[Quantum Information] Mixed States
Mixed States
“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|EAm⊗1AB|ψAB⟩=trAEAmρA≠⟨ψAB|EAm|ψAB⟩ρA=trB|ψAB⟩⟨ψAB|projective measurement on B in complete basis {|bB⟩} for B
Pb=1A⊗|bB⟩⟨bB|🧐 How does a mixed state evolve when system A evolves unitarily on its own?
- Unitarily evolve only subsystem A
|ψAB⟩↦(UA⊗1B)|ψAB⟩ - ρA↦trB(UA⊗1B)|ψAB⟩⟨ψAB|(U†A⊗1B)=UAρAU†A
- Unitarily evolve only subsystem A
🧐 How does a mixed state change by a projective measurement on system A only?
- |ψAB⟩↦1√pm(PAm⊗1B)|ψAB⟩pm=⟨ψAB|PAm⊗1B|ψAB⟩
- ρA↦1pmPAmρAPAm
- |ψAB⟩↦1√pm(PAm⊗1B)|ψAB⟩pm=⟨ψAB|PAm⊗1B|ψAB⟩
Density operator
- Hermicity: ρ†A=ρA
- Positivity (implies Hermicity): R∋⟨φ|ρA|φ⟩≥0,∀ |φ⟩
- Unit trace: trAρA=1
- 🧐 Are state-vector projections precisely the pure states?
- ⭕️ ρ=|ψ⟩⟨ψ| 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 b 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|ψAb⟩⟨ψAb|=ρA=∑bpb′|ψ′Ab⟩⟨ψ′Ab|
- 🧐 How are all ensemble decompositions of the same mixed state related ?
- Two basis related by an unitary V
- V†V=I
Ensemble v.s. Mixed state
- Ensemble: if A knows b then the basis in B does matter
ρAb=|ψAb⟩⟨ψAb|- 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|ψAb⟩⟨ψAb|- 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
- A canonical purified state |ψAB⟩ can be constructed directly from |ψAB⟩=∑k√λk|kA⟩|kB⟩
where ρA=∑kλk|kA⟩⟨kA| and choose {|kB⟩} for system B - {|kB⟩} are indeed orthogonal and can be normalized to ONB for B
- ⭕️ If you have a same Hilbert space size B as system A (or larget). dim HA=dim HB
⭐️⭐️⭐️ Every pure bipartite state |ψAB⟩ has a canonical/Schmidt decomposition |ψAB⟩=∑k√λk|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⟩=(1A⊗UB)|ψ′AB⟩
⭐️⭐️⭐️ No information without disturbance
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