Units#

Review & Summary#

Entanglement#

Non-separable correlations in composite systems. Bell states: \(|\Phi^\pm\rangle = \frac{1}{\sqrt{2}}(|\uparrow\uparrow\rangle \pm |\downarrow\downarrow\rangle)\).

Density matrix#

\(\rho = \sum_i p_i|\psi_i\rangle\langle\psi_i|\). Pure: \(\mathrm{Tr}(\rho^2)=1\). Mixed: \(\mathrm{Tr}(\rho^2)<1\).

Partial trace#

\(\rho_A = \mathrm{Tr}_B(\rho_{AB})\). Reduced state describing subsystem \(A\) alone.

Von Neumann entropy#

\(S(\rho) = -\mathrm{Tr}(\rho\ln\rho)\). Measures mixedness; equals entanglement entropy for bipartite pure states.

CHSH inequality#

\(|\langle\mathcal{B}\rangle| \le 2\) classically; quantum violation up to \(2\sqrt{2}\) (Tsirelson bound).

POVM#

Generalized measurement: \(\{M_i\}\) with \(\sum_i M_i = I\), \(p_i = \mathrm{Tr}(M_i\rho)\).

Quantum channel#

CPTP map in Kraus form: \(\mathcal{E}(\rho) = \sum_k K_k\rho K_k^\dagger\), \(\sum_k K_k^\dagger K_k = I\).

Lindblad equation#

\(\dot\rho = -\frac{i}{\hbar}[H,\rho] + \sum_k(L_k\rho L_k^\dagger - \tfrac{1}{2}\{L_k^\dagger L_k,\rho\})\).