Glossary

2.2 Quantum states and measurements

3 min read•july 23, 2024

are the foundation of quantum mechanics, describing a system's properties and behavior. Using , these states are represented mathematically in a complex , allowing for precise calculations and analysis of quantum systems.

and measurements are key concepts in quantum mechanics. Observables represent measurable quantities, while the measurement process is inherently probabilistic, collapsing quantum states to . Understanding these concepts is crucial for interpreting quantum phenomena and performing calculations.

Quantum States and Their Representation

Quantum states and ket notation

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  • Quantum states fully describe the state of a quantum system at a specific point in time
    • Encapsulate all the information about the system's properties and behavior
  • Represented mathematically using Dirac notation, specifically
    • Ket notation denotes a quantum state as ∣ψ⟩|\psi\rangle, where ψ\psi is a label identifying the state (e.g., ∣0⟩|0\rangle, ∣1⟩|1\rangle)
    • ⟨ψ∣\langle\psi| represents the conjugate transpose of the ket ∣ψ⟩|\psi\rangle
  • State vectors reside in a complex Hilbert space, a complete inner product space
    • Hilbert space provides a mathematical framework for describing quantum systems
  • Quantum states can be expressed as linear combinations of
    • Basis states form a set of orthonormal vectors that span the Hilbert space (e.g., ∣0⟩|0\rangle and ∣1⟩|1\rangle for a qubit)
    • Example: A general qubit state is written as ∣ψ⟩=α∣0⟩+β∣1⟩|\psi\rangle = \alpha|0\rangle + \beta|1\rangle
      • α\alpha and β\beta are representing the probability amplitudes of the basis states
      • The ∣α∣2+∣β∣2=1|\alpha|^2 + |\beta|^2 = 1 ensures the total probability sums to 1

Observables and Measurements

Observables and quantum states

  • Observables represent measurable physical quantities in quantum systems
    • Examples include position, momentum, energy, and spin
  • Mathematically described by Hermitian operators, denoted by A^\hat{A}
    • Hermitian operators satisfy A^†=A^\hat{A}^\dagger = \hat{A}, where A^†\hat{A}^\dagger is the adjoint of A^\hat{A}
  • Eigenstates of an observable are quantum states for which the observable has a definite value
    • The A^∣ψ⟩=a∣ψ⟩\hat{A}|\psi\rangle = a|\psi\rangle relates the eigenstate ∣ψ⟩|\psi\rangle to its corresponding eigenvalue aa
  • The of an observable, given by ⟨A^⟩=⟨ψ∣A^∣ψ⟩\langle\hat{A}\rangle = \langle\psi|\hat{A}|\psi\rangle, represents the average value obtained from multiple measurements on identical quantum states

Quantum measurement process

  • Quantum measurements are inherently probabilistic, collapsing the quantum state to an eigenstate of the measured observable
  • The probability of measuring an eigenvalue aia_i is given by P(ai)=∣⟨ψ∣ai⟩∣2P(a_i) = |\langle\psi|a_i\rangle|^2
    • ∣ai⟩|a_i\rangle represents the eigenstate corresponding to the eigenvalue aia_i
  • Repeated measurements on identical quantum states produce a distribution of eigenvalues
  • Individual measurement outcomes are fundamentally random and cannot be predicted with certainty

Probability calculations in quantum mechanics

  • To calculate the probabilities of different measurement outcomes for a given quantum state ∣ψ⟩|\psi\rangle and an observable A^\hat{A}:
    1. Expand the state in the eigenbasis of the observable: ∣ψ⟩=∑ici∣ai⟩|\psi\rangle = \sum_i c_i|a_i\rangle
      • ci=⟨ai∣ψ⟩c_i = \langle a_i|\psi\rangle are the
    2. Calculate the aia_i using P(ai)=∣ci∣2=∣⟨ai∣ψ⟩∣2P(a_i) = |c_i|^2 = |\langle a_i|\psi\rangle|^2
  • Example: For a qubit state ∣ψ⟩=12(∣0⟩+∣1⟩)|\psi\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)
    • The probability of measuring ∣0⟩|0\rangle is P(0)=∣⟨0∣ψ⟩∣2=12P(0) = |\langle0|\psi\rangle|^2 = \frac{1}{2}
    • The probability of measuring ∣1⟩|1\rangle is P(1)=∣⟨1∣ψ⟩∣2=12P(1) = |\langle1|\psi\rangle|^2 = \frac{1}{2}

Key Terms to Review (17)

Basis States: Basis states refer to a set of fundamental quantum states that can be used to describe the complete state of a quantum system. They form a basis for the state space, meaning any quantum state can be represented as a linear combination of these basis states. Understanding basis states is essential for grasping how quantum measurements work and how they relate to the probabilities of different outcomes when a quantum system is observed.
Bra notation: Bra notation is a mathematical notation used in quantum mechanics to represent quantum states in the context of linear algebra. It is part of the broader framework known as Dirac notation, where 'bra' refers to the left-hand side of an inner product and is denoted by a left angle bracket followed by a state label, such as $$\langle \psi |$$. This form is essential for understanding quantum states and their interactions during measurements.
Complex Amplitudes: Complex amplitudes are mathematical representations used in quantum mechanics to describe the state of a quantum system, characterized by both magnitude and phase. They provide a way to encapsulate the probabilities and relative phases associated with different outcomes of measurements, making them essential for understanding quantum states and how they evolve. By incorporating both real and imaginary components, complex amplitudes help to portray the full behavior of quantum systems, such as interference and superposition effects.
Dirac Notation: Dirac notation, also known as bra-ket notation, is a standard mathematical notation used in quantum mechanics to describe quantum states. This notation simplifies the representation of states and their relationships, allowing for concise expressions of inner products and outer products, which are fundamental in quantum computing and linear algebra.
Eigenstates: Eigenstates are specific quantum states of a quantum system that correspond to definite values of an observable quantity, meaning they are the states in which a measurement of that observable will yield a specific result. When a quantum system is in an eigenstate, the observable's associated operator acts on the state to produce the same state multiplied by a scalar, known as the eigenvalue. This concept is crucial in understanding how quantum measurements work and how they relate to the fundamental properties of quantum systems.
Eigenvalue Equation: The eigenvalue equation is a fundamental concept in linear algebra, represented as $$A\mathbf{v} = \lambda\mathbf{v}$$, where $$A$$ is a square matrix, $$\mathbf{v}$$ is a non-zero vector known as an eigenvector, and $$\lambda$$ is a scalar known as the eigenvalue. This equation describes how an eigenvector is transformed by the matrix $$A$$, showing that it only gets scaled by the eigenvalue, rather than changing direction. Understanding this relationship is crucial for analyzing quantum states and measurements, as well as solving problems in quantum computing using linear algebra techniques.
Expansion Coefficients: Expansion coefficients are numerical values that quantify how much a quantum state can be expressed in terms of a basis of states. In the context of quantum mechanics, they represent the probability amplitude for a system to be found in a specific state after measurement, connecting the abstract mathematical description of quantum states to measurable outcomes in experiments.
Expectation Value: The expectation value is a fundamental concept in quantum mechanics that represents the average outcome of a measurement for a given observable when considering a quantum state. It acts as a statistical average, providing insights into the behavior of quantum systems by summarizing the probabilities of various measurement results. This concept is pivotal when discussing quantum states and how measurements influence them, as well as in optimization algorithms where it helps evaluate potential solutions.
Hermitian Operator: A Hermitian operator is a linear operator that is equal to its own adjoint, meaning it has real eigenvalues and orthogonal eigenvectors. This property makes Hermitian operators fundamental in quantum mechanics, as they correspond to observable physical quantities and ensure that measurement outcomes are real numbers. Additionally, the eigenstates of Hermitian operators form a complete basis for the state space, which is crucial for quantum state representation and measurements.
Hilbert Space: Hilbert space is a complete vector space equipped with an inner product, allowing for the geometric interpretation of quantum states and operations. It serves as the foundational framework for quantum mechanics, facilitating the representation of quantum states as vectors and enabling measurements through projection operators.
Ket notation: Ket notation is a mathematical shorthand used in quantum mechanics to represent quantum states as vectors in a complex Hilbert space. It simplifies the representation of quantum states, allowing physicists and mathematicians to express and manipulate them conveniently, especially when dealing with measurements and multi-qubit systems.
Normalization Condition: The normalization condition is a requirement that ensures the total probability of all possible outcomes of a quantum state sums to one. This concept is essential in quantum mechanics, as it guarantees that when measuring a quantum state, the probabilities of all potential measurement results are valid and meaningful, reinforcing the probabilistic nature of quantum mechanics and its mathematical formulation through linear algebra.
Observables: Observables are physical quantities in quantum mechanics that can be measured, such as position, momentum, and energy. Each observable is associated with a mathematical operator that acts on the state of a quantum system, defined by wave functions. When a measurement is made, the outcome corresponds to one of the eigenvalues of this operator, collapsing the wave function into a definite state.
Probability of measuring eigenvalue: The probability of measuring an eigenvalue is the likelihood that a measurement of a quantum system will yield a specific value corresponding to an eigenstate of an observable. This concept is fundamental in quantum mechanics, linking quantum states to measurable outcomes through the mathematical framework of linear algebra, where observables are represented as operators acting on state vectors. It emphasizes the inherent randomness of quantum measurements and the role of the wave function in predicting results.
Quantum measurement process: The quantum measurement process refers to the procedure of observing and determining the state of a quantum system, which results in the collapse of its wave function into a definite outcome. This process is unique because, unlike classical measurements, the act of measuring affects the system being observed, often leading to outcomes that are probabilistic rather than deterministic. This highlights the fundamental differences between quantum mechanics and classical physics, especially regarding the nature of reality and information.
Quantum States: Quantum states refer to the mathematical descriptions of the physical properties of a quantum system, encapsulated within a complex vector in a Hilbert space. They provide a complete description of all possible outcomes of measurements made on the system. The behavior and characteristics of quantum states are pivotal when discussing phenomena such as superposition, entanglement, and the results of measurements in quantum mechanics.
State Vector: A state vector is a mathematical representation of a quantum state in a Hilbert space, typically represented as a column vector. It encapsulates all the information about a quantum system, including probabilities and possible measurement outcomes. State vectors can exist in superposition, allowing them to represent multiple states simultaneously, which is fundamental to the behavior of quantum systems.
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