Distinguish quantum states

Distinguish probability distributions over Quantum states;

We want a machine that prepares state ρ from our theory.



 ඌ 
/|\

There you go.



 ඌ      ----   
/|\     | S |   
        -----   

How to verify
it works?


 ඌ      ----   
/|\     | S |   
        -----   

How to verify
it doesn't produce some other state σ?


 ඌ      ----   
/|\     | S |  σ
        -----   
 

• “We are given ρ and σ each with probability 1/2. We are challenged to distinguish them.”

• “Perform a measurement. Modelled as a POVM. With elements Mreal and Mideal = I − Mreal.”

    /- "The probability of correctly identifying σ. -/
    p_succ^max = 1/2 + 1/2 * max {tr(Mreal (σ - ρ)) | 1 ≥ M ≥ 0}
  

• What is the optimal measurement to learn the identity of the true state?

  • “[Hel76] says: The optimal M is the projector onto the positive eigenspace of σ − ρ.”

  • “Diagonalize σ − ρ: D = ∑ i, di * |di⟩⟨di|. Denote the set S+ = { j | d j > 0 }. The optimal M is then given by M_opt = ∑ j, |di⟩⟨di|.”

---

• More generally:

  • “We have a probability distribution q: from Type finset quantum_state ρ to ℝ.

    • Distinguish probability distributions;

    • POVM tests partition the symbol space like the T parameter in the classical scenario

  • And a POVM Λ. What is it?

  • To identify the states ρ with high probability: We would like Tr((Λ i) (ρ i)) to be as high as possible.”

    /-- The expected success probability of the POVM: -/
    ∑ i, (p i) * Tr((Λ i) (ρ i))
    

---

• Quantum Stein’s lemma;

---

• Measures of distinguishability

  • Trace distance: “If two states are close in Trace distance, then there exists no measurement or process in the universe that can distinguish them very well.”

  • Quantum relative entropy

    • related to Trace distance?

---

• How is a large computation affected if it depends on this machine?

  • “If σ and ρ are nearly impossible to distinguish, then it should not matter much which one we use.”

  • “Otherwise, we could use the large protocol to tell the two states apart, but we know this cannot be.”

---

Can we use Randomized measurements to estimate the encoded phase?

References

• edX Quantum Cryptography Lecture 3 notes;