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‎doc/pub/week4/html/week4-bs.html‎

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@@ -43,6 +43,31 @@
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('Readings', 2, None, 'readings'),
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('Gates, the whys and hows', 2, None, 'gates-the-whys-and-hows'),
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('Structure of the lecture', 2, None, 'structure-of-the-lecture'),
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('Reminder on density matrices and traces',
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2,
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None,
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'reminder-on-density-matrices-and-traces'),
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('Definition of density matrix',
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2,
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None,
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'definition-of-density-matrix'),
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('Von Neumann entropy', 2, None, 'von-neumann-entropy'),
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('Schmidt decomposition', 2, None, 'schmidt-decomposition'),
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('Pure states and Schmidt decomposition',
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2,
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None,
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'pure-states-and-schmidt-decomposition'),
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('Proof of Schmidt decomposition',
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2,
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None,
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'proof-of-schmidt-decomposition'),
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('Further parts of proof', 2, None, 'further-parts-of-proof'),
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('SVD parts in proof', 2, None, 'svd-parts-in-proof'),
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('Slight rewrite', 2, None, 'slight-rewrite'),
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('Different dimensionalities',
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2,
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None,
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'different-dimensionalities'),
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('Part 1: Mathematical background',
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2,
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None,
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<!-- navigation toc: --> <li><a href="#readings" style="font-size: 80%;">Readings</a></li>
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<!-- navigation toc: --> <li><a href="#gates-the-whys-and-hows" style="font-size: 80%;">Gates, the whys and hows</a></li>
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<!-- navigation toc: --> <li><a href="#structure-of-the-lecture" style="font-size: 80%;">Structure of the lecture</a></li>
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<!-- navigation toc: --> <li><a href="#reminder-on-density-matrices-and-traces" style="font-size: 80%;">Reminder on density matrices and traces</a></li>
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<!-- navigation toc: --> <li><a href="#definition-of-density-matrix" style="font-size: 80%;">Definition of density matrix</a></li>
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<!-- navigation toc: --> <li><a href="#von-neumann-entropy" style="font-size: 80%;">Von Neumann entropy</a></li>
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<!-- navigation toc: --> <li><a href="#schmidt-decomposition" style="font-size: 80%;">Schmidt decomposition</a></li>
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<!-- navigation toc: --> <li><a href="#pure-states-and-schmidt-decomposition" style="font-size: 80%;">Pure states and Schmidt decomposition</a></li>
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<!-- navigation toc: --> <li><a href="#proof-of-schmidt-decomposition" style="font-size: 80%;">Proof of Schmidt decomposition</a></li>
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<!-- navigation toc: --> <li><a href="#further-parts-of-proof" style="font-size: 80%;">Further parts of proof</a></li>
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<!-- navigation toc: --> <li><a href="#svd-parts-in-proof" style="font-size: 80%;">SVD parts in proof</a></li>
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<!-- navigation toc: --> <li><a href="#slight-rewrite" style="font-size: 80%;">Slight rewrite</a></li>
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<!-- navigation toc: --> <li><a href="#different-dimensionalities" style="font-size: 80%;">Different dimensionalities</a></li>
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<!-- navigation toc: --> <li><a href="#part-1-mathematical-background" style="font-size: 80%;">Part 1: Mathematical background</a></li>
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<!-- navigation toc: --> <li><a href="#unitary-transformation" style="font-size: 80%;">Unitary transformation</a></li>
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<!-- navigation toc: --> <li><a href="#interaction-picture" style="font-size: 80%;">Interaction picture</a></li>
@@ -330,10 +365,181 @@ <h2 id="gates-the-whys-and-hows" class="anchor">Gates, the whys and hows </h2>
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<h2 id="structure-of-the-lecture" class="anchor">Structure of the lecture </h2>
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<ol>
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<li> First we review some of the basic ways of representing the solution to the Schr&#246;dinger equation, introducing the so-called Interaction, Heisenberg and Schr&#246;dinger prictures and unitary transformations. This part is meant mainly as background material</li>
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<li> The first part is a reminder from last week</li>
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<li> Thereafter we review some of the basic ways of representing the solution to the Schr&#246;dinger equation, introducing the so-called Interaction, Heisenberg and Schr&#246;dinger prictures and unitary transformations. This part is meant mainly as background material</li>
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<li> Secondly, we present examples of physical processes and how they can be represented as unitary operations on a given state.</li>
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<li> These unitary transformations are then represented as gates. Setting gates together gives us a final circuit which can represent a specific physical system</li>
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</ol>
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<!-- !split -->
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<h2 id="reminder-on-density-matrices-and-traces" class="anchor">Reminder on density matrices and traces </h2>
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<p>We have the spectral decomposition of a given operator \( \boldsymbol{A} \) given by</p>
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$$
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\boldsymbol{A}=\sum_i\lambda_i\vert i \rangle\langle i\vert,
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$$
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<p>with the ONB \( \vert i\rangle \) being eigenvectors of \( \boldsymbol{A} \) and \( \lambda_i \) being the eigenvalues. Similarly, a operator which is a function of \( \boldsymbol{A} \) is given by</p>
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$$
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f(\boldsymbol{A})=\sum_if(\boldsymbol{A})\vert i \rangle\langle i\vert.
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$$
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<p>The trace of a product of matrices is cyclic, that is</p>
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$$
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\mathrm{tr}[\boldsymbol{ABC}])=\mathrm{tr}[\boldsymbol{BCA}])=\mathrm{tr}[\boldsymbol{CBA}]),
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$$
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<p>and we have also </p>
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$$
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\mathrm{tr}[\boldsymbol{A}\vert \psi\rangle\langle\psi\vert])=\langle\psi\vert\boldsymbol{A}\vert\psi\rangle.
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$$
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<!-- !split -->
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<h2 id="definition-of-density-matrix" class="anchor">Definition of density matrix </h2>
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<p>Using the spectral decomposition we defined also the density matrix as</p>
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$$
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\rho = \sum_i p_i\vert i \rangle\langle i\vert,
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$$
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<p>where the probability \( p_i \) are the eigenvalues of the density linked with the ONB \( \vert i \rangle \).</p>
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<p>The trace of the density matrix is \( \mathrm{tr}\rho=1 \) and it is invariant under unitary transformations
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\( \vert \psi_i'\rangle = \boldsymbol{U}\vert \psi_i\rangle \).
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The unitary transformation of the density matrix gives, with
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\( \boldsymbol{U}^{\dagger}\boldsymbol{U}=\boldsymbol{U}^{T}\boldsymbol{U}=\boldsymbol{I} \),
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</p>
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$$
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\boldsymbol{U}\rho\boldsymbol{U}^{\dagger}=\sum_ipi_i\boldsymbol{U}\vert \psi_i\rangle\langle\psi_i\vert\boldsymbol{U}^{\dagger},
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$$
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<p>and with the unitary transformation it is easy to show that thet trace of the transformaed density matrix is equal to one,</p>
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$$
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\mathrm{tr}\left[ \boldsymbol{U}\rho\boldsymbol{U}^{\dagger}\right]=\mathrm{tr}\left[ \boldsymbol{U}\boldsymbol{U}^{\dagger}\rho\right]=1.
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$$
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<!-- !split -->
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<h2 id="von-neumann-entropy" class="anchor">Von Neumann entropy </h2>
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<p>Using the density matrix \( \rho \), we define the quantum mechanical equivalent of the classical entropy as </p>
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$$
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S=-\mathrm{Tr}[\rho\log_2{\rho}].
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$$
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<p>This is the so-called Von Neumann entropy. </p>
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<!-- !split -->
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<h2 id="schmidt-decomposition" class="anchor">Schmidt decomposition </h2>
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<p>A way to study entanglement is through the so-called Schmidt decomposition,
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which is essentially an application of the
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singular-value decomposition.
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</p>
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<!-- !split -->
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<h2 id="pure-states-and-schmidt-decomposition" class="anchor">Pure states and Schmidt decomposition </h2>
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<p>The Schmidt decomposition allows us to define a pure state in a
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bipartite Hilbert space composed of systems \( A \) and \( B \) as
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</p>
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$$
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\vert\psi\rangle=\sum_{i=0}^{d-1}\sigma_i\vert i\rangle_A\vert i\rangle_B,
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$$
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<p>where the amplitudes \( \sigma_i \) are real and positive and their
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squared values sum up to one, \( \sum_i\sigma_i^2=1 \). The states \( \vert
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i\rangle_A \) and \( \vert i\rangle_B \) form orthornormal bases for systems
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\( A \) and \( B \) respectively, the amplitudes \( \lambda_i \) are the so-called
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Schmidt coefficients and the Schmidt rank \( d \) is equal to the number
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of Schmidt coefficients and is smaller or equal to the minimum
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dimensionality of system \( A \) and system \( B \), that is \( d\leq
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\mathrm{min}(\mathrm{dim}(A), \mathrm{dim}(B)) \).
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</p>
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<!-- !split -->
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<h2 id="proof-of-schmidt-decomposition" class="anchor">Proof of Schmidt decomposition </h2>
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<p>The proof for the above decomposition is based on the singular-value
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decomposition. To see this, assume that we have two orthonormal bases
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sets for systems \( A \) and \( B \), respectively. That is we have two ONBs
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\( \vert i\rangle_A \) and \( \vert j\rangle_B \). We can always construct a
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product state (a pure state) as
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</p>
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$$ \vert\psi \rangle = \sum_{ij} c_{ij}\vert i\rangle_A\vert
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j\rangle_B,
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$$
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<p>where the coefficients \( c_{ij} \) are the overlap coefficients which
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belong to a matrix \( \boldsymbol{C} \).
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</p>
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<!-- !split -->
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<h2 id="further-parts-of-proof" class="anchor">Further parts of proof </h2>
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<p>If we now assume that the
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dimensionalities of the two subsystems \( A \) and \( B \) are the same \( d \),
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we can always rewrite the matrix \( \boldsymbol{C} \) in terms of a singular-value
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decomposition with unitary/orthogonal matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \)
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of dimension \( d\times d \) and a matrix \( \boldsymbol{\Sigma} \) which contains the
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(diagonal) singular values \( 0\leq \sigma_0\leq \sigma_1 \leq \dots \sigma_{d-1} \) as
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</p>
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$$
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\boldsymbol{C}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^{\dagger}.
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$$
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<p>Note that the singular values can organized either as a descending series or as an ascending series.</p>
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<!-- !split -->
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<h2 id="svd-parts-in-proof" class="anchor">SVD parts in proof </h2>
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<p>This means we can rewrite the coefficients \( c_{ij} \) in terms of the singular-value decomposition</p>
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$$
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c_{ij}=\sum_k u_{ik}\sigma_kv_{kj},
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$$
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<p>and inserting this in the definition of the pure state \( \vert \psi\rangle \) we have</p>
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$$
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\vert\psi \rangle = \sum_{ij} \left(\sum_k u_{ik}\sigma_kv_{kj} \right)\vert i\rangle_A\vert j\rangle_B.
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$$
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<!-- !split -->
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<h2 id="slight-rewrite" class="anchor">Slight rewrite </h2>
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<p>We rewrite the last equation as</p>
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$$
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\vert\psi \rangle = \sum_{k}\sigma_k \left(\sum_i u_{ik}\vert i\rangle_A\right)\otimes\left(\sum_jv_{kj}\vert j\rangle_B\right),
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$$
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<p>which we identify simply as, since the matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \) represent unitary transformations,</p>
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$$
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\vert\psi \rangle = \sum_{k}\sigma_k \vert k\rangle_A\vert k\rangle_B.
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$$
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<!-- !split -->
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<h2 id="different-dimensionalities" class="anchor">Different dimensionalities </h2>
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<p>It is straight forward to prove this relation in case systems \( A \) and
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\( B \) have different dimensionalities. Once we know the Schmidt
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decomposition of a state, we can immmediately say whether it is
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entangled or not. If a state \( \psi \) is entangled, then its Schmidt
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decomposition has more than one term. Stated differently, the state is
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entangled if the so-called Schmidt rank is greater than one.
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</p>
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<p>There
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is another important property of the Schmidt decomposition which is
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related to the properties of the density matrices and their trace
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operations and the entropies. These concepts can be studied for example
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by studying the two-qubit Hamiltonian described above.
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</p>
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<!-- !split -->
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<h2 id="part-1-mathematical-background" class="anchor">Part 1: Mathematical background </h2>
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X=HZH.
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$$
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<!-- ------------------- end of main content --------------- -->
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</div> <!-- end container -->
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<!-- include javascript, jQuery *first* -->

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