[Quantum Information] Quantum Instruments

 Quantum Instruments


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

Measurement Quantum Instruments

  1. Linear map
  2. Hermicity-preserving map (HP)
  3. Completely-positive maps (CP)
  4. Trace non-increasing (TNI) maps
    trAMAb(ρA)≤trAρA

EA=∑bMAb is a quantum channel with trace-preserving (TP) instead of trace non-increasing (TNI).

  • 🧐 Do measurement superoperators {Mb} describe any measurement?

    • ⭕️ Only two neccesary assumptions
    1. Outcome b known:
      valid state-update in presence of entanglement ⇔
      MAb=∑c∈bMAbc∙M†Abc(CP)∑c∈bM†AbcMAbc≤1A (TNI)
    2. Outcomes discarded:
      Valid evolution in presence of entanglement ⇔
      ∑bMb=∑b∑c∈bMAbc∙M†Abc=E (CP)∑b∑c∈bM†AbcMAbc=1A (TP)
    • This is not true for
      • POVM with ∑bM†AbMAb=1A
        • already assumes state-update to preserve purity
      • projective measurement
        • further assumes that state-updates produce distinguishable states (orthogonal)

  • What post-measurement states can we get by coupling to an environment (B) in a pure state, measuring it and partially discarding outcomes?

  • Partially discarding outcomes: Discarding information about distinctions between outcomes c∈b which form subsets (b) of different sizes coarse-grainin
    • 丟棄c與b的差異=使兩者相同方式計算
    • Example: Qubits with ONB {|1B⟩,|2B⟩,|3B⟩}
      • Coarse-graining {1} from {2,3}
      • Discard distinctions from {2,3}
      • relabeled {|bcB⟩}={|11B⟩}b=1,{|21B⟩,|22B⟩}b=2

  • measurement superoperator

    • MAb=∑c∈bMAbc∙M†AbcMAbc=⟨bcB|UABC|0B⟩
    • post-measurement state ρAb=1pbMAb(ρA)=1pb∑c∈bMAbcρAM†Abc
    • probability pb=trAMAb(ρA)=trA∑c∈bMAbcρAM†Abc

  • MAb=∑c∈bMAbc∙M†Abc
    This cannot describe by any POVM-measurements MAb=MAb∙M†Ab

  • incomplete measurement model with composite environment

    • 3 systems A,B,C
    • B,C initially pure (possibly entangled)
    • Local complete projective measurement {1A⊗|bBcC⟩⟨bBcC|}
    • commute b with A
    • discard c
      pb=trA∑ctrBC{∙}ρAb=1pb∑ctrBC{∙}

⭐️⭐️⭐️ This model is better than POVM
Example
Maximally depolarizing channel:

UABC=SAB⊗1C, 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 ρ

Axioms quantum theory (open system version)

  1. Preparation state
    ρ≥0, trρ=1

  2. Evolution CP-TP maps
    Eρ=∑bMbρM†b∑bM†bMb=1

  3. Measurement instrument CP, TNI maps
    pbρb=∑c∈bMbcρM†bcPOVM=MbρM†b∑b∑c∈bMbcρM†bc=1POVM=∑bMbρM†b
    Measurement effects
    pb=tr(Ebρb)Eb=∑c∈bM†bcMbcPOVM=M†bcMbc∑bEb=1


⭐️⭐️⭐️ POVM measurements are not general
Because it assumes max. information conservation (assume pure states) in evolution process
→ Incomplete information requires more general state-updates
⭐️⭐️⭐️ POVM measurements are fundamental
Every measurement can be simulated by POVM measurement {MAbc} by discarding information (∑c∈b)
→ Incomplete information allows effect description for probability functions

  • 🧐 Are there non-unitary evolution E that are physically reversible? i.e. whose mathematical inverse on all states is also a CP-TP map?
    • ❌ Reversibility of non-unitary evolution would violate no information without measurement disturbance.

Readings
Wilde (advanced)
 4.6.8 Quantum Instruments

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