Measurement Quantum Instruments

# Measurement Quantum Instruments 1. Linear map 2. Hermicity-preserving map (HP) 3. Completely-positive maps (CP) 4. Trace non-increasing (TNI) maps $$\text{tr}_A\mathcal{M}_{Ab}(\rho_A)\leq\text{tr}_A \rho_A$$ > $\mathcal{E}_A$$=\sum_b$$\mathcal{M}_{Ab}$ is a quantum channel with trace-preserving (TP) instead of trace non-increasing (TNI). - 🧐 *Do measurement superoperators $\{\mathcal{M}_b\}$ describe any measurement?* - ⭕️ Only two neccesary assumptions 1. Outcome $b$ known: valid *state-update* in presence of **entanglement** $\Leftrightarrow$ $$\mathcal{M}_{Ab}=\sum_{c\in b}M_{Abc}\bullet M_{Abc}^\dagger \text{(CP)}\\ \sum_{c\in b}M_{Abc}^\dagger M_{Abc}\leq \mathbb{1}_A\text{ (TNI)}$$ 2. Outcomes discarded: Valid *evolution* in presence of entanglement $\Leftrightarrow$ $$\sum_b\mathcal{M}_{b}=\sum_b\sum_{c\in b}M_{Abc}\bullet M_{Abc}^\dagger=\mathcal{E} \text{ (CP)}\\ \sum_b\sum_{c\in b}M_{Abc}^\dagger M_{Abc}=\mathbb{1}_A\text{ (TP)}$$ - This is not true for - **POVM** with $\sum_bM_{Ab}^\dagger M_{Ab}=\mathbb{1}_A$ - already assumes **state-update to preserve purity** - **projective measurement** - further assumes that **state-updates produce distinguishable states** (orthogonal) - **Partially discarding outcomes:** Discarding information about *distinctions between outcomes $c\in b$* which form subsets $(b)$ of different sizes coarse-graining - 丟棄$c$與$b$的差異=使兩者相同方式計算 - Example: Qubits with ONB $\{|1_B\rangle,|2_B\rangle,|3_B\rangle\}$ - Coarse-graining $\{1\}$ from $\{2,3\}$ - Discard distinctions from $\{2,3\}$ - relabeled $$\{|bc_B\rangle\}=\{|11_B\rangle\}_{b=1},\{|21_B\rangle,|22_B\rangle\}_{b=2}$$ - measurement superoperator - $$\mathcal{M}_{Ab}=\sum_{c\in b}M_{Abc}\bullet M_{Abc}^\dagger\\M_{Abc}=\langle bc_B|U_{ABC}|0_B\rangle$$ - post-measurement state $$\rho_{Ab}=\frac{1}{p_b}\mathcal{M}_{Ab}(\rho_{A})=\frac{1}{p_b}\sum_{c\in b}M_{Abc}\rho_{A} M_{Abc}^\dagger$$ - probability $$p_b=\text{tr}_A\mathcal{M}_{Ab}(\rho_{A})=\text{tr}_A\sum_{c\in b}M_{Abc}\rho_{A} M_{Abc}^\dagger$$ - $\mathcal{M}_{Ab}=$$\sum_{c\in b}$$M_{Abc}\bullet M_{Abc}^\dagger$ This cannot describe by any POVM-measurements $\mathcal{M}_{Ab}=M_{Ab}\bullet M_{Ab}^\dagger$ - *incomplete measurement model with composite environment* - 3 systems $A,B,C$ - $B,C$ initially pure (possibly entangled) - Local complete projective measurement $\{\mathbb{1}_A\otimes|b_Bc_C\rangle\langle b_Bc_C|\}$ - commute $b$ with $A$ - discard $c$ $$p_b=\text{tr}_A\sum_c \text{tr}_{BC}\{\bullet\}\\ \rho_{Ab}=\frac{1}{p_b}\sum_c \text{tr}_{BC}\{\bullet\}$$ > ⭐️⭐️⭐️ **This model is better than POVM** > Example > *Maximally depolarizing channel*: > $U_{ABC}=\mathbb{S}_{AB}\otimes\mathbb{1}_C$, $\mathbb{S}$ is a swap operator. > Probabilities for outcome c and outcome b are statistically **independent** > Post-POVM-measurement state depends only on control bit c, **not b or $\rho$** ## Axioms quantum theory (open system version) 1. Preparation state $$\rho\geq 0, \text{ tr}\rho=1$$ 2. Evolution CP-TP maps $$\mathcal{E}\rho=\sum_b M_b\rho M_b^\dagger\\ \sum_b M_b^\dagger M_b=\mathbb{1}$$ 3. Measurement instrument CP, TNI maps $$p_b\rho_b=\sum_{c\in b} M_{bc}\rho M_{bc}^\dagger\stackrel{POVM}{=}M_b\rho M_b^\dagger\\ \sum_b\sum_{c\in b} M_{bc}\rho M_{bc}^\dagger=\mathbb{1}\stackrel{POVM}{=}\sum_bM_b\rho M_b^\dagger$$ Measurement effects $$p_b=\text{tr}(E_b\rho_b)\\ E_b=\sum_{c\in b}M_{bc}^\dagger M_{bc}\stackrel{POVM}{=}M_{bc}^\dagger M_{bc}\\\sum_b E_b=\mathbb{1}$$ > ⭐️⭐️⭐️ POVM measurements are not *general* > Because it **assumes max. information conservation** (assume pure states) in evolution process > $\rightarrow$ Incomplete information requires more general *state-updates* > ⭐️⭐️⭐️ POVM measurements are *fundamental* > Every measurement can be simulated by POVM measurement $\{M_{Abc}\}$ by *discarding information* $(\sum_{c\in b})$ > $\rightarrow$ Incomplete information allows effect description for *probability functions* - 🧐 *Are there non-unitary evolution $\mathcal{E}$ that are physically inversible? i.e. whose mathematical inverse on all states is also a CP-TP map?* - ❌ Reversibility of *non-unitary* evolution would violate *no information withour measurement disturbance.*