Mathematics of Quantum Mechanics ii

Part 3: Quantum Mechanics

3.1 Mathematical Postulates of Quantum Mechanics

1. Quantum States: Quantum states are described by unit vectors in a complex Hilbert space. In the context of quantum mechanics, the state of a quantum system is represented as a vector \(\vert\psi\rangle\) in a complex - valued Hilbert space \(\mathcal{H}\), where \(\|\vert\psi\rangle\| = 1\).

2. Quantum Measurement: The probability of measuring a system in a given state \(\vert\varphi\rangle\) is given by \(P(\varphi)=\vert\langle\varphi\vert\psi\rangle\vert^{2}\) (Born’s rule). After measurement, the wave - function \(\vert\psi\rangle\) collapses into the state \(\vert\varphi\rangle\). If we have a set of orthonormal basis states \(\{\vert\varphi_i\rangle\}\) in the Hilbert space, and the system is in the state \(\vert\psi\rangle\), the probability of obtaining the state \(\vert\varphi_i\rangle\) upon measurement is calculated using this rule.

3. Quantum Operations: Quantum operations are represented by unitary operators \(U\) on the Hilbert space, i.e., \(UU^{\dagger}=U^{\dagger}U = I\), where \(U^{\dagger}\) is the conjugate - transpose of \(U\) and \(I\) is the identity operator. Unitary operations preserve the norm of the quantum state, \(\|\ U\vert\psi\rangle\|=\|\vert\psi\rangle\|\).

4. Multiple Quantum States: The Hilbert space of a composite system consisting of subsystems with Hilbert spaces \(\mathcal{H}_1\) and \(\mathcal{H}_2\) is given by the tensor product \(\mathcal{H}=\mathcal{H}_1\otimes\mathcal{H}_2\).

5. Observables: Physical observables are represented by the eigenvalues of a Hermitian operator \(H\) on the Hilbert space. A Hermitian operator satisfies \(H = H^{\dagger}\).、

• Common Conclusions:
  • Quantum states are normalized, \(\langle\psi\vert\psi\rangle = 1\), to ensure that \(\sum_iP(\varphi_i)=1\), where \(P(\varphi_i)\) is the probability of measuring the system in the state \(\vert\varphi_i\rangle\) as given by Born’s rule.
  • Unitary operations \(U\) satisfy \(\langle U\vert\psi\rangle\vert U\vert\varphi\rangle\rangle=\langle\vert\psi\rangle\vert\varphi\rangle\rangle\), which means they preserve the inner product and the norm of quantum states.

3.2 New Notation: The Bra - Ket Notation

• Ket: \(\vert v\rangle\) represents a vector \(v\) in the Hilbert space.
• Bra: \(\langle v\vert\) represents the conjugate - transpose of \(\vert v\rangle\). If \(\vert v\rangle=\sum_iv_i\vert e_i\rangle\), then \(\langle v\vert=\sum_i\overline{v_i}\langle e_i\vert\), where \(\{\vert e_i\rangle\}\) is an orthonormal basis of the Hilbert space.
• Bra - Ket: \(\langle v\vert w\rangle=\vert v\rangle\cdot\vert w\rangle\), which is the inner product of the vectors \(\vert v\rangle\) and \(\vert w\rangle\).
  • Example: Given \(\vert a\rangle=\begin{bmatrix}1\\0\end{bmatrix}\) and \(\vert b\rangle=\begin{bmatrix}0\\1\end{bmatrix}\), the inner product \(\langle a\vert b\rangle=\begin{bmatrix}1&0\end{bmatrix}\begin{bmatrix}0\\1\end{bmatrix}=0\).
• Common Conclusions:
  • The bra - ket notation simplifies the representation of quantum states and operations. For example, when calculating the probability of a measurement outcome using Born’s rule \(P(\varphi)=\vert\langle\varphi\vert\psi\rangle\vert^{2}\), the notation makes the operations more intuitive.
  • The inner product \(\langle v\vert w\rangle\) is a scalar that represents the overlap between two quantum states. A larger magnitude of \(\langle v\vert w\rangle\) indicates a greater similarity between the two states.

3.3 Single Quantum State and the Qubit

• Qubit: The basic unit of quantum information, with two exclusive states \(\vert0\rangle\) and \(\vert1\rangle\). These states form an orthonormal basis for the two - dimensional Hilbert space of a qubit, \(\langle0\vert0\rangle = 1\), \(\langle1\vert1\rangle = 1\), and \(\langle0\vert1\rangle = 0\).
• Quantum Superposition Principle: A quantum system can be in a superposition of \(\vert0\rangle\) and \(\vert1\rangle\), represented as \(\vert\psi\rangle=a\vert0\rangle + b\vert1\rangle\), where \(a,b\in\mathbb{C}\) and \(\vert a\vert^{2}+\vert b\vert^{2}=1\).
  • Example: A qubit in the state \(\vert\psi\rangle=\frac{1}{\sqrt{2}}(\vert0\rangle+\vert1\rangle)\) is in a superposition of \(\vert0\rangle\) and \(\vert1\rangle\) with equal coefficients. Here, \(a = b=\frac{1}{\sqrt{2}}\).
• Common Conclusions:
  • The coefficients \(a\) and \(b\) in the superposition state \(\vert\psi\rangle=a\vert0\rangle + b\vert1\rangle\) determine the probabilities of measuring the qubit in the \(\vert0\rangle\) or \(\vert1\rangle\) state. Specifically, \(P(0)=\vert a\vert^{2}\) and \(P(1)=\vert b\vert^{2}\).
  • The superposition principle is a fundamental feature of quantum mechanics, distinguishing it from classical physics. In classical physics, a system can be in only one state at a time, while in quantum mechanics, a qubit can be in a superposition of multiple states simultaneously.

3.4 Quantum Measurement

• Born’s Rule: The probability of measuring \(\vert\psi\rangle\) in the state \(\vert\varphi_i\rangle\) is \(P(\varphi_i)=\vert\langle\varphi_i\vert\psi\rangle\vert^{2}\).
  • Wave Function Collapse: After measurement, the state collapses to \(\vert\varphi_i\rangle\).
  • Example: Given \(\vert\psi\rangle=\frac{1}{\sqrt{2}}(\vert0\rangle+\vert1\rangle)\) and \(\vert\varphi_0\rangle=\vert0\rangle\), the probability of measuring \(\vert\psi\rangle\) in \(\vert0\rangle\) is \(P(0)=\vert\langle0\vert\psi\rangle\vert^{2}=\left|\frac{1}{\sqrt{2}}\right|^{2}=\frac{1}{2}\).
• Common Conclusions:
  • Quantum measurement is probabilistic, and the outcome is determined by the inner product of the measured state and the basis states. This is in contrast to classical measurements, which are usually deterministic.
  • The wave - function collapse is a non - reversible process that occurs upon measurement. Once the measurement is made, the state of the system changes from the superposition state to one of the eigenstates of the measured observable.

3.5 Quantum Operations

• Quantum Operations: Represented by unitary matrices \(U\) such that \(UU^{\dagger}=U^{\dagger}U = I\).
• Example: The Hadamard gate \(H\) is a unitary matrix \(H=\frac{1}{\sqrt{2}}\begin{bmatrix}1&1\\1& - 1\end{bmatrix}\). Applying \(H\) to \(\vert0\rangle\): \(H\vert0\rangle=\frac{1}{\sqrt{2}}\begin{bmatrix}1&1\\1& - 1\end{bmatrix}\begin{bmatrix}1\\0\end{bmatrix}=\frac{1}{\sqrt{2}}\begin{bmatrix}1\\1\end{bmatrix}=\vert+\rangle\).
• Common Conclusions:

  • Unitary operations are essential for manipulating quantum states without altering their norm. They can be used to change the state of a qubit or a multi - qubit system in a way that preserves the quantum information.

  • Quantum operations can be combined by matrix multiplication. For example, if \(U_1\) and \(U_2\) are two unitary operations, then applying \(U_1\) followed by \(U_2\) is equivalent to applying the product \(U_2U_1\). The order of operations matters because matrix multiplication is non - commutative in general.

3.6 Multiple Quantum States

• Tensor Product: The tensor product of two vectors \(v\) and \(w\) is \(v\otimes w\). If \(v=\sum_iv_i\vert e_i\rangle\) and \(w=\sum_jw_j\vert f_j\rangle\), then \(v\otimes w=\sum_{i,j}v_iw_j\vert e_i\rangle\otimes\vert f_j\rangle\).
• Bell States: A set of orthonormal entangled states for two qubits.
  • Example: The Bell state \(\vert\Phi^{+}\rangle=\frac{1}{\sqrt{2}}(\vert00\rangle+\vert11\rangle)\).

Entangled States and Separable States

Definitions:

• Separable State: A state of a composite quantum system that can be written as the tensor - product of states of its individual subsystems. Mathematically, a two - qubit state \(|\psi\rangle\) is separable if it can be expressed as:
  \(|\psi\rangle = |\phi_A\rangle\otimes|\phi_B\rangle\)
  where \(|\phi_A\rangle\) and \(|\phi_B\rangle\) are states of the individual subsystems.

• Entangled State: A state of a composite quantum system that cannot be written as the tensor - product of states of its individual subsystems. Mathematically, a two - qubit state \(|\psi\rangle\) is entangled if it cannot be expressed as:
  \(|\psi\rangle\neq|\phi_A\rangle\otimes|\phi_B\rangle\)

Examples:

• Separable State Example: Consider the state:
  \(|\psi\rangle = |0\rangle\otimes|+\rangle = |0\rangle|+\rangle\)
  where \(|0\rangle\) is the standard basis state and \(|+\rangle=\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\) is a superposition state. This state is separable because it can be written as the tensor - product of \(|0\rangle\) and \(|+\rangle\).

• Entangled State Example: Consider the Bell state:
  \(|\Phi^+\rangle=\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)\)
  This state is entangled because it cannot be written as the tensor - product of two single - qubit states.

  • To see this, assume for contradiction that \(|\Phi^+\rangle\) is separable:
    \(|\Phi^+\rangle=(a|0\rangle + b|1\rangle)\otimes(c|0\rangle + d|1\rangle)\)
    Expanding the right - hand side:
    \((a|0\rangle + b|1\rangle)\otimes(c|0\rangle + d|1\rangle)=ac|00\rangle + ad|01\rangle + bc|10\rangle + bd|11\rangle\)

  Comparing coefficients with \(|\Phi^+\rangle=\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)\), we get:\(ac = \frac{1}{\sqrt{2}}, ad = 0, bc = 0, bd = \frac{1}{\sqrt{2}}\)

  From \(ad = 0\) and \(bc = 0\), either \(a = 0\) or \(d = 0\), and either \(b = 0\) or \(c = 0\). However, if \(a = 0\) or \(b = 0\), then \(ac\) or \(bd\) cannot be \(\frac{1}{\sqrt{2}}\). Similarly, if \(c = 0\) or \(d = 0\), then \(ac\) or \(bd\) cannot be \(\frac{1}{\sqrt{2}}\). Therefore, no such \(a,b,c,d\) exist, and \(|\Phi^+\rangle\) is entangled.

• Common Conclusions:

  • The tensor product allows the representation of composite quantum systems. For a two - qubit system, the Hilbert space is the tensor product of the two - dimensional Hilbert spaces of each qubit.
  • Entangled states exhibit correlations that have no classical analog, making them a key resource in quantum information processing. For example, in quantum teleportation and quantum key distribution, entangled states play a crucial role.

3.7 Observables

• Observables: Represented by the eigenvalues of a Hermitian operator on the Hilbert space.
• Example: The Pauli - Z operator \(Z=\begin{bmatrix}1&0\\0& - 1\end{bmatrix}\) is a Hermitian matrix. The eigenvalues of \(Z\) are \(\lambda_1 = 1\) and \(\lambda_2=-1\), corresponding to the eigenstates \(\vert0\rangle\) and \(\vert1\rangle\) respectively. To find the eigenvalues, we solve the characteristic equation \(\det(Z - \lambda I)=0\), where \(I\) is the \(2\times2\) identity matrix.

Conclusions:

• Hermitian operators have real eigenvalues, which correspond to the possible measurement outcomes. This is important because physical observables, such as energy or spin, are real - valued.

• The expectation value of an observable \(A\) in a given quantum state \(\vert\psi\rangle\) is \(\langle A\rangle=\langle\psi\vert A\vert\psi\rangle\), which represents the average result of many measurements of the observable \(A\) on the state \(\vert\psi\rangle\).

Common natation

• Ket: \(\vert v\rangle\) represents a vector \(v\).
• Bra: \(\langle v\vert\) represents the conjugate - transpose of \(\vert v\rangle\).
• Bra - Ket: \(\langle v\vert w\rangle\) represents the inner product of \(\vert v\rangle\) and \(\vert w\rangle\).

Common Conclusion

1. Unitary Operator \(U\):
   A unitary operator \(U\) satisfies the condition:
   \(U^{\dagger}U = UU^{\dagger}=I\)
   where \(U^{\dagger}\) is the conjugate transpose of \(U\), and \(I\) is the identity matrix.

2. Hermitian Operator \(H\):
   A Hermitian operator \(H\) satisfies the condition:
   \(H = H^{\dagger}\)
   This means that for any vectors \(\vert\psi\rangle\) and \(\vert\phi\rangle\) in the Hilbert space:
   \(\langle\psi\vert H\vert\phi\rangle=\langle\phi\vert H\vert\psi\rangle^{*}\)

3. Identity Matrix \(I\):
   The identity matrix \(I\) is a square matrix with ones on the diagonal and zeros elsewhere:
   \(I=\begin{pmatrix} 1&0&\cdots&0\\ 0&1&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&1 \end{pmatrix}\)

4. Pauli Matrices \(X\), \(Y\), \(Z\):
   The Pauli matrices are a set of three \(2\times2\) Hermitian and unitary matrices:

\[X=\begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}, \quad Y=\begin{pmatrix} 0&i\\ - i&0 \end{pmatrix}, \quad Z=\begin{pmatrix} 1&0\\ 0& - 1 \end{pmatrix}\]

5. Hadamard Gate \(H\):
   The Hadamard gate \(H\) is represented by the matrix:
   \(H=\frac{1}{\sqrt{2}}\begin{pmatrix} 1&1\\ 1& - 1 \end{pmatrix}\)
   This operator satisfies \(H^{2}=I\).

6. Bell States \(\vert\Phi^{+}\rangle\), \(\vert\Phi^{-}\rangle\), \(\vert\Psi^{+}\rangle\), \(\vert\Psi^{-}\rangle\):
   The Bell states are four maximally entangled two - qubit states:
   \(\vert\Phi^{+}\rangle=\frac{1}{\sqrt{2}}(\vert00\rangle+\vert11\rangle)\)
   \(\vert\Phi^{-}\rangle=\frac{1}{\sqrt{2}}(\vert00\rangle - \vert11\rangle)\)
   \(\vert\Psi^{+}\rangle=\frac{1}{\sqrt{2}}(\vert01\rangle+\vert10\rangle)\)
   \(\vert\Psi^{-}\rangle=\frac{1}{\sqrt{2}}(\vert01\rangle - \vert10\rangle)\)