Quantum information theory

One or two assessed problems will be announced at the end of the problem sheets most week. However, these problems will not be formally set until we have completed the first half of the course and therefore they will not be due until later in term. Once formally set you will have 2 weeks to submit your answers. 

Aims for the week:

High level over view of the course

Core reading: Pgs.1 - 12 of typed notes. 

The exercises this week cover classical probability theory and an intro to mathematica. This material is intended to prepare you to tackle the rest of the course but is not strictly examinable... so if you don't have time for all the mathematica exercises no worries. 

If you need extra reading to catch up on the basics of quantum information and quantum computation I recommend:

Jonathan A. Jones and Dieter Jaksch: Quantum Information, Computation and Communication pg.5 - 17.

Michael A. Nielsen & Isaac L. Chuang: Quantum Computation and Quantum Information. Chpt 1. 

Vincenzo Savona's lecture course - weeks 2 and 3.

Aims for the week: 

Be able to:

• Represent pure and mixed states on the Bloch sphere.
• Determine if a state is pure or mixed.
• Describe the ensemble decomposition paradox 
• Explain the purpose of the partial trace operation
• Compute the partial trace of quantum states and operators
• Interpret the action of a single qubit unitary as a rotation on the Bloch sphere
• Draw and interpret quantum circuits 

Core reading: pgs. 17- 19 typed notes. pgs. 32 - 33 of typed notes. handwritten notes for the state space of a qubit and basic quantum circuits. 

For more background reading see again: 

Jonathan A. Jones and Dieter Jaksch: Quantum Information, Computation and Communication pg.5 - 17.

Michael A. Nielsen & Isaac L. Chuang: Quantum Computation and Quantum Information. Chpt 1.2,1.3, 2.1 and 2.4. 

Vincenzo Savona's lecture course - weeks 2 and 3.

Aim for the week:

Purifications

Be able to:

• Compute a purification of a generic mixed state
• Explain the difference between a proper and improper mixture

Measurements:

• Explain the difference between a POVM, projective measurement and a measurement of a Hermitian observable
• Check whether a given set of measurement operators is a valid POVM
• State the question asked by a given POVM / propose a POVM to ask a given question
• Propose a POVM to distinguish between non-orthogonal states 
• Propose a POVM for an informationally complete measurement 

Core reading: pgs. 19 - 30 of typed notes. 

For an experimental example of how POVMs can be realized see for example https://journals.aps.org/pra/pdf/10.1103/PhysRevA.63.040305. 

Aims for the week:

Measurements:

Be able to: 

• State Naimark's theorem
• Use Naimark's theorem to map a POVM to a projective measurement on a larger space

Channels:

Be able to:

• State the operational properties that uniquely define quantum evolutions
• Construct a set of Kraus operators corresponding to a given channel 
• Interpret the action of a channel corresponding to a given set of Kraus operators 

Core reading: For Naimark's theorem see handwritten notes the end of the 'Measurement' notes from last week. For channels see pgs. 31 - 38 of typed notes. 

Aims for the week:

Channels:

Be able to:

• Compute the channel induced on a system that interacts with an environment
• State and prove Stinespring's Dilation Theorem
• Compute the dilation of a generic channel
• Prove basic vectorization identities 
• Compute the Choi state assosiated with a channel 

• Compute Kraus operators for a generic channel
• Show that Kraus operators are related by unitary mixing 
• Determine whether a channel is positive and/or completely positive 
• State (but not prove) the representer theorem 

Core reading: pgs. 41 - 48. Handwritten notes on positivity versus complete positivity and the representer theorem.

 For further information you could look at N&C section 8.2.4 (but this is more detail than needed for the exam)  

Vectorization is great and you should all use it

The aim of this week is to get you more comfortable working in vectorized notation. By the end of the week you should be comfortable computing quantum information theoretic quantities (expectation values, averages, variances, higher order moments, etc... ).

Lecture instead of exercise this week. So two lectures this week (TUE and WED).

Shot noise and measurement problem.

Measurement problem is non-examinable.

Aims for the week:

Be able to:

• State the estimator for a measurement strategy
• Access whether a measurement strategy is biased or unbiased
• Use Chebyshev's inequality and Hoeffding's inequality to bound the convergence of a measurement strategy 

Core reading: handwritten notes (if anything is unclear please email me!)

Estimators and Hoeffding's inequality are also covered briefly in Section 9.1 of the typed notes. 

I also attach a set of notes I found which I think have a nice proof of Hoeffding's (non-examinable)

Aims for the week:

Be able to:

• State and derive the operational meaning of the trace distance between two quantum states
• Derive basic bounds on the distance between Hermitian and unitary matrices 
• Comment on the operational meaning of different matrix norms and on how they could be computed 

Aims for the week:

Be able to:

• Explain the content of Shannon's and Schumacher's noiseless coding theorems
• State the definitions of classical and quantum entropy, conditional entropy, mutual information and relative entropy. 
• Prove basic entropy equalities and inequalities 
  

Core reading: Handwritten notes (feel free to email me if anything is unclear). Pgs. 54 - 56 of typed notes. 

For further reading see N&C chapter 11 and 12.2 or Preskill's notes (https://www.lorentz.leidenuniv.nl/quantumcomputers/literature/preskill_1_to_6.pdf) pg. 167 - 194. 

DEBATE

A little bit on Fidelity:

• Compute the Fidelity between any two quantum states 
• State and prove Ulhmann's theorem 
• State and prove the relation between the operational definition of Fidelity as the POVM that minimizes distinguishability and the standard expression for fidelity in terms of matrix multiplication. 

Handwritten notes for fidelty. For more on quantum fidelity you could look at N&C. 

Entanglement Theory Part 1.

Be able to:

• Explain what is a resource theory and what they can be useful for (other than publishing easy papers)
• State the resource theory of entanglement 
• Prove basic majorization identities and bounds 
• State Nielson's Majorization Theorem and use it to determine what pure states can be interconverted via LOCC

Core reading: pgs. 1 - 17 of typed notes

Entanglement Theory (Part 2):

Be able to:

• State, prove and compute the number of maximally entangled states that can be distilled from N copies of a bipartite pure state in the limit of large N.
• Explain why von Neumann entropy is a natural measure of entanglement. 
• State the Hahn-Banach corollary and explain why it guarantees the existence of entanglement witnesses. 
• State the Peres-Horodecki criterion and explain its link to entanglement witnesses
• Use entanglement witnesses and the Peres-Horodecki criterion to discuss whether or not a given state is entangled. 
• Prove basic vectorization identities 

Core reading: Pgs. 20 - 40 of typed notes. 

EXAM DETAILS.

I have included a mock exam below to get a sense of the structure and type of questions of the exam. Please note I have given you four long questions to choose between in the mock- in the real exam you will have three to choose from. 

CHEAT SHEET.

You may communally compile a cheat sheet of useful definitions, equalities and inequalities on the following overleaf:  

Please do not add proofs or theorem statements. I will go through the cheat sheet a week before the exam to remove any content I think is inappropriate. So please add all that you would like to be added by June 29th. The cheat sheet will then be printed and provided in the exam. 

You may find a CALCULATOR helpful in the exam. However, any calculator used must not be capable to connecting to the internet.