Skip to content

Commit a6d8e79

Browse files
mhjensenmhjensen
committed
update week 3
1 parent da3130f commit a6d8e79

File tree

7 files changed

+121
-1119
lines changed

7 files changed

+121
-1119
lines changed

‎doc/pub/week3/html/week3-bs.html‎

Lines changed: 4 additions & 187 deletions
Original file line numberDiff line numberDiff line change
@@ -41,7 +41,6 @@
4141
2,
4242
None,
4343
'plans-for-the-week-of-february-3-7-entanglement-entropies-and-density-matrices'),
44-
('Motivation', 2, None, 'motivation'),
4544
('Reminder on density matrices and traces',
4645
2,
4746
None,
@@ -66,22 +65,6 @@
6665
('Examples of entanglement', 2, None, 'examples-of-entanglement'),
6766
('Ground state of helium', 2, None, 'ground-state-of-helium'),
6867
('Maximally entangled', 2, None, 'maximally-entangled'),
69-
('Schmidt decomposition', 2, None, 'schmidt-decomposition'),
70-
('Pure states and Schmidt decomposition',
71-
2,
72-
None,
73-
'pure-states-and-schmidt-decomposition'),
74-
('Proof of Schmidt decomposition',
75-
2,
76-
None,
77-
'proof-of-schmidt-decomposition'),
78-
('Further parts of proof', 2, None, 'further-parts-of-proof'),
79-
('SVD parts in proof', 2, None, 'svd-parts-in-proof'),
80-
('Slight rewrite', 2, None, 'slight-rewrite'),
81-
('Different dimensionalities',
82-
2,
83-
None,
84-
'different-dimensionalities'),
8568
('Entropies and density matrices',
8669
2,
8770
None,
@@ -95,12 +78,7 @@
9578
'exercise',
9679
2,
9780
None,
98-
'two-qubit-system-and-calculation-of-density-matrices-and-exercise'),
99-
('Exercise 1: Two-qubit Hamiltonian',
100-
2,
101-
None,
102-
'exercise-1-two-qubit-hamiltonian'),
103-
('The next lecture', 2, None, 'the-next-lecture')]}
81+
'two-qubit-system-and-calculation-of-density-matrices-and-exercise')]}
10482
end of tocinfo -->
10583

10684
<body>
@@ -136,7 +114,6 @@
136114
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
137115
<ul class="dropdown-menu">
138116
<!-- navigation toc: --> <li><a href="#plans-for-the-week-of-february-3-7-entanglement-entropies-and-density-matrices" style="font-size: 80%;">Plans for the week of February 3-7: Entanglement, entropies and density matrices</a></li>
139-
<!-- navigation toc: --> <li><a href="#motivation" style="font-size: 80%;">Motivation</a></li>
140117
<!-- navigation toc: --> <li><a href="#reminder-on-density-matrices-and-traces" style="font-size: 80%;">Reminder on density matrices and traces</a></li>
141118
<!-- navigation toc: --> <li><a href="#definition-of-density-matrix" style="font-size: 80%;">Definition of density matrix</a></li>
142119
<!-- navigation toc: --> <li><a href="#first-entanglement-encounter-two-qubit-system" style="font-size: 80%;">First entanglement encounter, two qubit system</a></li>
@@ -152,19 +129,10 @@
152129
<!-- navigation toc: --> <li><a href="#examples-of-entanglement" style="font-size: 80%;">Examples of entanglement</a></li>
153130
<!-- navigation toc: --> <li><a href="#ground-state-of-helium" style="font-size: 80%;">Ground state of helium</a></li>
154131
<!-- navigation toc: --> <li><a href="#maximally-entangled" style="font-size: 80%;">Maximally entangled</a></li>
155-
<!-- navigation toc: --> <li><a href="#schmidt-decomposition" style="font-size: 80%;">Schmidt decomposition</a></li>
156-
<!-- navigation toc: --> <li><a href="#pure-states-and-schmidt-decomposition" style="font-size: 80%;">Pure states and Schmidt decomposition</a></li>
157-
<!-- navigation toc: --> <li><a href="#proof-of-schmidt-decomposition" style="font-size: 80%;">Proof of Schmidt decomposition</a></li>
158-
<!-- navigation toc: --> <li><a href="#further-parts-of-proof" style="font-size: 80%;">Further parts of proof</a></li>
159-
<!-- navigation toc: --> <li><a href="#svd-parts-in-proof" style="font-size: 80%;">SVD parts in proof</a></li>
160-
<!-- navigation toc: --> <li><a href="#slight-rewrite" style="font-size: 80%;">Slight rewrite</a></li>
161-
<!-- navigation toc: --> <li><a href="#different-dimensionalities" style="font-size: 80%;">Different dimensionalities</a></li>
162132
<!-- navigation toc: --> <li><a href="#entropies-and-density-matrices" style="font-size: 80%;">Entropies and density matrices</a></li>
163133
<!-- navigation toc: --> <li><a href="#shannon-information-entropy" style="font-size: 80%;">Shannon information entropy</a></li>
164134
<!-- navigation toc: --> <li><a href="#von-neumann-entropy" style="font-size: 80%;">Von Neumann entropy</a></li>
165135
<!-- navigation toc: --> <li><a href="#two-qubit-system-and-calculation-of-density-matrices-and-exercise" style="font-size: 80%;">Two-qubit system and calculation of density matrices and exercise</a></li>
166-
<!-- navigation toc: --> <li><a href="#exercise-1-two-qubit-hamiltonian" style="font-size: 80%;">Exercise 1: Two-qubit Hamiltonian</a></li>
167-
<!-- navigation toc: --> <li><a href="#the-next-lecture" style="font-size: 80%;">The next lecture</a></li>
168136

169137
</ul>
170138
</li>
@@ -204,35 +172,17 @@ <h2 id="plans-for-the-week-of-february-3-7-entanglement-entropies-and-density-ma
204172
<div class="panel-body">
205173
<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
206174
<ol>
207-
<li> Reminder and review of density matrices and measurements from last week</li>
208-
<li> Schmidt decomposition and entanglement</li>
175+
<li> Reminder and review of density matrices and measurements from last week
176+
<!-- Postpone thiso Schmidt decomposition and entanglement --></li>
209177
<li> Discussion of entropies, classical information entropy (Shannon entropy) and von Neumann entropy</li>
210-
<li> Single and two-qubit gates</li>
178+
<li> Single-qubit and two-qubit gates and codes</li>
211179
<li> <a href="https://youtu.be/" target="_self">Video of lecture to be added</a>
212180
<!-- Chapters 3 and 4 of Scherer's text contains useful discussions of several of these topics. More reading suggestions will be added. --></li>
213181
</ol>
214182
</div>
215183
</div>
216184

217185

218-
<!-- !split -->
219-
<h2 id="motivation" class="anchor">Motivation </h2>
220-
221-
<p>In order to study entanglement and why it is so important for quantum
222-
computing, we need to introduce some basic measures and useful
223-
quantities. For these endeavors, we will use our two-qubit system from
224-
the second lecture in order to introduce and repeat, through examples, density
225-
matrices and entropy. These two quantities, together with
226-
technicalities like the Schmidt decomposition define important quantities in analyzing quantum computing examples.
227-
</p>
228-
229-
<p>The Schmidt decomposition is again a
230-
linear decomposition which allows us to express a vector in terms of
231-
tensor product of two inner product spaces. In quantum information
232-
theory and quantum computing it is widely used as away to define and
233-
describe entanglement.
234-
</p>
235-
236186
<!-- !split -->
237187
<h2 id="reminder-on-density-matrices-and-traces" class="anchor">Reminder on density matrices and traces </h2>
238188

@@ -492,113 +442,6 @@ <h2 id="maximally-entangled" class="anchor">Maximally entangled </h2>
492442
$$
493443

494444

495-
<!-- !split -->
496-
<h2 id="schmidt-decomposition" class="anchor">Schmidt decomposition </h2>
497-
<p>If we cannot write the density matrix in this form, we say the system
498-
\( AB \) is entangled. In order to see this, we can use the so-called
499-
Schmidt decomposition, which is essentially an application of the
500-
singular-value decomposition.
501-
</p>
502-
503-
<!-- !split -->
504-
<h2 id="pure-states-and-schmidt-decomposition" class="anchor">Pure states and Schmidt decomposition </h2>
505-
506-
<p>The Schmidt decomposition allows us to define a pure state in a
507-
bipartite Hilbert space composed of systems \( A \) and \( B \) as
508-
</p>
509-
510-
$$
511-
\vert\psi\rangle=\sum_{i=0}^{d-1}\sigma_i\vert i\rangle_A\vert i\rangle_B,
512-
$$
513-
514-
<p>where the amplitudes \( \sigma_i \) are real and positive and their
515-
squared values sum up to one, \( \sum_i\sigma_i^2=1 \). The states \( \vert
516-
i\rangle_A \) and \( \vert i\rangle_B \) form orthornormal bases for systems
517-
\( A \) and \( B \) respectively, the amplitudes \( \lambda_i \) are the so-called
518-
Schmidt coefficients and the Schmidt rank \( d \) is equal to the number
519-
of Schmidt coefficients and is smaller or equal to the minimum
520-
dimensionality of system \( A \) and system \( B \), that is \( d\leq
521-
\mathrm{min}(\mathrm{dim}(A), \mathrm{dim}(B)) \).
522-
</p>
523-
524-
<!-- !split -->
525-
<h2 id="proof-of-schmidt-decomposition" class="anchor">Proof of Schmidt decomposition </h2>
526-
527-
<p>The proof for the above decomposition is based on the singular-value
528-
decomposition. To see this, assume that we have two orthonormal bases
529-
sets for systems \( A \) and \( B \), respectively. That is we have two ONBs
530-
\( \vert i\rangle_A \) and \( \vert j\rangle_B \). We can always construct a
531-
product state (a pure state) as
532-
</p>
533-
534-
$$ \vert\psi \rangle = \sum_{ij} c_{ij}\vert i\rangle_A\vert
535-
j\rangle_B,
536-
$$
537-
538-
<p>where the coefficients \( c_{ij} \) are the overlap coefficients which
539-
belong to a matrix \( \boldsymbol{C} \).
540-
</p>
541-
542-
<!-- !split -->
543-
<h2 id="further-parts-of-proof" class="anchor">Further parts of proof </h2>
544-
545-
<p>If we now assume that the
546-
dimensionalities of the two subsystems \( A \) and \( B \) are the same \( d \),
547-
we can always rewrite the matrix \( \boldsymbol{C} \) in terms of a singular-value
548-
decomposition with unitary/orthogonal matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \)
549-
of dimension \( d\times d \) and a matrix \( \boldsymbol{\Sigma} \) which contains the
550-
(diagonal) singular values \( \sigma_0\leq \sigma_1 \leq \dots 0 \) as
551-
</p>
552-
553-
$$
554-
\boldsymbol{C}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^{\dagger}.
555-
$$
556-
557-
558-
<!-- !split -->
559-
<h2 id="svd-parts-in-proof" class="anchor">SVD parts in proof </h2>
560-
561-
<p>This means we can rewrite the coefficients \( c_{ij} \) in terms of the singular-value decomposition</p>
562-
$$
563-
c_{ij}=\sum_k u_{ik}\sigma_kv_{kj},
564-
$$
565-
566-
<p>and inserting this in the definition of the pure state \( \vert \psi\rangle \) we have</p>
567-
568-
$$
569-
\vert\psi \rangle = \sum_{ij} \left(\sum_k u_{ik}\sigma_kv_{kj} \right)\vert i\rangle_A\vert j\rangle_B.
570-
$$
571-
572-
573-
<!-- !split -->
574-
<h2 id="slight-rewrite" class="anchor">Slight rewrite </h2>
575-
<p>We rewrite the last equation as</p>
576-
577-
$$
578-
\vert\psi \rangle = \sum_{k}\sigma_k \left(\sum_i u_{ik}\vert i\rangle_A\right)\otimes\left(\sum_jv_{kj}\vert j\rangle_B\right),
579-
$$
580-
581-
<p>which we identify simply as, since the matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \) represent unitary transformations,</p>
582-
$$
583-
\vert\psi \rangle = \sum_{k}\sigma_k \vert k\rangle_A\vert k\rangle_B.
584-
$$
585-
586-
587-
<!-- !split -->
588-
<h2 id="different-dimensionalities" class="anchor">Different dimensionalities </h2>
589-
590-
<p>It is straight forward to prove this relation in case systems \( A \) and
591-
\( B \) have different dimensionalities. Once we know the Schmidt
592-
decomposition of a state, we can immmediately say whether it is
593-
entangled or not. If a state \( \psi \) has is entangled, then its Schmidt
594-
decomposition has more than one term. Stated differently, the state is
595-
entangled if the so-called Schmidt rank is is greater than one. There
596-
is another important property of the Schmidt decomposition which is
597-
related to the properties of the density matrices and their trace
598-
operations and the entropies. In order to introduce these concepts let
599-
us look at the two-qubit Hamiltonian described here.
600-
</p>
601-
602445
<!-- !split -->
603446
<h2 id="entropies-and-density-matrices" class="anchor">Entropies and density matrices </h2>
604447

@@ -633,8 +476,6 @@ <h2 id="two-qubit-system-and-calculation-of-density-matrices-and-exercise" class
633476

634477
<b>This part is best seen using the jupyter-notebook</b>.
635478

636-
<p>The system we discuss here is a continuation of the two qubit example from week 2.</p>
637-
638479
<p>This system can be thought of as composed of two subsystems
639480
\( A \) and \( B \). Each subsystem has computational basis states
640481
</p>
@@ -834,30 +675,6 @@ <h2 id="two-qubit-system-and-calculation-of-density-matrices-and-exercise" class
834675
</div>
835676

836677

837-
<!-- !split -->
838-
839-
<!-- --- begin exercise --- -->
840-
<h2 id="exercise-1-two-qubit-hamiltonian" class="anchor">Exercise 1: Two-qubit Hamiltonian </h2>
841-
842-
<p>Use the Hamiltonian for the two-qubit example to find the eigenpairs
843-
as functions of the interaction strength \( \lambda \) and study the final
844-
eigenvectors as functions of the admixture of the original basis
845-
states. Discuss the results as functions of the parameter \( \lambda \) and compute the von Neumann
846-
entropy and discuss the results. You will need to calculate the entropy of the subsystems \( A \) or \( B \).
847-
</p>
848-
849-
<!-- --- end exercise --- -->
850-
851-
<!-- !split -->
852-
<h2 id="the-next-lecture" class="anchor">The next lecture </h2>
853-
854-
<p>In our next lecture, we will discuss</p>
855-
<ol>
856-
<li> Reminder and review of entropy and entanglement</li>
857-
<li> Gates and circuits and how to perform operations on states</li>
858-
</ol>
859-
<a href="https://github.com/CompPhysics/QuantumComputingMachineLearning/blob/gh-pages/doc/Textbooks/Programming/chapter2.pdf" target="_self">Reading: Chapters 2.1-2.11 of Hundt's text</a>
860-
861678
<!-- ------------------- end of main content --------------- -->
862679
</div> <!-- end container -->
863680
<!-- include javascript, jQuery *first* -->

0 commit comments

Comments
 (0)