Postulates of Quantum Mechanics

Postulates

Associated to any isolated physical system is a complex vector space with inner product known as state space of the system. The system is completely describled by its state vector/density operator, which is a unit vector in the systems state
- state vector: $|\psi_i\rangle$
- density operator: $\rho=\sum_ip_i|\psi_i\rangle\langle\psi_i|$
The evolution of a closed quantum system is described by a unitary transformation
- $|\psi'\rangle=U|\psi\rangle$
- $\rho'=U\rho U^\dagger$
Quantum measuerments are described by a collection $\{M_m\}$ ${M_{m}}$ of measurement operator ( $m$ $m$ is number of possible outcome).
- measurement operator: $\sum_mM_m^\dagger M_m=I$ $\sum_{m} M_{m}^{†} M_{m} = I$
  - $M_0=|0\rangle\langle 0|,M_1=|1\rangle\langle 1|$
  - not unitary
- Original
  - $P(m)=\langle\psi|M_m^\dagger M_m|\psi\rangle$
  - state after the measurement: $\frac{M_m|\psi\rangle}{\sqrt{\langle\psi|M_m^\dagger M|\psi\rangle}}$
- Reformed
  - $P(m)=\text{Tr}(M_m^+M_m\rho)$
  - state after the measurement: $\frac{M_m\rho M_m^\dagger}{\text{Tr}(M_m^\dagger M_m\rho)}$
The state space of a composite physical system is the tensor prorduct of the state spaces of the component physical systems.
- $|\psi_1\rangle\otimes\cdots\otimes |\psi_n\rangle$
- $\rho_1\otimes\cdots\otimes\rho_n$
- 前提：两个系统独立

Use a set of quantum states $\{|\psi_1\rangle,\cdots,|\psi_m\rangle\}$ ${∣ ψ_{1} ⟩, \dots, ∣ ψ_{m} ⟩}$ and probability distribution $\{p_1,\cdots,p_m\}$ ${p_{1}, \dots, p_{m}}$
- Ensemble of pure state: $\{p_i,|\psi_i\rangle\}$
- Density operator (mixed state): $\rho=\sum_ip_i|\psi_i\rangle\langle\psi_i|$ $ρ = \sum_{i} p_{i} ∣ ψ_{i} ⟩ ⟨ ψ_{i} ∣$
  - $\text{Tr}(\rho)=1$
  - positive semidefinite
  - $|0\rangle\rightarrow |0\rangle\langle0|,|1\rangle\rightarrow |1\rangle\langle1|$
- Ensemble of mixed state: $\{p_i,\rho_i\}$ : $\rho=\sum_ip_i\rho_i$
POVM(Positive operator-valued measure): $E_m=M_m^\dagger M_m$ $E_{m} = M_{m}^{†} M_{m}$
- positive semidefinite and $\sum_mE_m=I$
- $P(m)=\text{Tr}(E_m\rho)$
Projective measurement: $p(m)=\text{Tr} (P_m\rho)$ $p (m) = Tr (P_{m} ρ)$
- post-measurement state: $\rho_m=\frac{P_m\rho P_m}{\text{Tr} (P_m\rho)}$
- 所有测量可以转化为投影测量
Partial trace: $tr_B\rho^{AB}=\sum_{i=0}^{d_B-1}(I\otimes\langle i|)\rho^{AB}(I\otimes|i\rangle)$ $t r_{B} ρ^{A B} = \sum_{i = 0}^{d_{B} - 1} (I \otimes ⟨ i ∣) ρ^{A B} (I \otimes ∣ i ⟩)$
- $tr_B(|a_1\rangle\langle a_2|\otimes|b_1\rangle\langle b_2|)=tr(|b_1\rangle\langle b_2|)|a_1\rangle\langle a_2|$
reduced density operator: $\rho^A=tr_B\rho^{AB}$ $ρ^{A} = t r_{B} ρ^{A B}$
- state of system $A$ : $\rho^A$

Bell States(EPR States): $|\beta_{xy}\rangle=\frac{|0y\rangle+(-1)^x|1,1-y\rangle}{\sqrt{2}}$
$|\psi\rangle=\alpha|0\rangle+\beta|1\rangle=\frac{\alpha+\beta}{\sqrt{2}}|+\rangle+\frac{\alpha-\beta}{\sqrt{2}}|-\rangle$
state vector coefficient $e^{i\theta}|\psi\rangle$ $e^{i θ} ∣ ψ ⟩$ : global phase, nonsense in physics
- relative phase: meaningful
Theroem(Distinguishing Quantum States): there is measurement distinguishing two states perfectly iif $|\psi_1\rangle$ and $|\psi_2\rangle$ are orthogonal
Principle of deferred measurement: Measurement can always be moved from an intermediate state of a quantum circuit to the end of the circuit
Principle of implicit measurement: Without loss of generality, any unterminated quantum wires (qubits which are not measured) at the end of a quantum circuit may be assumed to be measured

$V(s,t,a,b)=1$ if $s\cdot t=a\oplus b$
Classic: $P=\frac{3}{4}$
Quantum: $P\geq 0.8$ $P \geq 0.8$
- share an EPR state
- if $x=1$ , Alice rotate $\frac{\pi}{8}$
- if $y=1$ , Bob rotate $-\frac{\pi}{8}$
- both measure qubits and output $a,b$