[Quantum Information] Quantum Instruments
Quantum Instruments
Notes from RWTH Aachen University course
“Quantum Information” Summer semester 2020
professor: Wegewijs, Maarten
Measurement Quantum Instruments
- Linear map
- Hermicity-preserving map (HP)
- Completely-positive maps (CP)
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Trace non-increasing (TNI) maps
trAMAb(ρA)≤trAρA
EA=∑bMAb is a quantum channel with trace-preserving (TP) instead of trace non-increasing (TNI).
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🧐 Do measurement superoperators {Mb} describe any measurement?
- ⭕️ Only two neccesary assumptions
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Outcome b known:
valid state-update in presence of entanglement ⇔
MAb=∑c∈bMAbc∙M†Abc(CP)∑c∈bM†AbcMAbc≤1A (TNI) -
Outcomes discarded:
Valid evolution in presence of entanglement ⇔
∑bMb=∑b∑c∈bMAbc∙M†Abc=E (CP)∑b∑c∈bM†AbcMAbc=1A (TP)
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This is not true for
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POVM with ∑bM†AbMAb=1A
- already assumes state-update to preserve purity
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projective measurement
- further assumes that state-updates produce distinguishable states (orthogonal)
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POVM with ∑bM†AbMAb=1A
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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}
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relabeled {|bcB⟩}={|11B⟩}b=1,{|21B⟩,|22B⟩}b=2
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measurement superoperator
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MAb=∑c∈bMAbc∙M†AbcMAbc=⟨bcB|UABC|0B⟩
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post-measurement state ρAb=1pbMAb(ρA)=1pb∑c∈bMAbcρAM†Abc
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probability pb=trAMAb(ρA)=trA∑c∈bMAbcρAM†Abc
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MAb=∑c∈bMAbc∙M†AbcMAbc=⟨bcB|UABC|0B⟩
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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
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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)
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Preparation state
ρ≥0, trρ=1 -
Evolution CP-TP maps
Eρ=∑bMbρM†b∑bM†bMb=1 -
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.
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