Basics of Quantum Physics & Quantum Information Science

Dirac Formalism

$|x\rangle$ is called a ket viz. a vector with complex values in Hilbert space. It’s dual is called a bra, $\langle x|$

Let $\{|\phi_1\rangle, |\phi_2\rangle, ..., \phi_n\rangle\}$ be a basis for a Hilbert space, then any vector $|\psi\rangle$ can be written as a linear combination of the basis vectors:

\[ |\psi\rangle = \sum_{i=1}^n c_i |\phi_i\rangle\ ;\ c_i \in \mathbb{C} \]

We would in this course consider 2-dimensional Hilbert space with orthonormal basis $\{|\phi_1\rangle, |\phi_2\rangle\}$ unless otherwise stated.

If the basis follows $\langle \phi_i | \phi_j \rangle = \delta_{ij}$ then they are orthonormal. Furthermore $\sum_{i} |\phi_i\rangle \langle \phi_i | = I$ states that the basis is complete.

Block Sphere Representation

Any state $|\psi\rangle$ can be represented as a point on the block sphere. The state $|\psi\rangle$ is represented by the vector $\vec{r} = (r, \theta, \phi)$ where a general state is given by:

\[ |\psi\rangle = e^{i\gamma} \cos(\frac{\theta}{2}) |0\rangle + e^{i\gamma}e^{i\phi} \sin(\frac{\theta}{2}) |1\rangle \]

$\gamma$ is the global phase.

Operators

An operator is a linear map from one Hilbert space to another. An operator $\hat{A}$ acting on a state $|\psi\rangle$ gives another state $|\psi'\rangle$.

\[\hat{A}|\psi\rangle = |\psi'\rangle\]

The operator $\hat{A}$ can be represented as a matrix in the basis $\{|\phi_1\rangle, |\phi_2\rangle, ..., \phi_n\rangle\}$ as:

\[\hat{A} = \sum_{i,j} A_{ij} |\phi_i\rangle \langle \phi_j |\]

The matrix elements $A_{ij}$ are given by:

\[A_{ij} = \langle \phi_i | \hat{A} | \phi_j \rangle\]

The operator $\hat{A}$ is said to be Hermitian if $\hat{A} = \hat{A}^\dagger$ where $\hat{A}^\dagger$ is the adjoint of $\hat{A}$.

\[\hat{A}^\dagger = \sum_{i,j} A_{ij}^* |\phi_j\rangle \langle \phi_i |\]

The adjoint of an operator is the transpose of the matrix elements with complex conjugate.

Eigenvalues of an operator $\hat{A}$ are the values $\lambda$ such that:

\[\hat{A}|\psi\rangle = \lambda|\psi\rangle\]

The eigenvalues of a Hermitian operator are real. i.e. $\lambda \in \mathbb{R}$

Unitary operators are those that preserve the norm of the state. i.e. $\hat{U}^\dagger \hat{U} = \hat{U} \hat{U}^\dagger = I$

\[\hat{U}(t) = e^{-i\hat{H}t/\hbar}\]

The operator $\hat{H}$ is called the Hamiltonian and is Hermitian. The Hamiltonian is the energy operator.

Commutator of two operators $\hat{A}$ and $\hat{B}$ is defined as:

\[[\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}\]

$[\hat{A}, \hat{B}] = 0$ if $\hat{A}$ and $\hat{B}$ commute i.e. $\hat{A}\hat{B} = \hat{B}\hat{A}$

Uncertainty principle: $[\hat{A}, \hat{B}] \neq 0$ implies that the operators do not commute and the uncertainty in the measurement of $\hat{A}$ and $\hat{B}$ cannot be simultaneously zero.

\[\Delta A \Delta B \geq \frac{1}{2} |\langle [\hat{A}, \hat{B}] \rangle|\]

where $\Delta A = \sqrt{\langle \hat{A}^2 \rangle - \langle \hat{A} \rangle^2} = \sqrt{\langle \psi | \hat{A}^2 | \psi \rangle - \langle \psi | \hat{A} | \psi \rangle^2}$

Pauli Matrices

Defining $|0\rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$ and $|1\rangle = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$

The Pauli matrices are defined as:

Identity \[ I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = |0\rangle \langle 0 | + |1\rangle \langle 1 | \]

Pauli-X \[ \sigma_x = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} = |0\rangle \langle 1 | + |1\rangle \langle 0 | \]

Pauli-Y \[ \sigma_y = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix} = -i|0\rangle \langle 1 | + i|1\rangle \langle 0 | \]

Pauli-Z \[ \sigma_z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} = |0\rangle \langle 0 | - |1\rangle \langle 1 | \]

The Pauli matrices are Hermitian and unitary. The Pauli matrices are also traceless. They do not commute with each other.

$[\sigma_x, \sigma_y] = 2i\sigma_z\ ;\ [\sigma_y, \sigma_z] = 2i\sigma_x\ ;\ [\sigma_z, \sigma_x] = 2i\sigma_y$

On Bloch sphere, $\sigma_x$ is rotation about $x$-axis by $\pi$ radians, $\sigma_y$ is rotation about $y$-axis by $\pi$ radians and $\sigma_z$ is rotation about $z$-axis by $\pi$ radians.

\[ \begin{bmatrix} \langle \sigma_x \rangle \\ \langle \sigma_y \rangle \\ \langle \sigma_z \rangle \end{bmatrix} = \begin{bmatrix} \langle \psi | \sigma_x | \psi \rangle \\ \langle \psi | \sigma_y | \psi \rangle \\ \langle \psi | \sigma_z | \psi \rangle \end{bmatrix} = \begin{bmatrix} \cos(\phi)\sin(\theta) \\ \sin(\phi)\sin(\theta) \\ \cos(\theta) \end{bmatrix} \]

where $|\psi\rangle = \cos(\frac{\theta}{2}) |0\rangle + e^{i\phi} \sin(\frac{\theta}{2}) |1\rangle$

Eigenvectors of Pauli matrices are:

where $|+\rangle = \frac{1}{\sqrt{2}} (|0\rangle + |1\rangle)\ ;\ |-\rangle = \frac{1}{\sqrt{2}} (|0\rangle - |1\rangle)\ ;\ |R\rangle = \frac{1}{\sqrt{2}} (|0\rangle + i|1\rangle)$ and $|L\rangle = \frac{1}{\sqrt{2}} (|0\rangle - i|1\rangle)$

The transformations can be represented as: \[(r, \theta, \phi) \overset{\sigma_x}{\rightarrow} (r, \theta, \pi + \phi)\] \[(r, \theta, \phi) \overset{\sigma_y}{\rightarrow} (r, \pi - \theta, -\phi)\] \[(r, \theta, \phi) \overset{\sigma_z}{\rightarrow} (r, \pi - \theta, -\pi - \phi)\]

Single Photon Polarization

Let $|H|0$ and $|V\rangle \equiv |1\rangle$ be the horizontal and vertical polarization states of a photon.

The diagonal polarization states are $|D\rangle = \frac{1}{\sqrt{2}} (|H\rangle + |V\rangle)$ and $|A\rangle = \frac{1}{\sqrt{2}} (|H\rangle - |V\rangle)$

The circular polarization states are $|R\rangle = \frac{1}{\sqrt{2}} (|H\rangle + i|V\rangle)$ and $|L\rangle = \frac{1}{\sqrt{2}} (|H\rangle - i|V\rangle)$

Effect of optical elements on polarization states:

Linear Polarizer

A linear polarizer transmits light polarized along a particular direction. The transmission axis of the polarizer is the direction of polarization of the transmitted light.

\[|\psi\rangle \overset{P}{\rightarrow} |\psi'\rangle\]

\[ P = \begin{bmatrix} \cos^2(\theta) & \cos(\theta)\sin(\theta) \\ \cos(\theta)\sin(\theta) & \sin^2(\theta) \end{bmatrix} \]

where $\theta$ is the angle between the transmission axis of the polarizer and the horizontal axis.

Quarter Wave Plate

A quarter wave plate converts linear polarization to circular polarization and vice versa. The quarter wave plate is oriented at an angle of $\theta$ with respect to the horizontal axis.

\[ QWP = \begin{bmatrix} \exp(-i\pi/4) & 0 \\ 0 & \exp(i\pi/4) \end{bmatrix} = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & -i \\ -i & 1 \end{bmatrix} \]

Half Wave Plate

A half wave plate converts linear polarization to linear polarization and vice versa. The half wave plate is oriented at an angle of $\theta$ with respect to the horizontal axis.

\[HWP = \sigma_z\]

More Operators

Trace operator

The trace operator is defined as:

\[ Tr(\hat{A}) = \sum_i \langle \phi_i | \hat{A} | \phi_i \rangle \]

The inner product of two states $|\psi\rangle$ and $|\phi\rangle$ can be given by:

\[\langle \psi | \phi \rangle = Tr(|\psi\rangle \langle \phi|)\]

Density Operator

The density operator is defined as:

\[ \rho = |\psi\rangle \langle \psi| \]

where $\rho_{ij} = \langle \phi_i | \psi \rangle \langle \psi | \phi_j \rangle$

The expectation value of an operator $\hat{A}$ is given by:

$\langle \hat{A} \rangle = Tr(\rho \hat{A})\\Proof:\\$ $\langle \hat{A} \rangle = \langle \psi | \hat{A} | \psi \rangle = \sum_{i,j} \rho_{ij} \langle \phi_i | \hat{A} | \phi_j \rangle \\ = \sum_{i} \rho_{i} \langle \phi_i | \hat{A} \sum_{j} |\phi_j\rangle \langle \phi_j| \phi_j \rangle = \sum_{i} \rho_{i} \langle \phi_i | \hat{A} | \phi_i \rangle \\ = Tr(\rho \hat{A})$

The density operator can be written as:

\[\rho = \frac{1}{2} (I + \vec{r} \cdot \vec{\sigma})\]

where $\vec{r} = (r, \theta, \phi)$

\[Tr(\rho) = 1\]

Purity of a state is given by: \[Tr(\rho^2) = \sum_{i} \rho_{ii}^2 \leq 1\] if $Tr(\rho^2) = 1$ then the state is pure else it is mixed.
Purity = 1 - $Tr(\rho^2)$

Von Neumann entropy is given by: \[S(\rho) = -Tr(\rho \log_2 \rho)\] $S(\rho) = 0$ for pure states and $S(\rho) > 0$ for mixed states.

Measurement

Probability of measuring a state $|\psi\rangle$ in the state $|\phi\rangle$ is given by:

\[P(\phi) = |\langle \phi | \psi \rangle|^2\]

It can be also given using the density operator as:

\[P(\phi) = \langle \phi | \rho | \phi \rangle = Tr(\rho |\phi\rangle \langle \phi|)\]

\[ P(\phi) = Tr(\rho \Pi_\phi) \]

where $\Pi_\phi = |\phi\rangle \langle \phi|$ is the projection operator.

\[\sum_{\phi} \Pi_\phi = I\]

\[\Pi_\phi \Pi_\psi = \delta_{\phi\psi} \Pi_\phi\]

The expectation value of an operator $\hat{A}$ is given by:

\[\langle \hat{A} \rangle = \sum_{\phi} P(\phi) \langle \phi | \hat{A} | \phi \rangle = Tr(\rho \hat{A})\]

The density operator after measurement is given by:

\[ \rho' = \frac{\Pi_\phi \rho \Pi_\phi}{P(\phi)} = \frac{\Pi_\phi \rho \Pi_\phi}{Tr(\rho \Pi_\phi)} \]

Positive Operator Valued Measure (POVM)

A POVM is a set of operators $\{E_i\}$ such that $E_i \geq 0$ and $\sum_i E_i = I$ where $I$ is the identity operator.

The probability of measuring a state $|\psi\rangle$ in the state $|\phi\rangle$ is given by:

\[P(\phi) = \sum_i \langle \phi | E_i | \phi \rangle = Tr(\rho E_i)\]

Multiple Quantum Systems

2 level system => qubit

d level system => qudit

The Hilbert space of a composite system is the tensor product of the Hilbert spaces of the individual systems.

\[\mathcal{H} = \mathcal{H}_1 \otimes \mathcal{H}_2\]

The basis vectors of the composite system are given by:

\[|\phi_i\rangle \otimes |\psi_j\rangle = |\phi_i\psi_j\rangle\]

The basis vectors of the composite system are orthonormal if the basis vectors of the individual systems are orthonormal.

\[\langle \phi_i | \phi_j \rangle = \delta_{ij}\ ;\ \langle \psi_i | \psi_j \rangle = \delta_{ij}\]

The basis vectors of the composite system are complete if the basis vectors of the individual systems are complete.

\[\sum_{i,j} |\phi_i\rangle \langle \phi_i | \otimes |\psi_j\rangle \langle \psi_j | = I\]

The density operator of the composite system is given by:

\[\rho = \rho_1 \otimes \rho_2\]

The expectation value of an operator $\hat{A}$ is given by:

\[\langle \hat{A} \rangle = Tr(\rho \hat{A}) = Tr(\rho_1 \otimes \rho_2 \hat{A}_1 \otimes \hat{A}_2) = Tr(\rho_1 \hat{A}_1) Tr(\rho_2 \hat{A}_2)\]

Generic States:

\[|\psi\rangle_a = \sum_{i} c_{i} |\phi_i\rangle\ ;\ |\psi\rangle_b = \sum_{j} d_{j} |\psi_j\rangle\ where\ c_{ij} \in \mathbb{C}\]

\[ |\psi\rangle_{ab} = \sum_{i,j} c_{ij} |\phi_i\rangle \otimes |\psi_j\rangle \] e.g. $|\psi\rangle_{ab} = \frac{1}{2} (|00\rangle + |01\rangle + |10\rangle + |11\rangle)\\ = \frac{1}{\sqrt{2}} (|0\rangle_a + |1\rangle_a) \otimes \frac{1}{\sqrt{2}} (|0\rangle_b + |1\rangle_b)\\ = |\psi\rangle_a \otimes |\psi\rangle_b$

Non-generic States (non-separable): \[|\psi\rangle_{ab} \neq |\psi\rangle_a \otimes |\psi\rangle_b\]

e.g. $|\psi\rangle_{ab} = \alpha|00\rangle + \beta|11\rangle$

They are called entangled states.

Bell States

The Bell states are given by:

\[ |\Phi^+\rangle = \frac{1}{\sqrt{2}} (|00\rangle + |11\rangle)\] \[ |\Phi^-\rangle = \frac{1}{\sqrt{2}} (|00\rangle - |11\rangle)\] \[ |\Psi^+\rangle = \frac{1}{\sqrt{2}} (|01\rangle + |10\rangle)\] \[ |\Psi^-\rangle = \frac{1}{\sqrt{2}} (|01\rangle - |10\rangle)\]

$|\Phi^+\rangle$ and $|\Phi^-\rangle$ are symmetric under exchange of qubits.

$|\Psi^+\rangle$ and $|\Psi^-\rangle$ are anti-symmetric under exchange of qubits.

Probability of measuring a Bell state $|\Phi^+\rangle$ in the state $|00\rangle$ is given by: > $P(00) = |\langle 00 | \Phi^+ \rangle|^2 = \frac{1}{2}$

This randomness of Bell states is used in Quantum Technologies.

Bell states are eigenstates of the Pauli operators of the composite systems with eigenvalues = +-1

Entanglement persistes in different basis

Entanglement Quantification

Partial Projection

Partial projection of a state $|\psi\rangle$ is given by:

\[ \Pi_a = \sum_{i} |\phi_i\rangle \langle \phi_i | \otimes I_b \]

Partial Trace

Partial trace of a state $|\psi\rangle$ is given by:

\[ \rho_a = Tr_b(|\psi\rangle \langle \psi|) = \sum_{i} \langle \phi_i | \psi \rangle \langle \psi | \phi_i \rangle \] \[ \rho_b = Tr_a(|\psi\rangle \langle \psi|) = \sum_{j} \langle \psi_j | \psi \rangle \langle \psi | \psi_j \rangle \]

3 Qubit System

The basis vectors of the composite system are given by:

\[|\phi_i\rangle \otimes |\psi_j\rangle \otimes |\chi_k\rangle = |\phi_i\psi_j\chi_k\rangle\]

GHZ State

The GHZ state is given by:

\[ |\psi\rangle_{GHZ} = \frac{1}{\sqrt{2}} (|000\rangle + |111\rangle) \]

The state is maximally entangled.

W State

The W state is given by:

\[ |\psi\rangle_{W} = \alpha |001\rangle + \beta |010\rangle + \gamma |100\rangle \]

Entanglement Entropy / Von Neumann Entropy

The entanglement entropy of a state $|\psi\rangle$ is given by:

\[ S(\rho_a) = -Tr(\rho_a \log_2 \rho_a) \]

where $\rho_a = Tr_b(|\psi\rangle \langle \psi|)$

Schmidt Decomposition

The Schmidt decomposition of a state $|\psi\rangle$ is given by:

\[ |\psi\rangle = \sum_{i} \sqrt{\lambda_i} |\phi_i\rangle \otimes |\psi_i\rangle \]

where $\lambda_i$ are the eigenvalues of $\rho_a$ and $|\phi_i\rangle$ and $|\psi_i\rangle$ are the eigenvectors of $\rho_a$ and $\rho_b$ respectively.

\[\rho_a = \sum_{i} \lambda_i |\phi_i\rangle \langle \phi_i |\\ \rho_b = \sum_{i} \lambda_i |\psi_i\rangle \langle \psi_i |\]

\[ S(\rho_a) = -\sum_{i} \lambda_i \log_2 \lambda_i = S(\rho_b) \]