[Quantum Information] POVM measurements

POVM measurements

Notes from RWTH Aachen University course 
“Quantum Information” Summer semester 2020
professor: Wegewijs, Maarten

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  $$E_{A0}=|0_A\rangle\langle 0_A|+\mu_0|1_A\rangle\langle 1_A|\\p_{A0}=\langle 0_A|\rho_A|0_A\rangle+\mu_0\langle 1_A|\rho_A|1_A\rangle\\ \text{probability depends of }\rho_A\Rightarrow\text{ informative}$$
• Example of POVM improvement:
• 
  
  
  $p_\text{certain}=0.25$
• 
  
  
  $p_\text{certain}=\frac{1}{2}\varepsilon = 0.293$
  $17\%$ better than projective measurement

⭐️⭐️⭐️ 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.

Readings

Nielsen and Chuang

 2.2.6 POVM measurements

 2.2.8 Composite systems

Preskill

 3.1 Orthogonal measurements and beyond

Wilde (advanced)

 4.2 Measurement in the Noisy Quantum Theory