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:
\[\sigma_x |+\rangle = |+\rangle\ ;\ \sigma_x |-\rangle = -|-\rangle\] \[\sigma_y |R\rangle = |R\rangle\ ;\ \sigma_y |L\rangle = -|L\rangle\] \[\sigma_z |0\rangle = |0\rangle\ ;\ \sigma_z |1\rangle = -|1\rangle\]
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.
\[|\psi\rangle \overset{P}{\rightarrow} |\psi'\rangle\]
where \(|\psi\rangle = \alpha|H\rangle + \beta|V\rangle\) and \(|\psi'\rangle = \alpha|H\rangle\)
\[ P = \begin{bmatrix} \cos^2(\theta) & \cos(\theta)\sin(\theta) \\ \cos(\theta)\sin(\theta) & \sin^2(\theta) \end{bmatrix} \]
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.
\[\rho_a = \sum_{i} \lambda_i |\phi_i\rangle \langle \phi_i |\\ \rho_b = \sum_{i} \lambda_i |\psi_i\rangle \langle \psi_i |\]
\[ S(\rho_a) = -\sum_{i} \lambda_i \log_2 \lambda_i = S(\rho_b) \]