[Quantum Information] Mixed States

 Mixed States


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


⭐️⭐️⭐️ The essense of quantum theory is the probability of state purification.

  • What states can we prepare on system A together with extra system B in a pure state and completely measuring B projectively but discarding the outcome?


  • Quantum probability for outcome m for entangled state
    pm=ψAB|EAm1AB|ψAB=trAEAmρAψAB|EAm|ψABρA=trB|ψABψAB|

  • projective measurement on B in complete basis {|bB} for B
    Pb=1A|bBbB|

  • 🧐 How does a mixed state evolve when system A evolves unitarily on its own?

    • Unitarily evolve only subsystem A
      |ψAB(UA1B)|ψAB
    • ρAtrB(UA1B)|ψABψAB|(UA1B)=UAρAUA
  • 🧐 How does a mixed state change by a projective measurement on system A only?

    • |ψAB1pm(PAm1B)|ψABpm=ψAB|PAm1B|ψAB
    • ρA1pmPAmρAPAm

Density operator

  1. Hermicity: ρA=ρA
  2. Positivity (implies Hermicity): Rφ|ρA|φ0, |φ
  3. Unit trace: trAρA=1
  • 🧐 Are state-vector projections precisely the pure states?
    • ⭕️ ρ=|ψψ| is a pure state.
    • ρ2=ρ

  • Which ensembles can we prepare on system A together with extra system B in a pure state and completely measuring B projectively and communicating to A

Ensemble of pure state

  • commuting measured outcome b with A
  • A is given a state from ensemble {pb,|ψAb}
  • B can prepare different ensemble by performing a different measurement
  • If B doesn’t inform A which measurement he takes, the marginal state is the same

⭐️⭐️⭐️ Same mixed quantum state of A may have different ensemble preparations by B

⭐️⭐️⭐️ In fact, there are infinitely many ensembles for one non-pure density operator !
they can be non-orthogonal basis.
bpb|ψAbψAb|=ρA=bpb|ψAbψAb|

  • 🧐 How are all ensemble decompositions of the same mixed state related ?
    • Two basis related by an unitary V
    • VV=I

Ensemble v.s. Mixed state

  • Ensemble: if A knows b then the basis in B does matter
    ρAb=|ψAbψAb|
    • A is given an unknown state |ψAb with probability pb from known ensemble
    • Coherence/purity of A is maintained by access of complete outcome b on B
  • Mixed state: if A doesn’t know b, then basis in B does not matter
    ρA=bpb|ψAbψAb|
    • A is given an unknown state from unknown ensemble
    • Decoherence of A by entanglement with inaccessible/discarded system B

⭐️⭐️⭐️ Decoherence = Loss of purity ~ entanglement + inaccessible information

  • 🧐 Can a given mixed state ρA of some system A be prepared from some entangled bipartite state |ψAB by discarding some system B?

    • ⭕️ If you pick a sufficient large system B
  • 🧐 Can a given mixed state ρA of some system A be prepared from some entangled bipartite state |ψAB by discarding a fixed system B?

    • ⭕️ If you have a same Hilbert space size B as system A (or larget). dim HA=dim HB
    • canonical purified state |ψAB can be constructed directly from |ψAB=kλk|kA|kB

      where ρA=kλk|kAkA| and choose {|kB} for system B
    • {|kB} are indeed orthogonal and can be normalized to ONB for B

⭐️⭐️⭐️ Every pure bipartite state |ψAB has a canonical/Schmidt decomposition |ψAB=kλk|kA|kB with ONBs {|kA},{|kB}


nonzero eigenvalues of ρA and ρB are always equal because |ψAB is pure.

  • 🧐 How are the purifications of the same mixed state related ?
    • Purifications of the same state ρA are related by a unitary on the purification syste.
    • |ψAB=(1AUB)|ψAB
Purification



⭐️⭐️⭐️ No information without disturbance

 

Readings
Nielsen and Chuang
 2.4 The density operator
 2.5 The Schmidt decomposition and purifications
Preskill
 2.3 The density operator
 2.4 Schmidt decomposition
 2.5 Ambiguity of the ensemble interpretation
Wilde (advanced)
 4.1 Noisy Quantum States

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