[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=mMmρMm  m|MmMm|
We take maximally entangled pure state 1d|1AB=1di|iA|iB


σAB=(EAIB)1d|1AB1AB|
σABonetooneEA

example:
EA=λ01A1A+λ1ZAZA
σAB=12λ0|1AB1AB|+λ1|ZABZAB|

  • Finite-time evolution
    U(t)=eiHtH:Hamiltonian, H=HH and H+φ describes the same evolution for any φR

Evolution in presence of entanglementQuantum Channels

  1. Linear map
  2. Hermicity-preserving map (HP): ρA=ρAEA(ρA)=(EA(ρA))
  3. Completely-positive map(CP)ρAB0EAIB(ρAB)0

     PP  HP

  4. 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
      ρAB0EAIB(ρAB)0

PPCP

  • 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?
    • ρA=EA(ρA)=trB{UAB(ρA|0B0B|)UAB}
  • 🧐 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|bBbB|} in B-basis |bB
    • Discarding complete outcome b

  • EA(ρA)=bpbρAb=bMAbρAMAb
  • 🧐 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=bbB|bBMAb

  • 🧐 Given any CP-TP map E how to compute a set of measurement operators {Mk}?

    1. compute a bipartite state corresponding to E σAB=(EAIB)1d|1AB1AB|
    2. Diagonalize it 1dk|(Mk)AB(Mk)AB|
    3. write down operator sum with operators Mk E=kMkMk

These are called canonical measurement operator.

Purification of evolution = tomography+purification
1d1A=trC1d|1AC1AC|


Readings
Nielsen and Chuang
 8.2 Quantum operations
 8.3 Examples of quantum noise and quantum operations
Preskill
 3.2 Quantum channels
 3.3 State-channel duality and the dilation of a channel
 3.4 Three quantum channels
Wilde (advanced)
 4.4 Quantum Evolutions
 4.5 Interpretations of Quantum Channels

留言

這個網誌中的熱門文章