Introduction to Quantum Information (C7.4) Consultation Session #3 - - PowerPoint PPT Presentation

Introduction to Quantum Information (C7.4)

Consultation Session #3 Thursday 21 May 2020 Tutor: Dr J. Mur-Petit

Outline

• Classical communication checks
• Messages:

– Check www.arturekert.com/quantum for

• Lecture notes
• Examinable syllabus

– Exam: Tuesday 9 June 14:30 -> check changes at:

http://www.ox.ac.uk/students/academic/exams/timetables

• Questions pre-submitted

– 2018 Q1 (quantum algorithms) – 2017 Q1f (quantum error correction) – 2018 Q2a-d (stabiliser formalism) – 2019 Q2e (quantum algorithms / quantum circuits) – 2019 Q3e (CPTP maps)

• Closing remarks: Last consult. session: Fri 28th 17:00 BST

2018 Q1 (q. algorithms)

2018 Q1

(a) See lecture notes “quantum algorithms” -> phase kick-back. Interpretation: reflection in subspace

• rthogonal to all |x> such that f(x) = 1

Reflection in subspace

• rthogonal to U|a>
• HnV0Hn = reflection in subspace orthog. to Hn|0> ;

Uf = reflection in subsp. orthog. to |s> ;  G = rotation in subsp. spanned by Hn|0> and |s>

2018 Q1

► More details: J. Watrous https://cs.uwaterloo.ca/~watrous/CPSC519/LectureNotes/12.pdf

• HnV0Hn = reflection in subspace orthog. to Hn|0> ;

Uf = reflection in subsp. orthog. to |s> ;  G = rotation in subsp. spanned by Hn|0> and |s>

2018 Q1

We have that i.e., span { |ψg>, |ψb>} = span{Q|ψg>, Q|ψb>}, and the subspace is invariant under Q.

Using and trigonometric identities, we get From here, it’s easy to show that

2018 Q1

NP describes the class of problems that are difficult to solve but easy to verify. It does matter here that the problem is in NP for it can be efficiently verified by Uf.

2018 Q1

2017 Q1 f (Q.E.C.)

• Errors are unrecoverable if there is more than one error in

each cluster of three qubits.

2017 Q1 f (Q.E.C.)

X

• Errors are unrecoverable if there is more than one error in

each cluster of three qubits.

2017 Q1 f (Q.E.C.)

X X X

• Errors are unrecoverable if there is more than one error in

each cluster of three qubits.

2017 Q1 f (Q.E.C.)

X X X X X

• Errors are unrecoverable if there is more than one error in

each cluster of three qubits.

2017 Q1 f (Q.E.C.)

X X X X X X X X

• Errors are unrecoverable if there is more than one error in

each cluster of three qubits.

2017 Q1 f (Q.E.C.)

• Errors are unrecoverable if there is more than one error in

each cluster of three qubits. X X X X X X X X X X

► More info: Ch. 10 in Kay, Laflamme & Mosca, “An introduction to quantum information” (OUP, 2017)

• -> e-book available at the Bodleian/SOLO

2018 Q2 a-d

→Q. 4.3(7) (sheet 4)

→Q. 4.3(10) →Q. 4.3(2)

2018 Q2 a-d

Definition: Separable

2018 Q2 a-d

Definition: Separable

► More info: Hillery et al., Phys. Rev. A 59, 1829 (1999)

2019 Q2 e (q. algorithms)

See Q. 3.8 (sheet 3)

ρ ψ

2019 Q2 e

See Q. 3.8 (sheet 3)

ψ ρ

What reduced density matrix describes state of auxiliary system? Think it like this:

2019 Q3 e (CPTP maps)

Start applying the transpose to the density matrix: For result to be a density matrix: * Has trace 1? Yes. * Positive semi-def.? Calculate e-values: need p≤1/3

2019 Q3 e

Start applying the transpose to the density matrix: For result to be a density matrix: * Has trace 1? Yes. * Positive semi-def.? Calculate e-values: need p≤1/3 When does ρ represent an entangled state? Peres-Horodecki criterion: p>1/3 (a.k.a. PPT criterion = positive partial transpose criterion → see next slide)

2019 Q3 e

Peres-Horodecki criterion (a.k.a. PPT criterion = positive partial transpose criterion): ► More info: A. Peres, Phys. Rev. Lett. 77, 1413-1415 (1996); Horodecki, Horodecki & Horodecki, Phys. Lett. A 223, 1-8 (1996).

This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License. It makes use of the works of Mateus Machado Luna.