2.3 Quantum State Vectors and Dirac Notation

Quantum state vectors are the mathematical backbone of quantum mechanics. They represent quantum systems as complex-valued vectors in Hilbert space, allowing us to describe and manipulate quantum states with precision.

Dirac notation provides a powerful shorthand for working with quantum states. It simplifies calculations and helps visualize complex operations, making it an essential tool for understanding and manipulating quantum information.

Quantum State Vectors

Mathematical representation of quantum states

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  Tensor Product of 2-Frames in 2-Hilbert Spaces View original

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  Variational Quantum Singular Value Decomposition – Quantum View original

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• State vectors in quantum mechanics represent quantum systems as complex-valued vectors in Hilbert space
• Properties of state vectors include normalization condition $\langle\psi|\psi\rangle = 1$ ensures total probability is 1 and linear superposition principle allows combining states
• Basis states form foundation for representing qubits (computational basis $|0\rangle$ and $|1\rangle$)
• General qubit state expressed as $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ where $\alpha$ and $\beta$ are complex amplitudes
• Multi-qubit systems constructed using tensor product of individual qubit states leads to exponential growth in dimensionality (2^n for n qubits)

Dirac notation for quantum states

• Ket notation $|\psi\rangle$ represents column vectors describing quantum states
• Bra notation $\langle\psi|$ denotes row vectors (conjugate transpose of ket)
• Operators in Dirac notation written as $\hat{A}$ act on kets $\hat{A}|\psi\rangle$
• Expectation values calculated using $\langle\hat{A}\rangle = \langle\psi|\hat{A}|\psi\rangle$
• Projectors formed by outer product $|\psi\rangle\langle\psi|$ project onto specific states

Operations on quantum state vectors

• Inner products $\langle\phi|\psi\rangle$ measure overlap between states with properties (conjugate symmetry, linearity, positive-definiteness)
• Outer products $|\psi\rangle\langle\phi|$ create operators for state transformations
• Vector addition and scalar multiplication enable superposition $|\psi\rangle = c_1|\phi_1\rangle + c_2|\phi_2\rangle$
• Tensor products $|\psi\rangle \otimes |\phi\rangle$ combine separate quantum systems (entanglement)

Probabilities in quantum measurements

• Born rule determines probability of measuring state $|\phi\rangle$ as $P(\phi) = |\langle\phi|\psi\rangle|^2$
• Projection operators calculate probability of outcome $a$ using $P(a) = \langle\psi|\hat{P}_a|\psi\rangle$
• Expectation values of observables computed as $\langle\hat{A}\rangle = \sum_i a_i P(a_i)$
• Measurement causes collapse of wave function to observed state

Dirac notation vs matrix representations

• Single-qubit states in matrix form: $|0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$, $|1\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$
• General state $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle = \begin{pmatrix} \alpha \\ \beta \end{pmatrix}$ shows amplitudes
• Operators as matrices include Pauli matrices ($\sigma_x$, $\sigma_y$, $\sigma_z$) and Hadamard gate $H = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$
• Multi-qubit states represented using tensor product (2^n-dimensional vectors)
• Transforming between representations involves identifying basis states and coefficients (Dirac to matrix) or writing out components (matrix to Dirac)