Bra-ket notation, also known as Dirac notation, is a standard notation for describing quantum states in quantum mechanics. Introduced in 1939 by Paul Dirac, it simplifies mathematical manipulations within quantum mechanics by employing angle brackets and vertical bars to represent vectors and linear functionals. This notation is ubiquitous in quantum mechanics and quantum computing, providing a concise way to express quantum states and operations.
The Building Blocks: Kets, Bras, Brackets, and Operators
Bra-ket notation revolves around four fundamental components: kets, bras, brackets, and operators. These elements form the foundation for describing and manipulating quantum mechanical systems.
Kets: Representing Quantum States
A ket, denoted as (| \psi \rangle), represents a quantum state. It can be visualized as a column vector composed of complex numbers. The symbol inside the ket, such as (k) in (| k \rangle), serves as a label to identify the specific vector. Kets are vectors in a complex Hilbert space, where the exact structure of the Hilbert space depends on the quantum system being described. For instance, an electron's state might be represented as (| \psi \rangle).
Quantum superpositions, a key concept in quantum mechanics, can be described as vector sums of constituent states. For example, a qubit, the basic unit of quantum information, can exist in a superposition of the states (| 0 \rangle) and (| 1 \rangle), represented as:
[| \psi \rangle = \alpha | 0 \rangle + \beta | 1 \rangle]
Here, (\alpha) and (\beta) are complex numbers known as probability amplitudes. The probability of the qubit collapsing into state (| 0 \rangle) upon measurement is (| \alpha |^2), and the probability of collapsing into state (| 1 \rangle) is (| \beta |^2). Since the total probability must equal 1, we have (| \alpha |^2 + | \beta |^2 = 1).
Bras: Linear Functionals in Dual Space
A bra, denoted as (\langle \phi |), is the Hermitian conjugate of a ket. Mathematically, it represents a linear functional, a linear map from the vector space to the complex numbers. For a given ket, the corresponding bra is a row vector (the transpose of the ket), with its elements being complex conjugates.
In essence, a bra (\langle \phi |) can be thought of as a function that acts on a ket (| \psi \rangle) to produce a complex number. This is written as (\langle \phi | \psi \rangle), representing the inner product of the two states.
Brackets: Inner Products and Probabilities
A bracket, such as (\langle \Phi | \Psi \rangle), represents the inner product (or dot product) of two states. It consists of a bra (\langle \Phi |) and a ket (| \Psi \rangle). The inner product yields a complex number that quantifies the overlap between the two quantum states.
In quantum mechanics, the expression (\langle \phi | \psi \rangle) is typically interpreted as the probability amplitude for the state (| \psi \rangle) to collapse into the state (| \phi \rangle). Mathematically, it represents the coefficient for the projection of (| \psi \rangle) onto (| \phi \rangle).
Kets are often assumed to be normalized, meaning that the inner product of a ket with itself is equal to 1: (\langle \psi | \psi \rangle = 1). This implies that at least one element in the column vector of the ket must be nonzero.
The inner product can also be used to compute the probability of a qubit being in either of the possible states (| 0 \rangle) and (| 1 \rangle). In Chapter 10, we saw that the product of a wavefunction by its complex conjugate is a probability distribution and must integrate to one. The dot product can be used to compute the probability of a qubit of being in either one of the possible states (| 0\rangle) and (| 1\rangle).
Operators: Transformations of Quantum States
Operators, denoted with hats (e.g., (\hat{O})), are objects that transform one ket (| k \rangle) into another ket (| q \rangle). They are represented by matrices in a Hilbert space. Linear operators are ubiquitous in quantum mechanics, describing various physical processes.
For example, an operator (\hat{A}) acting on a ket (| \psi \rangle) results in a new ket:
[\hat{A} | \psi \rangle = | \phi \rangle]
Operators can also act on bras from the right-hand side, which is commonly written as (\langle \phi | \hat{A}).
A crucial property of quantum mechanical operators is that they must be unitary. A unitary operator (\hat{O}) satisfies the condition (\hat{O}^{\dagger} \hat{O} = \mathbb{I}), where (\hat{O}^{\dagger}) is the Hermitian adjoint of (\hat{O}) and (\mathbb{I}) is the identity operator. This property ensures that the normalization of the probability distribution is preserved under the transformation.
Constructing Operators with Exterior Products
If the input and output kets are known, an operator can be constructed using the exterior product (also called outer product). The exterior product is the product of a column vector (ket) by a row vector (bra). For example, given an input ket (| k \rangle) and an output ket (| q \rangle), the operator (\hat{O}) can be constructed as:
[\hat{O} = | q \rangle \langle k |]
Dirac's Motivation and the Riesz Representation Theorem
Well before Dirac's work it was known that the only continuous linear functionals on a Hilbert space have the form $F(x)=(x,y)$ for a unique $y$; this is one of several theorems known as the Riesz Representation Theorem. Mathematicians were coming to understand the importance of separating the space from the dual of linear functionals because of what happens in infinite dimensional spaces. Dirac was aware of this earlier work, and was probably motivated out of this understanding to separate the dual on a Hilbert space from the space. And he thought of the objects on the left of the inner product as functionals instead of vectors.
Advantages of Bra-Ket Notation
Bra-ket notation offers several advantages:
- Conciseness: It provides a compact way to represent quantum states and operations, simplifying complex equations.
- Abstraction: It abstracts away from specific coordinate systems or bases, allowing for general manipulations of quantum states.
- Ease of Manipulation: It facilitates the formal manipulation of linear-algebraic expressions, making it easier to perform calculations and derivations.
- Clarity: It enhances the clarity of quantum mechanical expressions by clearly distinguishing between states (kets), linear functionals (bras), and operators.
- Versatility: It is applicable to both finite-dimensional and infinite-dimensional vector spaces, making it suitable for a wide range of quantum mechanical problems.
Applications of Bra-Ket Notation
Bra-ket notation is used extensively throughout quantum mechanics and quantum computing. Some common applications include:
- Representing Quantum States: Describing the states of qubits, electrons, and other quantum systems.
- Calculating Probabilities: Determining the probabilities of measurement outcomes in quantum experiments.
- Describing Quantum Evolution: Representing the time evolution of quantum states using unitary operators.
- Quantum Algorithms: Developing and analyzing quantum algorithms, where the notation simplifies the representation of quantum gates and circuits.
- Quantum Entanglement: Describing entangled states of multiple qubits or quantum systems.
- Spectroscopy: Computing transition amplitudes between states.
Examples
Spin-1/2 Particle
A stationary spin-1⁄2 particle has a two-dimensional Hilbert space. The spin-up and spin-down states can be represented as:
[| \uparrow \rangle = \begin{pmatrix} 1 \ 0 \end{pmatrix}, \quad| \downarrow \rangle = \begin{pmatrix} 0 \ 1 \end{pmatrix}]
Matrix Elements of Operators
Given an operator (A) and two states (| r \rangle) and (| s \rangle), the matrix element of (A) between these states is written as (\langle r | A | s \rangle). This expression can be evaluated in two ways:
- Let (A) act on (| s \rangle) first, then calculate the inner product with (\langle r |).
- Let (A) act on (\langle r |) to obtain a bra vector (\langle r' | = \langle r | A), then evaluate (\langle r' | s \rangle). Note that the ket (| r' \rangle) conjugate to the bra (\langle r' |) is given by (| r' \rangle = A^{\dagger} | r \rangle), where (A^{\dagger}) is the Hermitian adjoint of (A).
Outer Product and Projection Operators
One of the uses of the outer product is to construct projection operators. For example, the projection operator onto the state (| \psi \rangle) is given by:
[\hat{P}_{\psi} = | \psi \rangle \langle \psi |]
This operator projects any arbitrary state onto the state (| \psi \rangle).
Common Misconceptions and Clarifications
Bra vs. Ket
It's important to distinguish between bras and kets. A ket (| \psi \rangle) represents a vector in a complex Hilbert space, while a bra (\langle \psi |) is a linear functional on vectors in that space. The bra is the Hermitian conjugate of the ket.
Inner Product vs. Outer Product
The inner product of two states results in a complex number, representing the overlap between the states. The outer product of two states results in a matrix (an operator), which can be used to transform one state into another.
Operator Acting on Bra
Operators can act on both kets (from the left) and bras (from the right). When an operator acts on a bra, it transforms the linear functional.
Tensor Products
Two Hilbert spaces (V) and (W) may form a third space (V \otimes W) by a tensor product. In quantum mechanics, this is used for describing composite systems. If a system is composed of two subsystems described in (V) and (W) respectively, then the Hilbert space of the entire system is the tensor product of the two spaces.
If (| \psi \rangle) is a ket in (V) and (| \phi \rangle) is a ket in (W), the tensor product of the two kets is a ket in (V \otimes W). In quantum mechanics, it often occurs that little or no information about the inner product (\langle \psi | \phi \rangle) of two arbitrary (state) kets is present, while it is still possible to say something about the expansion coefficients (\langle \psi | ei \rangle = \langle ei | \psi \rangle^*) and (\langle e_i | \phi \rangle) of those vectors with respect to a specific (orthonormalized) basis.
Practical Examples
Normalization
Kets are always assumed to be normalized, which means that the dot product of a ket by itself is equal to 1. This implies that at least one of the elements in the column vector of the ket must be nonzero. This is precisely the result we postulated when we introduced the qubit as a superposition of wave functions.
Dot Product and Qubit States
The dot product can be used to compute the probability of a qubit being in either one of the possible states (| 0\rangle) and (| 1\rangle).