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update 2nd week
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‎doc/pub/week2/html/week2-bs.html‎

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2,
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None,
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'extending-the-expressions'),
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('Density Matrices', 2, None, 'density-matrices'),
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('Density matrices/operators',
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2,
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None,
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2,
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None,
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'properties-of-density-matrices'),
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('Important properties of the density matrix',
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2,
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'important-properties-of-the-density-matrix'),
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('More on the density matrix',
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None,
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'more-on-the-density-matrix'),
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('First entanglement encounter, two qubit system',
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None,
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('Examples of entanglement', 2, None, 'examples-of-entanglement'),
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('Ground state of helium', 2, None, 'ground-state-of-helium'),
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('Maximally entangled', 2, None, 'maximally-entangled'),
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('Density matrices in more detail',
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2,
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None,
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'density-matrices-in-more-detail'),
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('Second exercise set', 2, None, 'second-exercise-set'),
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('Ex1: One-qubit basis and Pauli matrices',
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2,
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2,
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None,
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'ex6-reduced-density-operators-ii'),
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('The next lecture, February 5, 2025',
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('The next lecture, February 4, 2026',
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2,
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None,
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'the-next-lecture-february-5-2025')]}
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'the-next-lecture-february-4-2026')]}
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end of tocinfo -->
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<body>
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<!-- navigation toc: --> <li><a href="#taking-the-trace" style="font-size: 80%;">Taking the trace</a></li>
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<!-- navigation toc: --> <li><a href="#outcome-probability" style="font-size: 80%;">Outcome probability</a></li>
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<!-- navigation toc: --> <li><a href="#extending-the-expressions" style="font-size: 80%;">Extending the expressions</a></li>
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<!-- navigation toc: --> <li><a href="#density-matrices" style="font-size: 80%;">Density Matrices</a></li>
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<!-- navigation toc: --> <li><a href="#density-matrices-operators" style="font-size: 80%;">Density matrices/operators</a></li>
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<!-- navigation toc: --> <li><a href="#properties-of-density-matrices" style="font-size: 80%;">Properties of density matrices</a></li>
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<!-- navigation toc: --> <li><a href="#important-properties-of-the-density-matrix" style="font-size: 80%;">Important properties of the density matrix</a></li>
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<!-- navigation toc: --> <li><a href="#more-on-the-density-matrix" style="font-size: 80%;">More on the density matrix</a></li>
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<!-- navigation toc: --> <li><a href="#first-entanglement-encounter-two-qubit-system" style="font-size: 80%;">First entanglement encounter, two qubit system</a></li>
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<!-- navigation toc: --> <li><a href="#computational-basis" style="font-size: 80%;">Computational basis</a></li>
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<!-- navigation toc: --> <li><a href="#bell-states" style="font-size: 80%;">Bell states</a></li>
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<!-- navigation toc: --> <li><a href="#examples-of-entanglement" style="font-size: 80%;">Examples of entanglement</a></li>
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<!-- navigation toc: --> <li><a href="#ground-state-of-helium" style="font-size: 80%;">Ground state of helium</a></li>
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<!-- navigation toc: --> <li><a href="#maximally-entangled" style="font-size: 80%;">Maximally entangled</a></li>
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<!-- navigation toc: --> <li><a href="#density-matrices-in-more-detail" style="font-size: 80%;">Density matrices in more detail</a></li>
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<!-- navigation toc: --> <li><a href="#second-exercise-set" style="font-size: 80%;">Second exercise set</a></li>
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<!-- navigation toc: --> <li><a href="#ex1-one-qubit-basis-and-pauli-matrices" style="font-size: 80%;">Ex1: One-qubit basis and Pauli matrices</a></li>
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<!-- navigation toc: --> <li><a href="#ex2-hadamard-and-phase-gates" style="font-size: 80%;">Ex2: Hadamard and Phase gates</a></li>
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<!-- navigation toc: --> <li><a href="#ex3-traces-of-operators" style="font-size: 80%;">Ex3: Traces of operators</a></li>
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<!-- navigation toc: --> <li><a href="#ex4-exponentiated-operators" style="font-size: 80%;">Ex4: Exponentiated operators</a></li>
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<!-- navigation toc: --> <li><a href="#ex5-reduced-density-operators-i" style="font-size: 80%;">Ex5: Reduced density operators I</a></li>
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<!-- navigation toc: --> <li><a href="#ex6-reduced-density-operators-ii" style="font-size: 80%;">Ex6: Reduced density operators II</a></li>
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<!-- navigation toc: --> <li><a href="#the-next-lecture-february-5-2025" style="font-size: 80%;">The next lecture, February 5, 2025</a></li>
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<!-- navigation toc: --> <li><a href="#the-next-lecture-february-4-2026" style="font-size: 80%;">The next lecture, February 4, 2026</a></li>
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</ul>
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</li>
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<li> introduced the inner product and showed how to calculate it in an orthonormal basis</li>
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<li> introduced outer products and projection operators</li>
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<li> introduced tensor products and showed how to construct state vectors for multiple qubits</li>
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<li> introduced the spectral decomposition of operators</li>
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</ol>
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</div>
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</div>
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<p>With these prerequisites we are now ready to introduce the density matrices, or density operators.</p>
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<!-- !split -->
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<h2 id="density-matrices" class="anchor">Density Matrices </h2>
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<p>As opposed to the wavefunction (a so-called pure state) in closed
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(Hermitian) quantum mechanics, the density matrix uniquely describes
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the state of a quantum system in an open system, as well as in closed
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systems.
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</p>
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<p>With pure states, the state of a quantum system is deterministic and
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leads to probabilities of outcomes of measurements. In contrast, what
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if the state itself is uncertain and therefore probabilistic? This is
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what density matrices describe.
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</p>
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<!-- !split -->
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<h2 id="density-matrices-operators" class="anchor">Density matrices/operators </h2>
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$$
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<!-- !split -->
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<h2 id="important-properties-of-the-density-matrix" class="anchor">Important properties of the density matrix </h2>
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<div class="panel panel-default">
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<div class="panel-body">
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<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
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<p>The density matrix of a pure state \( \vert\psi\rangle \) is \( \rho = \vert\psi\rangle\langle\psi\vert \).</p>
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</div>
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</div>
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<div class="panel panel-default">
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<div class="panel-body">
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<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
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$$
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\rho = \rho^\dagger, \quad \lambda(\rho) \ge 0, \quad \mathrm{Tr}(\rho) = 1.
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$$
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</div>
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</div>
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<div class="panel panel-default">
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<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
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<p>Given a probabilistic mixture of states with probabilities \( \{p_i\} \) and states \( \{\vert\psi_i\rangle\} \),</p>
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$$
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\rho = \sum_i p_i \vert \psi_i\rangle\langle\psi_i\vert.
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$$
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</div>
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</div>
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<!-- !split -->
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<h2 id="more-on-the-density-matrix" class="anchor">More on the density matrix </h2>
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<div class="panel panel-default">
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<div class="panel-body">
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<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
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<p>For a \( d \times d \) density matrix \( \rho \), there exists an orthonormal basis \( \{\vert b_i\rangle\} \) such that</p>
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$$
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\rho = \sum_i \lambda_i \vert b_i\rangle\langle b_i\vert,
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$$
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<p>where \( \lambda_i \) are probabilities.</p>
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</div>
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</div>
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<div class="panel panel-default">
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<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
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<p>The expectation value of an observable \( X \) is</p>
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$$
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\langle X \rangle = \mathrm{Tr}(\rho X) = \sum_i \lambda_i \langle b_i\vert X \vert b_i\rangle.
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$$
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<p>The time evolution in a closed system with Hamiltonian \( H \) is</p>
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$$
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\rho(t) = \exp{-\imath Ht} \rho(0) \exp{\imath Ht}.
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$$
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</div>
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</div>
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<!-- !split -->
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<h2 id="first-entanglement-encounter-two-qubit-system" class="anchor">First entanglement encounter, two qubit system </h2>
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$$
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<!-- !split -->
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<h2 id="density-matrices-in-more-detail" class="anchor">Density matrices in more detail </h2>
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<h2 id="second-exercise-set" class="anchor">Second exercise set </h2>
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</p>
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<!-- !split -->
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<h2 id="the-next-lecture-february-5-2025" class="anchor">The next lecture, February 5, 2025 </h2>
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<h2 id="the-next-lecture-february-4-2026" class="anchor">The next lecture, February 4, 2026 </h2>
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<p>In our next lecture, we will discuss</p>
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<ol>
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<li> Discussion of entropy and entanglement</li>
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<li> Gates and circuits and how to perform operations on states</li>
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</ol>
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<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>
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</footer>
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<center style="font-size:80%">
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<!-- copyright --> &copy; 1999-2025, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
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<!-- copyright --> &copy; 1999-2026, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
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‎doc/pub/week2/html/week2-reveal.html‎

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<center style="font-size:80%">
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<!-- copyright --> &copy; 1999-2025, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
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<!-- copyright --> &copy; 1999-2026, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
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</section>
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<p><li> introduced the inner product and showed how to calculate it in an orthonormal basis</li>
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<p><li> introduced outer products and projection operators</li>
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<p><li> introduced tensor products and showed how to construct state vectors for multiple qubits</li>
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<p><li> introduced the spectral decomposition of operators</li>
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</ol>
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</div>
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</section>
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<p>With these prerequisites we are now ready to introduce the density matrices, or density operators.</p>
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</section>
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<section>
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<h2 id="density-matrices">Density Matrices </h2>
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<p>As opposed to the wavefunction (a so-called pure state) in closed
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(Hermitian) quantum mechanics, the density matrix uniquely describes
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the state of a quantum system in an open system, as well as in closed
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systems.
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</p>
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<p>With pure states, the state of a quantum system is deterministic and
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leads to probabilities of outcomes of measurements. In contrast, what
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if the state itself is uncertain and therefore probabilistic? This is
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what density matrices describe.
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</p>
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</section>
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<h2 id="density-matrices-operators">Density matrices/operators </h2>
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<p>&nbsp;<br>
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<h2 id="important-properties-of-the-density-matrix">Important properties of the density matrix </h2>
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<div class="alert alert-block alert-block alert-text-normal">
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<b></b>
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<p>The density matrix of a pure state \( \vert\psi\rangle \) is \( \rho = \vert\psi\rangle\langle\psi\vert \).</p>
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<b>\( \rho \) is Hermitian, positive semi-definite (all eigenvalues non-negative), and has unit trace</b>
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$$
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\rho = \rho^\dagger, \quad \lambda(\rho) \ge 0, \quad \mathrm{Tr}(\rho) = 1.
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$$
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<p>&nbsp;<br>
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<b></b>
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<p>Given a probabilistic mixture of states with probabilities \( \{p_i\} \) and states \( \{\vert\psi_i\rangle\} \),</p>
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$$
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<p>&nbsp;<br>
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</section>
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<b></b>
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<p>
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<p>For a \( d \times d \) density matrix \( \rho \), there exists an orthonormal basis \( \{\vert b_i\rangle\} \) such that</p>
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<p>&nbsp;<br>
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$$
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\rho = \sum_i \lambda_i \vert b_i\rangle\langle b_i\vert,
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$$
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<p>&nbsp;<br>
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<p>where \( \lambda_i \) are probabilities.</p>
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</div>
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<div class="alert alert-block alert-block alert-text-normal">
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<b></b>
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<p>The expectation value of an observable \( X \) is</p>
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<p>&nbsp;<br>
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$$
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\langle X \rangle = \mathrm{Tr}(\rho X) = \sum_i \lambda_i \langle b_i\vert X \vert b_i\rangle.
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$$
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<p>&nbsp;<br>
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<p>The time evolution in a closed system with Hamiltonian \( H \) is</p>
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<p>&nbsp;<br>
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$$
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$$
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<p>&nbsp;<br>
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</div>
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</section>
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<h2 id="first-entanglement-encounter-two-qubit-system">First entanglement encounter, two qubit system </h2>
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<p>&nbsp;<br>
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</section>
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<h2 id="density-matrices-in-more-detail">Density matrices in more detail </h2>
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</section>
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<section>
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<h2 id="second-exercise-set">Second exercise set </h2>
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<section>
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<h2 id="the-next-lecture-february-5-2025">The next lecture, February 5, 2025 </h2>
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<h2 id="the-next-lecture-february-4-2026">The next lecture, February 4, 2026 </h2>
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<p>In our next lecture, we will discuss</p>
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<ol>

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