text:
## Instructor:

Debbie Leung

Email: wcleung(at)uwaterloo(dot)ca

## Lectures:

Starting Sept 08: Tue/Thur 1:00-2:20pm, QNC 1201

## Instructor office hours:

In general: after class Tue/Thur. After answering quick questions at the classroom, I will move to my office QNC 3313. Students having a class that time can arrange alternative office hours on an individual basis.

## Goal:

Communication is an integral part of information processing. For instance, data compression, data storage, gates acting on multiple systems, as well as sending data over a distance are all special cases of communication.

This course is concerned with the fundamental limits of communication, whereby we learn about communication and also the physical theory behind:

1. We cover important results in the theory of quantum communication.

For example: superdense coding, teleportation, data compression, noisy channel coding theorems (both classical and quantum settings), how auxiliary resources like entanglement or feedback can improve communication, and superactivation, zero-error communication.

2. We study fundamental concepts widely used in quantum communication that applies broadly to other aspects of quantum information. Examples include:

• how impossibility of superluminal communication implies various fundamental principles in quantum mechanics (Helstrom's bound, no cloning etc) and how it rules out many extensions of quantum mechanics.
• the equipartition theorem, typicality events, and the properties of entropy, mutual information, and relative entropy (both quantum and classical)
• how the above limits entanglement transformation
• (if time permits) how teleportation gives rise to quantum encryption and quantum message authentication, gate teleportation and consequently fault-tolerant quantum computation and measurement based quantum computation, and the equivalence of entanglement purification and error correction.
• ## References:

General:

• Richard Cleve's lecture notes for QIC 710, Fall 2015
• John Watrous' lecture notes for QIC 820 / CS 766, Fall 2011
• John Preskill's lecture ntoes on quantum computation
• Textbook by Nielsen and Chuang, Cambridge University Press 2000
• Specific to the course:

• My lecture notes for QIC 890 / CO 781 (Fall 2012)
• John Preskill's notes Chapter 10
• John Watrous' book Chapter 8
• Mark Wilde's text book
• ## Prerequisites and relations to QIC 710 and QIC 820:

Based on the above links for the courses:

Students should have good understanding of 4 lectures in QIC 710 before the term starts:

• Lectures 1-2 (p8-42 in the pdf file for slides for Lectures 1-3)
• Lectures 10-12 (entire pdf file)

Students who wish to take QIC 710 and this course simultaneously are encouraged to self-study the above.

The current course will have partial overlaps with the following:

In QIC 710:

• Lecture 13-14 (pdf file p30-35)
• Lecture 15 (pdf file p1-11, 23)
• Lecture 16 (entire pdf file)

Lecture 15 will be given as reading assignment as the course progress, and we will fully cover the materials in lectures 13-14, and 16.

In QIC 820:

• Lectures 1-6, 9-12

This more mathematically advanced treatment of the subjects can deepen your understanding of the subject, but will not be assumed as pre-requisite to the course.

## Requirements:

4 Assignments (total 60%). Each will be given out upon completion of a topic, and due 7-10 days later.

1 term project resulting in a term paper and a presentation second half of November (40%).

## Announcements:

Posted Aug 12, 2016, 23:59

Webpage was set up.

Posted Sept 07, 2016, 18:00

Topics and tentative plan included.

Posted Sept 08, 2016, 12:30

Notes for lecture 1 posted.

Posted Sept 13, 2016, 12:30

Notes for lecture 2 posted.

Posted Sept 13, 2016, 15:20

Notes for lecture 2 modified (we now distinguish m from M, b from B etc in the proof of C2).

Posted Sept 14, 2016, 21:40

Notes for lecture 3 posted.

Posted Sept 15, 2016, 17:30

Notes for last part of discussion for lecture 3 posted.

Posted Sept 19, 2016, 18:55

Notes for lectures 4-5 posted.

Posted Sept 22, 2016, 12:35

Additional notes for lectures 5 posted.

Posted Sept 22, 2016, 12:35

Preliminary version of A1 posted.

Posted Sept 22, 2016, 17:15

Small changes to the RSP and SD notes made, and they're now merged into one file.

Posted Sept 23, 2016, 03:15

Updated version of A1 posted.

Posted Sept 24, 2016, 01:15

3rd version of A1 posted, including a little more information for solving the problems, and tweaking some notations for better clarity.

Posted Sept 24, 2016, 18:00

In Q2a, each G_k is CP (completely positive) but not necessarily trace preserving.

Posted Sept 27, 2016, 12:25

Notes for lecture 6 posted.

Posted Sept 27, 2016, 18:15

Notes for lecture 6 re-posted.

Further clarifications and corrections to assignment 1.

• It is intended that in Q1, the given cbits and the desired qbit are in the same direction (from Alice to Bob), while in Q2, the proof is structured to apply even with cbits in both directions. If you try hard enough, you can extend the proof in Q1 for two way cbits. But the real value of Q1 is an extension to channels in which Eve has an output that can be turned into Bob's output; these channels are called antidegradable and we will come back to this topic later.
• For Q3, the state $|\tilde{\phi}>$ transmitted to Bob should be $\rho_{\phi}$ (a state that depends on $|\phi>$). It can be mixed. My apologies.
• Posted Sept 28, 2016, 17:00

Notes for lecture 7 posted.

Posted Sept 29, 2016, 15:30

Additional notes for lecture 7 posted.

Posted Oct 03, 2016, 18:15

Notes for lecture 8 posted.

Posted Oct 04, 2016, 12:25

Assignment 2 was posted.

Posted Oct 04, 2016, 16:15

Notes called "additional notes on entanglement dilution and concentration" reposted, with the circuit diagram for entanglement dilution included. The diagram is color-coded to make the resource transformation more obvious.

Posted Oct 04, 2016, 16:20

There is no Oct 11 lecture due to fall study period. The tentative plan of the course has been adjusted.

Posted Oct 05, 2016, 23:00

Notes for lecture 9 posted.

Posted Oct 13, 2016, 12:20

Notes for lecture 9 reposted (with an additional last page on the intuition).

Notes for the converse posted under "lecture 9" (but will be covered in lecture 10).

Notes for properties of quantum entropies posted as "lecture 10".

Posted Oct 17, 2016, 18:20

Notes for lecture 11 posted.

Lecture 11 starts 12:30pm Oct 18, 2016.

A2 due latest in class tomorrow.

Posted Oct 18, 2016, 18:40

NO LECTURE ON THUR OCT 20.

Posted Oct 25, 2016, 12:50

Notes for lecture 12 posted.

Posted Oct 26, 2016, 18:30

Notes for lecture 12 reposted, continuing to lecture 13.

Posted Oct 26, 2016, 18:35

I believe the last two lectures are Nov 29 and Dec 01. The plan is to have student presentations those two days, starting 12:30pm (to also make up for the missing lecture time on Oct 20). Please contact me for a topic if you haven't done so, and especially if you'd like to have a project on topic 5 or 6.

Posted Nov 02, 2016, 17:15

Notes for lecture 15 posted. We will finish off the last 5 pages of the previous lecture notes (repeated in the new pdf file) concerning the optimal ensemble for Holevo information, followed by a discussion of additivity. Depending on time and interest, we will cover parts of the notes uploaded (largely borrowed from previous years).

Posted Nov 03, 2016, 01:15

Assignment 3 posted.

Posted Nov 08, 2016, 12:50

Notes for lecture 16 posted.

Posted Nov 10, 2016, 12:50

Notes for lecture 17 posted, has some overlap with last set of notes.

Posted Nov 22, 2016, 12:40

Assignment 4 and notes for lecture 20-21 posted. Details for the presentation will be available later today.

Posted Nov 29, 2016, 12:50

Notes for a 20 min discussion of superactivation for lecture 22 posted, where the notes for lectures 20 and 21 are.

## Tentative plan:

### Topic 1 -- There is no free lunch [2.5 weeks]

• Superdense coding (SD) and teleportation (TP), optimality and duality.
• The no signalling principle and consequences.
• Optimality of SD and TP.
• Cobits, SD and TP as inverses of one another.
• Simulations and resource inequalities.
• Remote state preparation (RSP), the curious cousin of teleportation.
• Quantum encryption, the unexpected twin of teleportation.
• Why QM better be linear, and we should avoid time travel.
• ~ Assignment 1 out Sept 20, due Sept 30.~

### Topic 2 -- Entropy and data Compression [1 week]

• IID source
• Asymptotic equipartition theorem (AEP)
• Shannon entropy
• Data compression (Shannon's noiseless coding theorem)
• Quantum ensembles and quantum data compression
• Von Neumann entropy
• Entanglement concentration and dilution
• ### Topic 3 -- Classical communication via classical channels [1 week]

• Conditional entropy, relative entropy, mutual information, and joint typicality
• Distributed source coding
• Classical iid channels
• Error correcting codes.
• Shannon's noisy coding theorem.
• The direct coding theorem and the converse
• ~ No lecture on Oct 11 due to fall study break. ~

~ Assignment 2 out Oct 04, due Oct 14 (a little aheading of the lectures) ~

### Topic 4 -- Classical communication via quantum channels [3 weeks]

• Extractive classical information from quantum states
• Accessible information
• Holevo information, Holevo bound
• Entanglement and back communication cannot increase communication rates (beyond SD)
• Locking
• The HSW theorem for the classical capacity of a quantum channel
• Omitted 2016: (Tools: pretty good measurement, gentle measurement lemma, conditional typicality, random codes)
• Additivity issues, the existence of counterexamples.
• ~ Class cancelled Oct 20. Discuss make up class. ~

~ Assignment 3 out Nov 03, due Nov 11 ~

### Topic 5 -- Quantum communication via quantum channel [2-3 weeks]

• Definition of the quantum capacity and different measures for transmission with vanishing error.
• Quantum error correcting codes
• Coherent information of quantum states and quantum channels
• The LSD theorem for the quantum capacity of a quantum channel
• (Tools: Fannes inequality (converse), decoupling lemma (direct coding), Ulhmann's theorem, random codes)
• Different approaches and coding methods for the LSD theorem
• Isometric extensions and complementary channels
• Upper and lower bounds of quantum capacities, additive extensions, zero capacity conditions
• Degenerate codes and statement of superactivation
• ~ This topic starts Nov 8, with 4.5 weeks of lecture remaining. Term project should be chosen no later than Nov 8.

~ Assignment 4 out Nov 17, due Nov 24 ~

### Topic 6 -- Other capacities [? week]

• Private capacity
• Entanglement assisted quantum/classical capacity
• Quantum capacity assisted by free classical communication
• No-go for catalysis of capacities by noiseless resources
• Separations of capacities
• Superactivation
• Rocket channel
• Entanglement assisted zero-error communication
• ~ No assignment ~

Student presentations -- To be scheduled. Term paper due Dec 5.

## Lecture Notes and Assignments:

Communication setting, superdense coding and teleportation, circuit diagrams, and resource inequalities.
The non-signalling principle as a basic principle in communication theory, consequences.
Optimality of superdense coding and teleportatio, cobits, superdense coding and teleportation with cobits as inverses of one another. Cobits and bipartite quantum gates.

One to one correspondence between generalized teleportation and generalized quantum encryption.

Lecture 5, Sept 22, 2016. (1) remote state preparation & superdense coding. (2) Notes on beyond QM.
An example of non composably-good qbit: remote state preparation. Additional discussion on superdense coding. Beyond QM.

Further corrections:

• In Q2a, each G_k is a CP map that is not necessary trace preserving.
• It is intended that in Q1, the given cbits and the desired qbit are in the same direction (from Alice to Bob), while in Q2, the proof is structured to apply even with cbits in both directions. If you try hard enough, you can extend the proof in Q1 for two way cbits. But the real value of Q1 is an extension to channels in which Eve has an output that can be turned into Bob's output; these channels are called antidegradable and we will come back to this topic later.
• For Q3, the state $|\tilde{\phi}>$ transmitted to Bob should be $\rho_{\phi}$ (a state that depends on $|\phi>$). It can be mixed. My apologies.
• Asymptotic equipartition theorem, Shannon entropy, data compression.
von Neumann entropy, quantum sources, and quantum data compression.
Conditional entropy, relative entropy, mutual information, and joint typicality.

Lecture 9, Oct 06, 2016.

Shannon's noisy channel coding theorem, up to direct coding theorem. Converse, covered on Oct 13.
Properties of quantum entropies.
Encoding classical data in quantum states, and retrieving it. Accessible information, Holevo bound, locking, and subentropy.
Capacity of Q boxes (updated). Gentle measurement lemma, PGM, packing lemma, direct coding theorem, converse.
Part 1: example of Q-box capacity. Part 2: HSW theorem for the classical capacity of quantum channels.
Optimal ensemble for Holevo information and nonadditivity, equivalences of additivity conjectures (if time permits).
Transmitting quantum data through quantum channels, coherent information, LSD theorem, decoupling approach. Proof for Thm 3 of 0702005.
Decoupling approach with approximations, direct coding for the LSD theorem.
Consequencs for the LSD theorem (I). Degradable channels.
Consequencs for the LSD theorem (II). Random Pauli channels, non-degenerate random stabilizer code, depolarizing channel, degenerate QECCs, and non-additivity of 1-shot coherent information.
Upper bounds for quantum capacities using additive extensions. Assisted capacities. Quantum reverse Shannon theorem. Rocket channel and the separation of assisted capacities. Private vs quantum capacity, and superactivation.

Lecture 21-24. Student presentations.

Nov 24

Nicholas Funai: Quantum State Cloning Using Deutschian Closed timelike curves?

Nov 29

Ink Tansuwannont: Approximate quantum error correction and 4-bit code for amplitude damping channel

Dec 01

Ala Shayeghi: Approximate degradable quantum channels

Benjamin Lovitz: Limitations on separable measurements by convex optimization

Jason LeGrow: Information Locking

Dec 06

Hamoon Mousavi: Entanglement assisted classical capacity

Maria Kieferova: Noisy Interactive Quantum Communication

Maria Sobchuk: Zero error communication assisted by entanglement and nonlocal correlations


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# Proof of Harten’s Lemma re the Convergence of TVNI Finite-Difference Schemes

An academic paper with a rather unusual history is that of the Israeli mathematician Ami Harten’s “High Resolution Schemes for Hyperbolic Conservation Laws.”  First published in the Journal of Computation Physics in 1983, it was republished in 1997 in the same journal, and is often cited with the later date.

This paper was a landmark; it introduced a new design principle—total variation diminishing schemes—that led, in Harten’s hands, and subsequently in the hands of others, to an efficient, robust, highly accurate class of schemes for shock capturing free of oscillations. The citation index lists 429 references to it, not only in journals of numerical analysis and computational fluid dynamics, but also in journals devoted to mechanical engineering, astronautics, astrophysics, geophysics, nuclear science and technology, spacecraft and rockets, plasma physics, sound and vibration, aerothermodynamics, hydraulics, turbo and jet engines, and computer vision and imaging.

One point in the paper was a lemma concerning the convergence of TVNI (total variation nonincreasing) finite difference schemes.  Concerning the name of these schemes, Lax points out the following:

Harten originally called his schemes variation diminishing, abbreviated TVD; when Osher pointed out the usual meaning of these initials, the name was switched to total variation nonincreasing (TVNI), but was eventually settled on the more euphonious TVD.

The following is an expansion of Harten’s proof of the lemma.

Schemes which are total variation nonincreasing (TVNI) can be characterized as follows:
$\sum_{j=-\infty}^{\infty}\left|u_{{j+1,n+1}}-u_{{j,n+1}}\right|\leq\sum_{j=-\infty}^{\infty}\left|\left(u_{{j+1,n}}-u_{{j,n}}\right)\right|$

We can thus define
$TV\left(u^{n}\right)=\sum_{j=-\infty}^{\infty}\left|\left(u_{{j+1,n}}-u_{{j,n}}\right)\right|$

$TV\left(u^{n+1}\right)=\sum_{j=-\infty}^{\infty}\left|u_{{j+1,n+1}}-u_{{j,n+1}}\right|$

and substituting
$TV\left(u^{n+1}\right)\leq TV\left(u^{n}\right)$

Consider the general expression
$u_{{j,n+1}}=u_{{j,n}}-C_{{j-1,n}}\left(u_{{j,n}}-u_{{j-1,n}}\right)+D_{{j,n}}\left(u_{{j+1,n}}-u_{{j,n}}\right)$

where
$C_{{j-1,n}}\geq 0$

$D_{{j,n}}\geq 0$ $C_{{j-1,n}}+D_{{j,n}}\leq 1$

We should observe that our ultimate goal is to sum these values from negative infinity to positive infinity; thus, we can shift the index at will. The inequalities will still hold but the specific location in space may change. It is also worth noting that the coefficients may themselves change at different points in space.

Let us consider the next spatial step, to wit
$u_{{j+1,n+1}}=u_{{j+1,n}}-C_{{j,n}}\left(u_{{j+1,n}}-u_{{j,n}}\right)+D_{{j+1,n}}\left(u_{{j+2,n}}-u_{{j+1,n}}\right)$

Subtracting the previous spatial step from this yields
$u_{{j+1,n+1}}-u_{{j,n+1}}=u_{{j+1,n}}-C_{{j,n}}\left(u_{{j+1,n}}-u_{{j,n}}\right)+D_{{j+1,n}}\left(u_{{j+2,n}}-u_{{j+1,n}}\right)-u_{{j,n}}+C_{{j-1,n}}\left(u_{{j,n}}-u_{{j-1,n}}\right)-D_{{j,n}}\left(u_{{j+1,n}}-u_{{j,n}}\right)$

Some rearranging yields
$u_{{j+1,n+1}}-u_{{j,n+1}}=\left(u_{{j+1,n}}-u_{{j,n}}\right)\left(1-D_{{j,n}}-C_{{j,n}}\right)+C_{{j-1,n}}\left(u_{{j,n}}-u_{{j-1,n}}\right)+D_{{j+1,n}}\left(u_{{j+2,n}}-u_{{j+1,n}}\right)$

Taking the absolute value of both sides, we have
$\left|u_{{j+1,n+1}}-u_{{j,n+1}}\right|=\left|\left(u_{{j+1,n}}-u_{{j,n}}\right)\left(1-D_{{j,n}}-C_{{j,n}}\right)+C_{{j-1,n}}\left(u_{{j,n}}-u_{{j-1,n}}\right)+D_{{j+1,n}}\left(u_{{j+2,n}}-u_{{j+1,n}}\right)\right|$

At this point we observe that
$C_{{j-1,n}}\geq 0$
$D_{{j+1,n}}\geq 0$
$D_{{j,n}}+C_{{j,n}}\leq 1$
$1-D_{{j,n}}-C_{{j,n}}\geq 0$

We can thus limit the absolute values and write the expression as follows:
$\left|u_{{j+1,n+1}}-u_{{j,n+1}}\right|\leq\left|\left(u_{{j+1,n}}-u_{{j,n}}\right)\right|\left(1-D_{{j,n}}-C_{{j,n}}\right)+\left|\left(u_{{j,n}}-u_{{j-1,n}}\right)\right|C_{{j-1,n}}+\left|\left(u_{{j+2,n}}-u_{{j+1,n}}\right)\right|D_{{j+1,n}}$

Taking the summation for both sides,
$\sum_{j=-\infty}^{\infty}\left|u_{{j+1,n+1}}-u_{{j,n+1}}\right|\leq \sum_{j=-\infty}^{\infty}\left|\left(u_{{j+1,n}}-u_{{j,n}}\right)\right|\left(1-D_{{j,n}}-C_{{j,n}}\right)+\sum_{j=-\infty}^{\infty}\left|\left(u_{{j,n}}-u_{{j-1,n}}\right)\right|C_{{j-1,n}}+\sum_{j=-\infty}^{\infty}\left|\left(u_{{j+2,n}}-u_{{j+1,n}}\right)\right|D_{{j+1,n}}$

Since, as we observed before, we can shift the indices (as the “centre” of the system is arbitrary with infinite boundaries) we can rewrite the above as follows
$\sum_{j=-\infty}^{\infty}\left|u_{{j+1,n+1}}-u_{{j,n+1}}\right| \leq \sum_{j=-\infty}^{\infty}\left|\left(u_{{j+1,n}}-u_{{j,n}}\right)\right|\left(1-D_{{j,n}}-C_{{j,n}}\right)+\sum_{j=-\infty}^{\infty}\left|\left(u_{{j+1,n}}-u_{{j,n}}\right)\right|C_{{j,n}}+\sum_{j=-\infty}^{\infty}\left|\left(u_{{j+1,n}}-u_{{j,n}}\right)\right|D_{{j,n}}$

in which case
$\sum_{j=-\infty}^{\infty}\left|u_{{j+1,n+1}}-u_{{j,n+1}}\right| \leq \sum_{j=-\infty}^{\infty}\left|\left(u_{{j+1,n}}-u_{{j,n}}\right)\right|$

Substituting, we have at last
$TV\left(u^{n+1}\right)\leq TV\left(u^{n}\right)$

Citation: Harten, A. (1997) “High Resolution Schemes for Hyperbolic Conservation Laws.” Journal of Computational Physics, Vol. 135, pp. 260-278.

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# fourier / convolution

#### hamza324

Joined Jul 10, 2011
33
HI
My prof gave a question

$$x= \delta(t-1) + 5Cos(2t) \ \ \ \longrightarrow \ \ \ \ \ \fbox{H( \omega)= \frac {1}{j\omega +1}} \ \ \ \longrightarrow \ \ \ y=find \ using \ super \ position$$

Now for the solution , he treats each of the input functions separately with the boxed function.

he writes

$$x_1 = \delta(t-1) \longrightarrow \fbox{h(t)=e^{-t} u(t)} \longrightarrow h(t-1)=e^{-(t-1)}u(t-1) = y_1$$

Then,

$$x_2 = 5Cos(2t) \longrightarrow \fbox {H(\omega)=\frac{1}{j\omega+1}} \longrightarrow 5\ \mid H(\omega)\mid \ Cos(2t + \theta)$$

$$where \mid H(\omega) \mid=( \frac{1}{(2)^2+1})^{\frac{1}{2}}and \theta =- tan^{-1} ( \frac{2}{1})=-63^o$$

So using super sposition , we get

$$y(t)=\ e^{-(t-1)} u(t-1)\ \ \ + \ \ \ \frac{5}{\sqrt{5}} Cos(2t-63^o)$$

Now my question is that,

1 - Why did he used time domain for the impulse function and frequency domain for cos(2t).

2 - How did he get expression for $$y_2$$ , the one with magnitude of H(w).

thnx

#### steveb

Joined Jul 3, 2008
2,436
HI
My prof gave a question

$$x= \delta(t-1) + 5Cos(2t) \ \ \ \longrightarrow \ \ \ \ \ \fbox{H( \omega)= \frac {1}{j\omega +1}} \ \ \ \longrightarrow \ \ \ y=find \ using \ super \ position$$

Now for the solution , he treats each of the input functions separately with the boxed function.

he writes

$$x_1 = \delta(t-1) \longrightarrow \fbox{h(t)=e^{-t} u(t)} \longrightarrow h(t-1)=e^{-(t-1)}u(t-1) = y_1$$

Then,

$$x_2 = 5Cos(2t) \longrightarrow \fbox {H(\omega)=\frac{1}{j\omega+1}} \longrightarrow 5\ \mid H(\omega)\mid \ Cos(2t + \theta)$$

$$where \mid H(\omega) \mid=( \frac{1}{(2)^2+1})^{\frac{1}{2}}and \theta =- tan^{-1} ( \frac{2}{1})=-63^o$$

So using super sposition , we get

$$y(t)=\ e^{-(t-1)} u(t-1)\ \ \ + \ \ \ \frac{5}{\sqrt{5}} Cos(2t-63^o)$$

Now my question is that,

1 - Why did he used time domain for the impulse function and frequency domain for cos(2t).

2 - How did he get expression for $$y_2$$ , the one with magnitude of H(w).

thnx
For question 1, one is free to use either the frequency domain or them time domain, depending on which is easier. Note that the first input is a delayed impulse function, so the output of that will be a delayed impulse response. So, he transforms the H(w) of the system and then delays the time domain impulse response function.

For question 2, he simply noted that H(w) is the frequency domain response and the input is a pure cosine wave, with frequency 2 and no phase delay. So, he can quickly calculate that the output will be scaled in magnitude and phase shifted by using H(w).

#### hamza324

Joined Jul 10, 2011
33
Thanks for explanation.

I have another similar question regarding discrete time

I have an expression

$$e^{j \Omega n} \longrightarrow \fbox{h[n] \longleftrightarrow H(e^{j \Omega})} \longrightarrow H(e^{j \Omega}) e^{j \Omega n}$$

I know this is frequeny response but i dont understand the expression at the output

Can you please provide me with some simpler explanation for this expression and some kind of example if possible.

#### steveb

Joined Jul 3, 2008
2,436
Thanks for explanation.

I have another similar question regarding discrete time

I have an expression

$$e^{j \Omega n} \longrightarrow \fbox{h[n] \longleftrightarrow H(e^{j \Omega})} \longrightarrow H(e^{j \Omega}) e^{j \Omega n}$$

I know this is frequeny response but i dont understand the expression at the output

Can you please provide me with some simpler explanation for this expression and some kind of example if possible.
This is similar to the analysis of the second component of the input signal above (i.e. the 5cos(2t) part). That example was a continuous time cosine wave going through a system with known frequency response.

This problem is pretty much the same, only it is a complex sine wave, and the problem is in discrete time.

For background, in continuous time the frequency response is obtained by taking the Laplace s-domain transfer function $$H(s)$$, where $$s=\sigma+j\omega$$, and substituting $$s=j\omega$$, which puts you on the jw-axis. In discrete time, the frequency response is obtained by taking the Laplace z-domain transfer function $$H(z)$$, where $$z=r\; {\rm e}^{j\Omega}$$, and substituting $$z={\rm e}^{j\Omega}$$, which puts you on the unit circle.