Bra-ket notation, also known as Dirac notation, provides a concise and powerful way to represent quantum states and linear operators within the framework of linear algebra on complex vector spaces. This notation, introduced by Paul Dirac in 1939, has become ubiquitous in quantum mechanics and quantum computing due to its ease of manipulation and its ability to abstract away from specific representations.
What are Bras and Kets?
At its heart, bra-ket notation uses two primary symbols:
- Ket (|ψ⟩): Represents a quantum state vector. The symbol inside the ket, such as ψ, is a label that identifies the state. You can use any word or symbol inside the ket. In some cases numbers are also used, but they are used as labels, so don't try to do arithmetic with them. A ket can be thought of as a column vector.
- Bra (⟨ψ|): Represents a linear functional, or a linear map from the vector space to the complex numbers, which is associated with a specific ket. It is the complex conjugate transpose of the ket. A bra can be thought of as a row vector. The values are in a row, and each element is the complex conjugate of the ket's elements. For example, if an element in the ket is 2 - 3i, the corresponding element in the bra is 2 + 3i.
For the vector space , kets can be identified with column vectors, and bras with row vectors. Combinations of bras, kets, and linear operators are interpreted using matrix multiplication.
The Significance of the Notation
The power of bra-ket notation lies in its ability to represent quantum states abstractly, without committing to a specific basis. This is particularly useful in quantum mechanics, where calculations often involve switching between different bases (e.g., position basis, momentum basis, energy eigenbasis).
Symbols, letters, numbers, or even words-whatever serves as a convenient label-can be used as the label inside a ket, with the making clear that the label indicates a vector in vector space. In other words, the symbol "|A⟩" has a recognizable mathematical meaning as to the kind of variable being represented, while just the "A" by itself does not. For example, |1⟩ + |2⟩ is not necessarily equal to |3⟩. Nevertheless, for convenience, there is usually some logical scheme behind the labels inside kets, such as the common practice of labeling energy eigenkets in quantum mechanics through a listing of their quantum numbers.
Inner Products: Measuring Overlap
An inner product measures the overlap between two states. The inner product of two states, |ψ⟩ and |φ⟩, is written as ⟨φ|ψ⟩ (note the order). This represents the projection of the state |ψ⟩ onto the state |φ⟩. The result is a complex number, often interpreted as a probability amplitude.
Mathematically, this means the coefficient for the projection of ψ onto φ. In quantum mechanics the expression ⟨φ|ψ⟩ is typically interpreted as the probability amplitude for the state ψ to collapse into the state φ.
Calculating Inner Products
To calculate the inner product, we first find the bra of |ϕ>, i.e. ⟨ϕ∣. Next, we perform matrix multiplication between ⟨ϕ∣ and ∣ψ⟩. This results in a single complex number.
For example, consider two kets:
- |ϕ⟩ = [1, 2, 3]
- |ψ⟩ = [7, 9, 11]
To calculate their inner product, we first find the bra of |ϕ>, i.e. ⟨ϕ∣ : [1, 2, 3]. We then multiply them as shown below:
(1 * 7) + (2 * 9) + (3 * 11) = 7 + 18 + 33 = 58
This results in the final answer 58.
Length of a Vector
The dot product of a vector with itself is the length of the vector times the length of the vector.
Example: What is the length of the vector [1, 2, −2, 5]?
√((11)+(22)+(-2-2)+(55)) = √(1+4+4+25) = √34
Orthonormal Basis
The vectors "1, 0, 0", "0, 1, 0" and "0, 0, 1" form the basis: the vectors that we measure things against. In this case they are simple unit vectors, but any set of vectors can be used when they are independent of each other (being at right angles achieves this) and can together span every part of the space. Matrix Rank has more details about linear dependence, span and more.
Let's check it!Is the dot product zero? Is each length 1? So yes it is an orthonormal basis!
Outer Products: Building Operators
Unlike a single number, as with the inner product result, the result of the outer product is a matrix.
The outer product of two states, |ψ⟩ and |φ⟩, is written as |ψ⟩⟨φ|. The outer product results in a linear operator. One of the uses of the outer product is to construct projection operators.
Calculating Outer Products
To calculate an outer product of two quantum states ∣ϕ⟩ and ∣ψ⟩ , we first calculate the bra of ∣ψ⟩, i.e. we obtain ⟨ψ∣. Next, the outer product is calculated as follows: |ϕ⟩⟨ψ∣.
For example, consider two kets:
- |ϕ⟩ = [1, 2]
- |ψ⟩ = [3, 4]
To calculate their outer product, we first find the bra of |ψ>, i.e. ⟨ψ∣ : [3, 4]. Next, we multiply these as follows:
[1, 2] * [3, 4] = [[1*3, 1*4], [2*3, 2*4]] = [[3, 4], [6, 8]]
The answer is a matrix, as shown below.
Projection Operators
Projection operators are a specific type of linear operator that projects a vector onto a particular subspace. They can be constructed using the outer product of a state with itself.
Linear Operators: Transforming States
A linear operator is a map that inputs a ket and outputs a ket. Linear operators are ubiquitous in the theory of quantum mechanics. Operators can also be viewed as acting on bras from the right hand side (in other words, a function composition).
In an -dimensional Hilbert space, we can impose a basis on the space and represent in terms of its coordinates as a column vector. Using the same basis for , it is represented by an complex matrix.
Hermitian Conjugate
Just as kets and bras can be transformed into each other (making |ψ⟩ into ⟨ψ|), the element from the dual space corresponding to A|ψ⟩ is ⟨ψ|A†, where A† denotes the Hermitian conjugate (or adjoint) of the operator A.
Rules for Manipulating Bra-Ket Notation
Bra-ket notation was designed to facilitate the formal manipulation of linear-algebraic expressions. Some of the properties that allow this manipulation are listed herein. Given any expression involving complex numbers, bras, kets, inner products, outer products, and/or linear operators (but not addition), written in bra-ket notation, the parenthetical groupings do not matter (i.e., the associative property holds).
and so forth. The expressions on the right (with no parentheses whatsoever) are allowed to be written unambiguously because of the equalities on the left.
Bra-ket notation makes it particularly easy to compute the Hermitian conjugate (also called dagger, and denoted †) of expressions.
Tensor Products: Combining Systems
Two Hilbert spaces V and W may form a third space V ⊗ 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. (The exception to this is if the subsystems are actually identical particles.
If |ψ⟩ is a ket in V and |φ⟩ is a ket in W, the tensor product of the two kets is a ket in V ⊗ W.
Common Confusions and Clarifications
- Inner Product vs. Ket Multiplication: It is important to remember that you cannot directly multiply two kets. The inner product, denoted by ⟨φ|ψ⟩, is the correct way to combine a bra and a ket to obtain a complex number.
- Labels vs. Mathematical Objects: The symbols inside the kets are labels and not mathematical objects on which operations can be performed. For instance, if the vector is scaled by , it may be denoted . This can be ambiguous since is simply a label for a state, and not a mathematical object on which operations can be performed. This usage is more common when denoting vectors as tensor products, where part of the labels are moved outside the designed slot, e.g.
- The Importance of Context: The interpretation of bra-ket notation depends on the context. In quantum mechanics, it represents quantum states and operations, while in mathematics, it can represent abstract vectors and linear functionals.
Examples of Bra-Ket Notation in Action
Qubit Superposition
A qubit, the basic unit of quantum information, can exist in a superposition of two states, |0⟩ and |1⟩. This superposition can be represented as:
|ψ⟩ = α|0⟩ + β|1⟩
where α and β are complex numbers called probability amplitudes. The probability of measuring the qubit in the state |0⟩ is |α|², and the probability of measuring it in the state |1⟩ is |β|². Since the total probability must equal 1, we have |α|² + |β|² = 1.
Schrödinger's Cat
A famous example is "Schrödinger's Cat": a thought experiment where a cat is in a box with a quantum-triggered container of gas. There is an equal chance of it being alive or dead (until we open the box). It says the state of the cat is in a superposition of the two states "alive" and "dead".
The basis is the two vectors alive and dead. The cat is shown in that probability space as a vector with equal components a and d. A normalized vector has a length of 1.
Quantum Dice
We can easily have many dimensions. Imagine "Quantum Dice" that are in a superposition of 1, 2, 3, 4, 5 and 6. For a fair die all elements (a, b, c, d, e, f) are equal, but your dice may be loaded!
Why Use Bra-Ket Notation?
So we can "map" some real world case (usually one with probabilities) onto a well-defined mathematical basis. This then gives us the power to use all the math tools to study it.
Bra-ket notation was designed to facilitate the formal manipulation of linear-algebraic expressions.