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.