The Dirac notation, also known as bra-ket notation, is a mathematical notation used broadly in quantum mechanics and quantum computing to denote quantum states and linear operators. Introduced by Paul Dirac, it offers a short and intellectual way to work with the complex vector spaces that form the foundation of quantum theory, providing advantages over traditional matrix and vector representations, especially when dealing with multi-qubit systems.
Dirac Notation Symbols
Dirac notation uses two primary symbols: the ket and the bra.
Kets (|ψ⟩)
- A ket, denoted by |ψ⟩, denotes a quantum state vector. The symbol ψ inside the ket identifies the specific quantum state. Mathematically, a ket agrees to a column vector in a complex vector space called a Hilbert space. For a quantum system with N possible classical states, these states can be denoted by an orthonormal basis {|0⟩, |1⟩, …, |N-1⟩}. A general pure quantum state |φ⟩ can be stated as a superposition of these basis states, written as a linear combination:
|φ⟩= α₀|0⟩ + α₁|1⟩ + … + α<0xE2><0x82><0x90>−₁|N-1⟩
where αᵢ are complex numbers called amplitudes. These amplitudes fix the probability of finding the system in the corresponding basis state upon measurement. The ket |ψ⟩ can also be treated as a column vector of these amplitudes.
Example: A single qubit, having two basis states |0⟩ and |1⟩, can be in a state |ψ⟩ = α|0⟩ + β|1⟩, which agrees to the column vector (α β) ᵀ.
Bras (⟨ψ|)
A bra, denoted by ⟨ψ|, is the dual vector corresponding to the ket |ψ⟩. Mathematically, a bra corresponds to the conjugate transpose (Hermitian adjoint) of the ket vector, which is a row vector.
If |ψ⟩ = (α₀ α₁ … α<0xE2><0x82><0x90>−₁)ᵀ, then
⟨ψ| = (α₀* α₁* … α<0xE2><0x82><0x90>−₁*),
where αᵢ* denotes the complex conjugate of αᵢ. The set of all linear maps from a Hilbert space H to the complex numbers C forms the dual space H*, and bras are elements of this dual space. The action of a bra ⟨χ| on a ket |ψ⟩ is defined as
⟨χ|: |ψ⟩ ↦ ⟨χ|ψ⟩ ∈ C.
The Dirac notation becomes seeming when considering the inner product and the outer product.
Inner Product (⟨φ|ψ⟩):
- The expression ⟨φ|ψ⟩, known as a bra-ket or bracket, represents the inner product between the quantum states |φ⟩ and |ψ⟩. If |φ⟩ corresponds to the row vector
(φ₀* φ₁* … φ<0xE2><0x82><0x90>−₁*)
And |ψ⟩ corresponds to the column vector
(ψ₀ ψ₁ … ψ<0xE2><0x82><0x90>−₁)ᵀ,
then their inner product is given by
⟨φ|ψ⟩ = φ₀*ψ₀ + φ₁*ψ₁ + … + φ<0xE2><0x82><0x90>−₁*ψ<0xE2><0x82><0x90>−₁.
This is equal to the dot product of the bra vector with the ket vector. The inner product ⟨φ|ψ⟩ yields a complex number that can be understood as the amplitude of transition from state |ψ⟩ to state |φ⟩ or the overlap between the two states. For orthonormal basis states,
⟨ψ<0xE2><0x82><0x9Bm>|ψ<0xE2><0x82><0x9Bn>⟩ = 1
if n = m, and 0 otherwise. The squared magnitude of the amplitude, |⟨φ|ψ⟩|², gives the probability of finding the system in state |φ⟩ if it was initially in state |ψ⟩ upon measurement.
Outer Product (|ψ⟩⟨φ|)
- The expression |ψ⟩⟨φ| represents the outer product of the ket |ψ⟩ and the bra ⟨φ|. Mathematically, if |ψ⟩ is an n × 1 column vector and ⟨φ| is a 1 × n row vector, their outer product results in an n × n matrix, which is a linear operator acting on the Hilbert space. A particularly important case is when |ψ⟩ = |φ⟩, in which case |ψ⟩⟨ψ| is an estimate operator that projects any quantum state onto the state |ψ⟩. Estimate operators are critical for telling measurements in quantum mechanics.
Example, if P is a plan onto a subspace, it can be written in Dirac notation as
P = ∑ᵢ |vᵢ⟩⟨vᵢ|,
where {|vᵢ⟩} is an orthonormal basis for that subspace.
One of the advantages of Dirac notation is its shortness, particularly when dealing with multi-qubit systems. If we have two quantum systems with state spaces A and B, the joint system’s state space is given by the tensor product A ⊗ B. If system A is in state |i⟩ and system B is in state |j⟩, the combined state is written as |i⟩ ⊗ |j⟩, which is abbreviated as |i⟩|j⟩ or even |ij⟩. If system A has dimension N and system B has dimension M, the combined system has a state space of dimension NM. Representing an n-qubit basis state using Dirac notation involves a binary string of length n, e.g., |010…1⟩, while the equivalent column vector would have 2ⁿ components. This exponential increase in the size of the matrix representation makes Dirac notation meaningfully more practical for relating states of a large number of qubits.
Dirac notation operating concepts in quantum computing
- Qubits and Superposition: A qubit’s state |φ⟩ = α|0⟩ + β|1⟩ shows its nature as a superposition of the basis states |0⟩ and |1⟩, with amplitudes α and β.
- Quantum Gates: Quantum gates, which are unitary operators, act on quantum states. In Dirac notation, applying a gate U to a state |ψ⟩ results in a new state U|ψ⟩. For example, a controlled-U gate acting on a two-qubit state is defined by its action on the basis states in Dirac notation.
- Quantum Algorithms: Algorithms like Shor’s algorithm for factorization and Grover’s search algorithm are described using Dirac notation to denote the quantum states at various stages of the computation. The Quantum Fourier Transform (QFT), a critical component in many quantum algorithms, is also naturally expressed using this notation.
- Measurement: The process of measurement in quantum mechanics is defined in terms of quantum state onto the eigenstates of the measured noticeable. Dirac notation is critical for representing these eigenstates and the resulting post-measurement states.
- Entanglement: Entangled states, where the state of one qubit is connected with the state of another in a non-classical way, are willingly denoted using tensor products in Dirac notation, such as (|00⟩ + |11⟩)/√2.
- While Heisenberg notation uses vector representations, Dirac notation is preferred in quantum computing for its practicality in proving facts and its solidity. The intellectual nature of bras and kets allows for operations and derivations without the need to clearly write out large matrices or vectors. It provides a higher-level, more in-built way to reason about quantum systems.
- The tensor product symbol ⊗ is often omitted for brevity, so |ψ⟩⊗|φ⟩ is written as |ψ⟩|φ⟩ or |ψφ⟩. However, it’s important to remember that these are still elements of the tensor product space. Additionally, it is generally not meaningful to write kets inside other kets or bras.
Advantages of Dirac Notation
• Compactness: It provides a more short representation of quantum states and operators, especially for multi-qubit systems where the vector and matrix representations can become very large. For example, an n-qubit basis state can be written as a binary string of length n in Dirac notation, while the corresponding column vector would have 2ⁿ components .
• Abstraction: It allows for abstract operation of quantum states and operators without needing to clearly write out their matrix or vector components. This is particularly useful for proving theorems and relating quantum algorithms.
• Clarity: It clearly differentiates between quantum states (kets), their dual vectors (bras), and operations on them (bra-kets and outer products) .
Dirac notation is an essential tool in quantum mechanics and quantum computing. Its use of bras and kets provides a short, abstract, and powerful way to represent quantum states, dual vectors, and their products. This notation simplifies the mathematical formalism, improves conceptual clarity, and is particularly advantageous for dealing with the increasing complexity of multi-qubit systems, making it an important language for understanding and developing quantum technologies.