[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
  $$p_m=\langle\psi_{AB}|E_{Am}\otimes\mathbb{1}_{AB}|\psi_{AB}\rangle=\text{tr}_AE_{Am}\rho_A\neq\langle\psi_{AB}|E_{Am}|\psi_{AB}\rangle \\ \rho_A=\text{tr}_B|\psi_{AB}\rangle\langle\psi_{AB}|$$

• projective measurement on $B$ in complete basis $\{|b_B\rangle\}$ for $B$
  $$P_b=\mathbb{1}_{A}\otimes|b_B\rangle\langle b_B|$$

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

  • Unitarily evolve only subsystem $A$
    $$|\psi_{AB}\rangle\mapsto(U_{A}\otimes\mathbb{1}_B)|\psi_{AB}\rangle$$
  • $$\rho_A\mapsto\text{tr}_B(U_{A}\otimes\mathbb{1}_B)|\psi_{AB}\rangle\langle\psi_{AB}|(U_{A}^\dagger\otimes\mathbb{1}_B)=U_A\rho_A U_A^\dagger$$

• 🧐 How does a mixed state change by a projective measurement on system $A$ only?

  • $$|\psi_{AB}\rangle\mapsto\frac{1}{\sqrt{p_m}}(P_{Am}\otimes\mathbb{1}_B)|\psi_{AB}\rangle\\ p_m=\langle\psi_{AB}|P_{Am}\otimes\mathbb{1}_B|\psi_{AB}\rangle$$
  • $$\rho_A\mapsto \frac{1}{p_m}P_{Am}\rho_AP_{Am}$$

Density operator

1. Hermicity: $\rho_A^\dagger=\rho_A$
2. Positivity (implies Hermicity): $\mathbb{R}\ni\langle\varphi|\rho_A|\varphi\rangle\geq0, \forall\ |\varphi\rangle$
3. Unit trace: $\text{tr}_A \rho_A=1$

• 🧐 Are state-vector projections precisely the pure states?
• ⭕️ $\rho=|\psi\rangle\langle\psi|$ is a pure state.
• $\rho^2=\rho$

• 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 $\{p_b,|\psi_{Ab}\rangle\}$
• $B$ 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.
$$\sum_bp_b|\psi_{Ab}\rangle\langle\psi_{Ab}|=\rho_A=\sum_bp_{b'}|\psi_{Ab}'\rangle\langle\psi_{Ab}'|$$

• 🧐 How are all ensemble decompositions of the same mixed state related ?
  • Two basis related by an unitary $V$
  • $V^\dagger V=\mathbb{I}$

Ensemble v.s. Mixed state

• Ensemble: if $A$ knows $b$ then the basis in $B$ does matter
  $$\rho_{Ab}=|\psi_{Ab}\rangle\langle\psi_{Ab}|$$
  • $A$ is given an unknown state $|\psi_{Ab}\rangle$ with probability $p_b$ 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
  $$\rho_{A}=\sum_bp_b|\psi_{Ab}\rangle\langle\psi_{Ab}|$$
  • $A$ 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 $\rho_A$ of some system $A$ be prepared from some entangled bipartite state $|\psi_{AB}\rangle$ by discarding some system $B$?

  • ⭕️ If you pick a sufficient large system $B$

• 🧐 Can a given mixed state $\rho_A$ of some system $A$ be prepared from some entangled bipartite state $|\psi_{AB}\rangle$ by discarding a fixed system $B$?

  • ⭕️ If you have a same Hilbert space size $B$ as system $A$ (or larget).  $$\text{dim }\mathcal{H}_A=\text{dim }\mathcal{H}_B$$
  • A canonical purified state $|\psi_{AB}\rangle$ can be constructed directly from  $$|\psi_{AB}\rangle=\sum_k\sqrt{\lambda_k}|k_A\rangle|k_B\rangle$$
    where $\rho_A=\sum_k\lambda_k|k_A\rangle\langle k_A|$ and choose $\{|k_B\rangle\}$ for system $B$
  • $\{|k_B\rangle\}$ are indeed orthogonal and can be normalized to ONB for $B$

⭐️⭐️⭐️ Every pure bipartite state $|\psi_{AB}\rangle$ has a canonical/Schmidt decomposition  $$|\psi_{AB}\rangle=\sum_k\sqrt{\lambda_k}|k_A\rangle|k_B\rangle\text{ with ONBs }\{|k_A\rangle\}, \{|k_B\rangle\}$$
nonzero eigenvalues of $\rho_A$ and $\rho_B$ are always equal because $|\psi_{AB}\rangle$ is pure.

• 🧐 How are the purifications of the same mixed state related ?
• Purifications of the same state $\rho_A$ are related by a unitary on the purification syste.
• $$|\psi_{AB}\rangle=(\mathbb{1}_A\otimes U_B)|\psi_{AB}'\rangle$$

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