-th row and \(j\)-th column of \(M_A\) is \(A_{ij}\).

We can find the matrix elements \(A_{ij}\) by using the inner product. Taking the inner product of \(\left|{d_i}\right\rangle\) with \(A\left|{d_j}\right\rangle\): \[\left\langle{d_i}\middle|{A\left|{d_j}\right\rangle}\right\rangle = \left\langle{d_i}\middle|{\sum_{k=1}^{n} A_{kj} \left|{d_k}\right\rangle}\right\rangle\] Using the linearity of the inner product in the second argument, we get: \[\left\langle{d_i}\middle|{A\left|{d_j}\right\rangle}\right\rangle = \sum_{k=1}^{n} A_{kj} \left\langle{d_i}\middle|{d_k}\right\rangle\] Since \(\{\left|{d_i}\right\rangle\}\) is an orthonormal basis, \(\left\langle{d_i}\middle|{d_k}\right\rangle = \delta_{ik}\), where \(\delta_{ik}\) is the Kronecker delta. Thus, the sum simplifies to: \[\left\langle{d_i}\middle|{A\left|{d_j}\right\rangle}\right\rangle = \sum_{k=1}^{n} A_{kj} \delta_{ik} = A_{ij}\] Therefore, the matrix element \(A_{ij}\) is given by: \[A_{ij} = \left\langle{d_i}\middle|{A\left|{d_j}\right\rangle}\right\rangle\] This formula is fundamental for finding the matrix representation of an operator in a given orthonormal basis. It tells us that to find the \((i, j)\)-th entry of the matrix for operator \(A\), we need to compute the inner product of the \(i\)-th basis vector \(\left|{d_i}\right\rangle\) with the vector obtained by applying \(A\) to the \(j\)-th basis vector \(\left|{d_j}\right\rangle\).

In bra-ket notation, we can express the operator \(A\) in terms of its matrix elements and the basis vectors as: \[A = \sum_{i=1}^{n} \sum_{j=1}^{n} A_{ij} \left|{d_i}\middle\rangle\middle\langle{d_j}\right| = \sum_{i=1}^{n} \sum_{j=1}^{n} \left\langle{d_i}\middle|{A\left|{d_j}\right\rangle}\right\rangle \left|{d_i}\middle\rangle\middle\langle{d_j}\right|\] This is a powerful way to represent any linear operator in terms of the chosen basis \(\{\left|{d_i}\right\rangle\}\). It expresses \(A\) as a sum of outer products of basis vectors, weighted by the matrix elements \(A_{ij}\).

Composition of Operators and Matrix Multiplication

In quantum computing and quantum mechanics, we often apply sequences of operations. The composition of two linear operators \(A\) and \(B\), denoted as \(C = AB\), is defined as applying operator \(B\) first, followed by operator \(A\). That is, for any vector \(\left|{\psi}\right\rangle\), \(C\left|{\psi}\right\rangle = A(B\left|{\psi}\right\rangle)\).

If we have matrix representations \(M_A = (A_{ij})\) and \(M_B = (B_{ij})\) for operators \(A\) and \(B\) in the same basis \(\{\left|{d_i}\right\rangle\}\), then the matrix representation \(M_C = (C_{ij})\) of the composite operator \(C = AB\) is given by the product of the matrices \(M_A\) and \(M_B\): \(M_C = M_A M_B\). The matrix elements of \(C\) are obtained by matrix multiplication: \[C_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj}\] This is the standard rule for matrix multiplication: the \((i, j)\)-th entry of the product matrix \(M_C\) is the sum of products of the \(i\)-th row of \(M_A\) with the \(j\)-th column of \(M_B\).

Operator composition is associative, i.e., \((AB)C = A(BC)\), which corresponds to the associativity of matrix multiplication.

Commutation of Operators

In general, the order in which we apply operators matters. Operator multiplication is not always commutative. That is, for two operators \(A\) and \(B\), in general, \(AB \neq BA\).

Definition 17 (Commutator). Two operators \(A\) and \(B\) are said to commute* if their order of application does not matter, i.e., \(AB = BA\). The commutator of two operators \(A\) and \(B\) is defined as: \[[A, B] = AB - BA\] Operators \(A\) and \(B\) commute if and only if their commutator is the zero operator: \([A, B] = \hat{0}\).*

The concept of commutation is fundamentally important in quantum mechanics. It has deep implications for the simultaneous measurability of physical observables. In quantum mechanics, physical observables are represented by Hermitian operators. If two observables are represented by commuting operators, it means they can be measured simultaneously with arbitrary precision. However, if they are represented by non-commuting operators, they cannot be simultaneously measured with arbitrary precision; this is related to Heisenberg’s uncertainty principle.

Example: Non-Commutation of Pauli Operators \(\sigma_Z\) and \(\sigma_X\)

Let’s illustrate the concept of non-commuting operators with the example of the Pauli operators \(\sigma_Z\) and \(\sigma_X\). These are fundamental operators in quantum computing, particularly for qubit manipulations. In the computational basis \(\{\left|{0}\right\rangle, \left|{1}\right\rangle\}\), their matrix representations are: \[\sigma_Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \quad \sigma_X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\]

Using Matrix Multiplication to Show Non-Commutation

Let’s compute the products \(\sigma_Z \sigma_X\) and \(\sigma_X \sigma_Z\) using matrix multiplication: \[\sigma_Z \sigma_X = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} (1 \cdot 0 + 0 \cdot 1) & (1 \cdot 1 + 0 \cdot 0) \\ (0 \cdot 0 + (-1) \cdot 1) & (0 \cdot 1 + (-1) \cdot 0) \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\] \[\sigma_X \sigma_Z = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} (0 \cdot 1 + 1 \cdot 0) & (0 \cdot 0 + 1 \cdot (-1)) \\ (1 \cdot 1 + 0 \cdot 0) & (1 \cdot 0 + 0 \cdot (-1)) \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\] Comparing the resulting matrices, we observe that: \[\sigma_Z \sigma_X = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \neq \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} = \sigma_X \sigma_Z\] Since \(\sigma_Z \sigma_X \neq \sigma_X \sigma_Z\), the Pauli operators \(\sigma_Z\) and \(\sigma_X\) do not commute.

Using Bra-Ket Notation to Show Non-Commutation

Alternatively, we can demonstrate the non-commutation of \(\sigma_Z\) and \(\sigma_X\) by examining their action on the basis states \(\left|{0}\right\rangle\) and \(\left|{1}\right\rangle\) using bra-ket notation. Recall the actions of \(\sigma_Z\) and \(\sigma_X\): \[\sigma_Z \left|{0}\right\rangle = \left|{0}\right\rangle, \quad \sigma_Z \left|{1}\right\rangle = -\left|{1}\right\rangle\] \[\sigma_X \left|{0}\right\rangle = \left|{1}\right\rangle, \quad \sigma_X \left|{1}\right\rangle = \left|{0}\right\rangle\] Let’s compute \(\sigma_Z \sigma_X \left|{0}\right\rangle\) and \(\sigma_X \sigma_Z \left|{0}\right\rangle\): \[\sigma_Z \sigma_X \left|{0}\right\rangle = \sigma_Z (\sigma_X \left|{0}\right\rangle) = \sigma_Z \left|{1}\right\rangle = -\left|{1}\right\rangle\] \[\sigma_X \sigma_Z \left|{0}\right\rangle = \sigma_X (\sigma_Z \left|{0}\right\rangle) = \sigma_X \left|{0}\right\rangle = \left|{1}\right\rangle\] Since \(\sigma_Z \sigma_X \left|{0}\right\rangle = -\left|{1}\right\rangle\) and \(\sigma_X \sigma_Z \left|{0}\right\rangle = \left|{1}\right\rangle\), we see that \(\sigma_Z \sigma_X \left|{0}\right\rangle \neq \sigma_X \sigma_Z \left|{0}\right\rangle\). This difference in their action on the basis state \(\left|{0}\right\rangle\) is sufficient to conclude that \(\sigma_Z \sigma_X \neq \sigma_X \sigma_Z\), and thus \(\sigma_Z\) and \(\sigma_X\) do not commute.

Similarly, we can check their action on \(\left|{1}\right\rangle\): \[\sigma_Z \sigma_X \left|{1}\right\rangle = \sigma_Z (\sigma_X \left|{1}\right\rangle) = \sigma_Z \left|{0}\right\rangle = \left|{0}\right\rangle\] \[\sigma_X \sigma_Z \left|{1}\right\rangle = \sigma_X (\sigma_Z \left|{1}\right\rangle) = \sigma_X (-\left|{1}\right\rangle) = -\sigma_X \left|{1}\right\rangle = -\left|{0}\right\rangle\] Again, \(\sigma_Z \sigma_X \left|{1}\right\rangle = \left|{0}\right\rangle\) and \(\sigma_X \sigma_Z \left|{1}\right\rangle = -\left|{0}\right\rangle\), confirming that \(\sigma_Z \sigma_X \left|{1}\right\rangle \neq \sigma_X \sigma_Z \left|{1}\right\rangle\).

Finally, let’s use the bra-ket representation of \(\sigma_Z\) and \(\sigma_X\) to compute their products directly: \[\sigma_Z = \left|{0}\middle\rangle\middle\langle{0}\right| - \left|{1}\middle\rangle\middle\langle{1}\right|, \quad \sigma_X = \left|{0}\middle\rangle\middle\langle{1}\right| + \left|{1}\middle\rangle\middle\langle{0}\right|\] Then, \[\begin{aligned}\sigma_Z \sigma_X &= (\left|{0}\middle\rangle\middle\langle{0}\right| - \left|{1}\middle\rangle\middle\langle{1}\right|)(\left|{0}\middle\rangle\middle\langle{1}\right| + \left|{1}\middle\rangle\middle\langle{0}\right|) \\&= \left|{0}\middle\rangle\middle\langle{0}\right|\left|{0}\middle\rangle\middle\langle{1}\right| + \left|{0}\middle\rangle\middle\langle{0}\right|\left|{1}\middle\rangle\middle\langle{0}\right| - \left|{1}\middle\rangle\middle\langle{1}\right|\left|{0}\middle\rangle\middle\langle{1}\right| - \left|{1}\middle\rangle\middle\langle{1}\right|\left|{1}\middle\rangle\middle\langle{0}\right| \\&= \left|{0}\middle\rangle\middle\langle{1}\right|\left\langle{0}\middle|{0}\right\rangle + \left|{0}\middle\rangle\middle\langle{0}\right|\left|{1}\middle\rangle\middle\langle{0}\right| - \left|{1}\middle\rangle\middle\langle{1}\right|\left|{0}\middle\rangle\middle\langle{1}\right| - \left|{1}\middle\rangle\middle\langle{0}\right|\left\langle{1}\middle|{1}\right\rangle \\&= \left|{0}\middle\rangle\middle\langle{1}\right| + 0 - 0 - \left|{1}\middle\rangle\middle\langle{0}\right| \\&= \left|{0}\middle\rangle\middle\langle{1}\right| - \left|{1}\middle\rangle\middle\langle{0}\right|\end{aligned}\] \[\begin{aligned}\sigma_X \sigma_Z &= (\left|{0}\middle\rangle\middle\langle{1}\right| + \left|{1}\middle\rangle\middle\langle{0}\right|)(\left|{0}\middle\rangle\middle\langle{0}\right| - \left|{1}\middle\rangle\middle\langle{1}\right|) \\&= \left|{0}\middle\rangle\middle\langle{1}\right|\left|{0}\middle\rangle\middle\langle{0}\right| - \left|{0}\middle\rangle\middle\langle{1}\right|\left|{1}\middle\rangle\middle\langle{1}\right| + \left|{1}\middle\rangle\middle\langle{0}\right|\left|{0}\middle\rangle\middle\langle{0}\right| - \left|{1}\middle\rangle\middle\langle{0}\right|\left|{1}\middle\rangle\middle\langle{1}\right| \\&= \left|{0}\middle\rangle\middle\langle{0}\right|\left\langle{1}\middle|{0}\right\rangle - \left|{0}\middle\rangle\middle\langle{1}\right|\left|{1}\middle\rangle\middle\langle{1}\right| + \left|{1}\middle\rangle\middle\langle{0}\right|\left|{0}\middle\rangle\middle\langle{0}\right| - \left|{1}\middle\rangle\middle\langle{0}\right|\left|{1}\middle\rangle\middle\langle{1}\right| \\&= 0 - \left|{0}\middle\rangle\middle\langle{1}\right| + \left|{1}\middle\rangle\middle\langle{0}\right| - 0 \\&= \left|{1}\middle\rangle\middle\langle{0}\right| - \left|{0}\middle\rangle\middle\langle{1}\right| = - (\left|{0}\middle\rangle\middle\langle{1}\right| - \left|{1}\middle\rangle\middle\langle{0}\right|)\end{aligned}\] Thus, we find \(\sigma_X \sigma_Z = - \sigma_Z \sigma_X\), which explicitly shows that \(\sigma_Z\) and \(\sigma_X\) do not commute. In fact, their commutator is \([\sigma_X, \sigma_Z] = \sigma_X \sigma_Z - \sigma_Z \sigma_X = -2(\left|{0}\middle\rangle\middle\langle{1}\right| - \left|{1}\middle\rangle\middle\langle{0}\right|) = -2i\sigma_Y\) (where \(\sigma_Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} = i(\left|{1}\middle\rangle\middle\langle{0}\right| - \left|{0}\middle\rangle\middle\langle{1}\right|)\) is the Pauli-Y operator, though the imaginary unit factor was missed in the transcript and should be ignored). The key point is the non-zero commutator, indicating non-commutability.

This example highlights the non-commutative nature of quantum operators, a property that distinguishes quantum mechanics from classical mechanics and has profound implications for quantum phenomena and quantum computing.

Conclusion

In this lecture, we have established a robust foundation in the linear algebra principles that are indispensable for understanding quantum computing. We began by introducing Dirac notation (bra-ket notation), emphasizing its utility in simplifying the representation and manipulation of quantum states and operators. We explored complex vector spaces and the crucial concept of the inner product, which allows us to define orthogonality, normalization, and probabilities in quantum systems. We then formally defined Hilbert spaces as the abstract vector spaces that provide the mathematical arena for quantum mechanics, highlighting their key properties: linearity, inner product, completeness, and separability.

We delved into linear operators, which represent transformations of quantum states and physical observables. We focused particularly on projection operators, understanding their role in quantum measurements and their defining property of idempotency (\(P^2 = P\)). We also explored the completeness relation, which expresses the identity operator as a sum of projection operators onto the vectors of an orthonormal basis, providing a crucial tool for decomposing operators and states. Through concrete examples, we illustrated the projection of quantum states and verified the orthonormality of superposition states. Finally, we demonstrated the concept of operator composition and its correspondence to matrix multiplication, and crucially, we highlighted the non-commutativity of quantum operators using the Pauli operators \(\sigma_Z\) and \(\sigma_X\) as a prime example. We showed non-commutativity both through matrix multiplication and using bra-ket notation, underscoring the fundamental difference between quantum and classical operations.

Key takeaways from this lecture are:

• Quantum states as vectors: Quantum states are represented as vectors in complex vector spaces, specifically Hilbert spaces, allowing for superposition and linear combinations.

• Dirac Notation: Bra-ket notation is a powerful and convenient tool for representing quantum states, operators, and inner products, simplifying quantum mechanical expressions and calculations.

• Projection Operators: Projection operators, defined by \(\left|{u}\middle\rangle\middle\langle{u}\right| = \left|{u}\middle\rangle\middle\langle{u}\right|\) for a normalized state \(\left|{u}\right\rangle\), are idempotent (\(P^2 = P\)) and are essential for describing quantum measurements and extracting state components.

• Completeness Relation: For any orthonormal basis \(\{\left|{v_i}\right\rangle\}\), the completeness relation \(\sum_{i} \left|{v_i}\middle\rangle\middle\langle{v_i}\right| = \mathbb{I}\) holds, expressing the identity operator as a sum of projections and providing a valuable tool for calculations.

• Non-Commutativity of Operators: Quantum operators, in general, do not commute (\(AB \neq BA\)), a key feature distinguishing quantum mechanics from classical mechanics, with significant implications for simultaneous measurability and the order of quantum operations. The Pauli operators \(\sigma_Z\) and \(\sigma_X\) serve as a fundamental example of non-commuting operators.

• Matrix Representation and Composition: Linear operators can be represented as matrices, and the composition of operators corresponds to matrix multiplication, providing a computational framework for quantum operations.

Building upon this solid foundation of linear algebra, in the upcoming lectures, we will explore the fascinating world of quantum gates and quantum circuits. We will see how these linear algebra concepts directly translate into the building blocks of quantum algorithms and quantum computations, enabling us to harness the unique power of quantum mechanics for information processing. The mathematical tools and notations introduced in this lecture will be continuously utilized and expanded upon as we delve deeper into the exciting realm of quantum computing. Specifically, in our next session, we will see how these linear algebraic constructs become the very language for describing quantum gates and circuits, the fundamental building blocks of quantum algorithms. Understanding linear algebra is not just helpful, but absolutely essential to grasp the power and potential of quantum computation. This foundational knowledge will empower you to analyze, design, and ultimately implement quantum algorithms, paving the way for exploring the transformative capabilities of quantum computers in solving complex problems across various scientific and technological domains. As we move forward, remember that linear algebra is the bedrock upon which the edifice of quantum computing is built. Mastering these concepts is your first crucial step towards unlocking the quantum future.