Formula Sheet 2 god help me

Checks

Pure state

$$\begin{array}{rl}\mathrm{Tr}({\rho }^2)=1& \Rightarrow \text{pure}\\ \mathrm{Tr}({\rho }^2)<1& \Rightarrow \text{mixed}\\ \mathrm{Tr}({\rho }^2)=\frac{1}{d}& \Rightarrow \text{max. mixed}\end{array}$$

Hermitian

$$A^{†}=A$$

Unitary

$$U^{†}U=I$$

Entangled

State is entangled iff it cannot be written as   $|\psi ⟩\ne |{\psi }_1⟩\otimes |{\psi }_2⟩$.

---

Tensor Product

Vectors  

$$\left(\begin{array}{c}a\\ b\end{array}\right)\otimes \left(\begin{array}{c}c\\ d\end{array}\right)=\left(\begin{array}{c}ac\\ ad\\ bc\\ bd\end{array}\right)$$

Matrices  

$$(A\otimes B)(C\otimes D)=(AC)\otimes (BD)$$

---

Dirac & Completeness

$$|a⟩=\sum _i{a}_i|v_i⟩,⟨a|={|a⟩}^{†},\sum _i|v_i⟩⟨v_i|=I$$

---

Complex Numbers / Euler

$$e^{i\theta }=\cos \theta +i\sin \theta ,\cos \theta =\frac{e^{i\theta }+e^{-i\theta }}{2},\sin \theta =\frac{e^{i\theta }-e^{-i\theta }}{2i}$$

Modulus: $|z{|}^2=z{z}^{\ast }$. if something is observable the eigenvalues must be real

---

Qubit States & Bloch Sphere

$$|\psi (\theta ,\varphi )⟩=\cos \frac{\theta }{2}|0⟩+e^{i\varphi }\sin \frac{\theta }{2}|1⟩$$

Bloch vector $\vec{r}=(\sin \theta \cos \varphi ,\sin \theta \sin \varphi ,\cos \theta )$,   $\rho =\frac{1}{2}(I+\vec{r}\cdot \vec{\sigma })$.

---

Pauli Matrices & Spin-½

$${\sigma }_x=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\sigma }_y=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\sigma }_z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$$ $$S_i=\frac{\hslash }{2}{\sigma }_i,[S_x,S_y]=i\hslash S_z(\text{cyclic})$$

Spin length: $S^2=S_x^2+S_y^2+S_z^2=\frac{3}{4}{\hslash }^2$.

---

Measurement (Born Rule)

Pure: $P(a_j)=|⟨a_j|\psi ⟩{|}^2$   Mixed: $P(a_j)=\mathrm{Tr}(\rho P_j)$ where $P_j=|a_j⟩⟨a_j|$. Expectation: $⟨A⟩=\mathrm{Tr}(\rho A)$,   Variance $(\Delta A{)}^2=⟨A^2⟩-⟨A{⟩}^2$.

---

Density Matrix Tools

• Purity $\gamma =\mathrm{Tr}({\rho }^2)$ (see checks above).  
• Partial trace: ${\rho }_A={\mathrm{Tr}}_B({\rho }_{AB})$.

---

Uncertainty Principle

$$\Delta A\Delta B\ge \frac{1}{2}|⟨[A,B]⟩|$$

For spin: $\Delta S_x\Delta S_y\ge \frac{\hslash }{2}|⟨S_z⟩|$.

---

Time Evolution

$$i\hslash \frac{d}{dt}|\psi (t)⟩=H|\psi (t)⟩,|\psi (t)⟩=e^{-iHt/\hslash }|\psi (0)⟩$$

Density matrix: $\rho (t)=U(t)\rho (0)U^{†}(t)$.

---

Bell States

$$\begin{array}{rl}|{\Phi }^{\pm }⟩& =\frac{1}{\sqrt{2}}(|00⟩\pm |11⟩),\\ |{\Psi }^{\pm }⟩& =\frac{1}{\sqrt{2}}(|01⟩\pm |10⟩)\end{array}$$

---

50 : 50 Beam-Splitter (for photon path qubit)

$$U_{\text{BS}}=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)$$

---

Quick Identities

• $(AB{)}^{†}=B^{†}{A}^{†}$
• $[A,B]=0\Rightarrow$ common eigenbasis.
• $\mathrm{Tr}(A\otimes B)=\mathrm{Tr}A\mathrm{Tr}B$
• Mod-squared amplitude: $|⟨a|\psi ⟩{|}^2\to$ probability.