[Quantum Information] Evolution
Evolution
Notes from RWTH Aachen University course
“Quantum Information” Summer semester 2020
professor: Wegewijs, Maarten
Evolution
probability should stay probability ⇒E is a stochastic map.
Positivity Preservation (PP): E(ρ)≥0
Trace Preservation (TP): tr E(ρ)=1
🧐 These are general restrictions for quantum evolutions?
- ❌ Difficult in principle: because of entanglement
- ❌ Difficult in practice: PP is more complicated than complete positivity
⭐️⭐️⭐️ Evolutions (superoperators) correspond to states (operators)
ρ′=∑mMmρM†m ⟷ ∑m|Mm⟩⟨Mm|
We take maximally entangled pure state 1√d|1AB⟩=1√d∑i|iA⟩⊗|iB⟩
σAB=(EA⊗IB)1d|1AB⟩⟨1AB|
σABone−←to−one→EA
example:
EA=λ01A∙1A+λ1ZA∙ZA
σAB=12λ0|1AB⟩⟨1AB|+λ1|ZAB⟩⟨ZAB|
- Finite-time evolution
U(t)=e−iHtH:Hamiltonian, H†=HH and H+φ describes the same evolution for any φ∈R
Evolution in presence of entanglement: Quantum Channels
- Linear map
- Hermicity-preserving map (HP): ρA=ρ†A⇒EA(ρA)=(EA(ρA))†
- Completely-positive map(CP): ρAB≥0⇒EA⊗IB(ρAB)≥0
⇒ PP ⇒ HP
- Trace-preserving map(TP): tr E(ρ)=tr ρ=1
- Complete positivity preservation: How is the input state prepared?
- Any mixed input state on A ultimately arises from entanglement with some preparation system B:
- A superoperator EA is called completely positive (CP) when it preserves positivity even when extended to act "trivially" on any preparation ancilla B in any joint state
ρAB≥0⇒EAIB(ρAB)≥0
PP≠CP
- PP-non-CP: negative probabilities
- example: transposition superoperator EA(ρA)=ρTAA
⭐️⭐️⭐️ The difference between PP and CP constitudes what is quantum about evolution: Entanglement
Measurement model for evolution
- 🧐 What state-evolutions can we get by coupling to an inaccessible system (B) in a pure state?
- ρ′A=EA(ρA)=trB{UAB(ρA⊗|0B⟩⟨0B|)U†AB}
- 🧐 What state-evolutions can we get by coupling to an extra system (B) in a pure state, measuring it, and discarding the complete outcomes?
- Projective measurement {1A⊗|bB⟩⟨bB|} in B-basis |bB⟩
- Discarding complete outcome b
- EA(ρA)=∑bpbρAb=∑bMAbρAM†Ab
🧐 Are evolutions derived from measurement models indeed CP-TP maps?
- ⭕️
🧐 Does every operator-sum evolution EA correspond to a unique model {UAB,|0B⟩}?
- ❌
- Nonuniqueness comes from
- ⟨bB| UAB|cB⟩
- MAb=∑b′⟨bB|b′B⟩M′Ab
🧐 Given any CP-TP map E how to compute a set of measurement operators {Mk}?
- compute a bipartite state corresponding to E σAB=(EA⊗IB)1d|1AB⟩⟨1AB|
- Diagonalize it 1d∑k|(Mk)AB⟩⟨(Mk)AB|
- write down operator sum with operators Mk E=∑kMk∙M†k
- compute a bipartite state corresponding to E σAB=(EA⊗IB)1d|1AB⟩⟨1AB|
These are called canonical measurement operator.
Purification of evolution = tomography+purification
1d1A=trC1d|1AC⟩⟨1AC|
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