POVM

# POVM - Effect operator $E_m$ discribes **outcome** of measurements without *only neccesary restrictions* $$E_m\geq 0\ \Leftrightarrow\ p_m\geq 0\\\sum_m E_m=1\ \Leftrightarrow\ \sum_mp_m=1$$ - Effect operator $E_m$: care about outcome $p_m$ - Measurement operator $M_m$: care about state $|\psi_m\rangle$ ### Effect operator $E_m$ - $\Leftrightarrow$ Positivity $\langle\psi|E_m|\psi\rangle\geq 0,\ \forall\ |\psi\rangle $ - $\Leftrightarrow$ Sumrule $\sum_m E_m=\mathbb{1}$ ### Measurement operator $M_{Ab}$ - operators on $A$:$$M_{Ab}=\langle b_B|U_{AB}|0_B\rangle$$ - post-measurement states on $A$: $$\rho_{Ab}=\frac{1}{p_b}M_{Ab}\rho_AM_{Ab}^\dagger$$ - The measurement probability $$p_b=\text{tr}_A(M_{Ab}\rho_AM_{Ab}^\dagger)=\text{tr}_A(M_{Ab}^\dagger M_{Ab}\rho_A)$$ where $$E_{Ab}=M_{Ab}^\dagger M_{Ab}$$ > ⭐️⭐️⭐️ A set of $\{E_{Ab}\}$ is called a **POVM** for system $A$ if $$E_{Ab}\geq 0\\ \sum_b E_{Ab}=\mathbb{1}_A$$ > ⭐️⭐️⭐️ A set of measurement operators $\{M_{Ab}\}$ is called a **POVM-measurement** if $$\sum_b M_{Ab}^\dagger M_{Ab} = \mathbb{1}_A$$ - $E_{Ab}$ and $M_{Ab}$ are not uniquely corresponding. $$M_{Ab}=V_{Ab}\sqrt{E_{Ab}}$$ - 🧐 *Does a POVM-measurement $\{M_{Ab}\}$ discribe a posterior states of general measurement?* - ❌ A POVM-measurement is constrained to *preserve purity*. - It cannot describe decoherence of pure state to mixed post-measurement state. - 🧐 *Does every POVM-measurement $\{M_{Ab}\}$ derive from a pure, complete-measurement model $(U_{AB},|0_B\rangle)$?* - ⭕️ $$M_{Ab}=\langle b_B|U_{AB}|0_B\rangle \Leftrightarrow U_{AB}=\sum_b M_{Ab}|b_B\rangle\langle 0_B|+...$$ - Nonuniqueness comes from - $\langle b_B|$ $U_{AB}|c_B\rangle$ - $M_{Ab}=$$V_{Ab}$$\sqrt{E_{Ab}}$ - Example: 1. $$U_{AB}=|0_A\rangle\langle 0_A|\otimes e^{-itX_B}+|1_A\rangle\langle 1_A|\otimes e^{itX_B}$$ - POVM-measurement operator: non-projective $$M_{A0}=\langle 0_B|U_{AB}|0_B\rangle=\sqrt{\lambda_0}\mathbb{I}_A\neq (M_{A0})^2$$ - POVE effects: non-informative $$E_{A0}=\lambda_0\mathbb{I}_A\leftarrow\text{ independent of }\rho_A$$ 2. $$U_{AB}=|0_A\rangle\langle 0_A|\otimes \mathbb{I}_B+|1_A\rangle\langle 1_A|\otimes e^{itY_B}$$ - POVM-measurement operator: non-projective $$M_{A0}=|0_A\rangle\langle 0_A|+\sqrt{\mu_0}|1_A\rangle\langle 1_A|\neq (M_{A0})^2$$ - POVE effects: informative $$E_{A0}=|0_A\rangle\langle 0_A|+\mu_0|1_A\rangle\langle 1_A|\\p_{A0}=\langle 0_A|\rho_A|0_A\rangle+\mu_0\langle 1_A|\rho_A|1_A\rangle\\ \text{probability depends of }\rho_A\Rightarrow\text{ informative}$$ > ⭐️⭐️⭐️ Optimized POVM has better certainty than projective measurement! > i.e. more probability to measure with certainty (100%) - **Pretty good measurement (PGM)** - Ensemble $\{p_i\rho_i\}$ with full-rank marginal state $\rho=\sum_i p_i\rho_i$ (all eigenvalues $\geq0$ ) has a *canonical* POVM called square-root measurement, or Pretty good measurement $$E_i=\frac{1}{\sqrt{\rho}}p_i\rho_i\frac{1}{\sqrt{\rho}}$$ > ⭐️⭐️⭐️ Pretty good measurement: Optimized probability of detecting *unknown* state > POVM: detecting *known* state - 🧐 *Can evolution in presence of coupling/correlation to an observed environment be reversed? i.e.* $\mathcal{D}$$\mathcal{E}$$(\rho)\stackrel{?}{=}\rho,\ \forall\ \rho$ - ❌ **No information without disturbance.**