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<h2 class="hd hd-2 unit-title">Schmidt decomposition</h2>
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S1. Schmidt decomposition (1 of 2)
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Complete the following sequence of logic, to derive the <em>Schmidt decomposition</em> of a pure entangled state [mathjaxinline]|{\Psi }\rangle \in \mathbb {C}^{d_ A} \otimes \mathbb {C}^{d_ B}[/mathjaxinline]: </p>
<table id="a0000000002" class="eqnarray" cellspacing="0" cellpadding="7" width="100%" style="table-layout:auto">
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<td style="width:40%; border:none">&#160;</td>
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[mathjaxinline]\displaystyle |{\Psi }\rangle = \sum _{j=1}^{n} \sigma _{j} |{\alpha _ j}\rangle \otimes |{\beta _ j}\rangle ,[/mathjaxinline]
</td>
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.1)</td>
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<p>
where [mathjaxinline]n = \min (d_ A, d_ B)[/mathjaxinline], and the sets of vectors [mathjaxinline]\{ \alpha _ j\}[/mathjaxinline] and [mathjaxinline]\{ \beta _ j\}[/mathjaxinline] are orthogonal. </p>
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<h4 onclick="hideshow(this);" style="margin: 0px">Hint<span class="icon-caret-down toggleimage"/></h4>
<div class="hideshowcontent">Recall the <a href="https://en.wikipedia.org/wiki/Singular-value_decomposition" target="_blank">singular value decomposition</a> of a matrix [mathjaxinline]X[/mathjaxinline]: <table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]X = U \Sigma V^\dagger[/mathjax]</td><td class="eqnnum" style="width:20%; border:none">&#160;</td></tr></table> where [mathjaxinline]U[/mathjaxinline] and [mathjaxinline]V[/mathjaxinline] are unitaries and [mathjaxinline]\Sigma[/mathjaxinline] is diagonal, with real nonnegative entries. </div>
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<h3 class="hd hd-3 problem-header" id="ps1_schmidt_decomp_3-problem-title" aria-describedby="block-v1:MITx+8.371.1x+2T2018+type@problem+block@ps1_schmidt_decomp_3-problem-progress" tabindex="-1">
S2. Schmidt decomposition (2 of 2)
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Two bipartite entangled states [mathjaxinline]|{\alpha }\rangle[/mathjaxinline] and [mathjaxinline]|{\beta }\rangle[/mathjaxinline] are equivalent under local unitaries if there exist unitaries [mathjaxinline]U, V[/mathjaxinline] such that [mathjaxinline]|{\alpha }\rangle = (U \otimes V) |{\beta }\rangle[/mathjaxinline]. </p>
<p>
Complete the following sequence of logic to prove that [mathjaxinline]|{\alpha }\rangle[/mathjaxinline] and [mathjaxinline]|{\beta }\rangle[/mathjaxinline] are equivalent under local unitaries if and only if they have the same Schmidt coefficients. </p>
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<h2 class="hd hd-2 unit-title">Trace distance</h2>
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TD1. Trace distance (1 of 4)
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Suppose that you are given one of two possible [mathjaxinline]d[/mathjaxinline]-dimensional states [mathjaxinline]\sigma _1[/mathjaxinline] or [mathjaxinline]\sigma _2[/mathjaxinline], with probabilities [mathjaxinline]p_1[/mathjaxinline] and [mathjaxinline]p_2=1-p_1[/mathjaxinline] respectively. Your task is to perform a two-outcome measurement and then try to guess which state you had been given, minimizing the probability of error. </p>
<p>
If the measurement elements are nonnegative Hermitian matrices [mathjaxinline]M_1 = M[/mathjaxinline] and [mathjaxinline]M_2=I-M[/mathjaxinline] then the probability of guessing wrong is </p>
<table id="a0000000005" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]P_{\text err} = p_1 {\rm Tr}(\sigma _1 M_2) + p_2 {\rm Tr}(\sigma _2 M_1).[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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Find an operator [mathjaxinline]\Delta[/mathjaxinline] such that </p>
<table id="a0000000006" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]P_{\text err} = p_1 + {\rm Tr}[M\Delta ].[/mathjax]</td>
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Please enter math symbols using <a href="http://asciimath.org/" target="_blank">asciimath-like conventions</a>, e.g. [mathjaxinline]\sigma _1[/mathjaxinline] as [mathjaxinline]{\tt sigma\_ 1}[/mathjaxinline]. Please make sure to explicitly denote multiplication operations, e.g. [mathjaxinline]x + 2y[/mathjaxinline] should be entered as [mathjaxinline]{\tt x + 2 * y}[/mathjaxinline], and [mathjaxinline]x(1-y)[/mathjaxinline] should be entered as [mathjaxinline]{\tt x * (1-y)}[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\Delta =[/mathjaxinline]</p>
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TD2. Trace distance (2 of 4)
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This step of the problem is a mathematical interlude. You may consider using the result of the problem (the Schatten norms), and returning to complete the proofs of the norms later. </p>
<p>
Consider Hermitian matrices [mathjaxinline]X,\Lambda[/mathjaxinline]. Let [mathjaxinline]\| X\| _1[/mathjaxinline] denote the Schatten 1-norm, i.e. the sum of the absolute values of the eigenvalues of [mathjaxinline]X[/mathjaxinline], and let [mathjaxinline]\| \Lambda \| _\infty[/mathjaxinline] denote the Schatten [mathjaxinline]\infty[/mathjaxinline]-norm, i.e. the maximum absolute value of the eigenvalues of [mathjaxinline]\Lambda[/mathjaxinline]. </p>
<p>
Complete the following sequences of logic to prove that </p>
<table id="a0000000007" class="eqnarray" cellspacing="0" cellpadding="7" width="100%" style="table-layout:auto">
<tr id="a0000000008">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \| X\| _1[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \max _{ \| \Lambda \| _\infty \leq 1} {\rm Tr}[X\Lambda ][/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.2)</td>
</tr>
<tr id="a0000000009">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \| \Lambda \| _\infty[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \max _{\| X\| _1\leq 1} {\rm Tr}[X\Lambda ][/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.3)</td>
</tr>
</table>
<p>
(These relations say that the Schatten [mathjaxinline]1[/mathjaxinline]-norm and the Schatten [mathjaxinline]\infty[/mathjaxinline]-norm are <em>dual</em> to each other.) </p>
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TD4. Trace distance (4 of 4)
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Evaluate [mathjaxinline]P_{\text err,opt}[/mathjaxinline] in the following four cases. Please enter math symbols using <a href="http://asciimath.org/" target="_blank">asciimath-like conventions</a>, e.g. [mathjaxinline]\sigma _1[/mathjaxinline] as [mathjaxinline]{\tt sigma\_ 1}[/mathjaxinline], and [mathjaxinline]\sqrt{1-x}[/mathjaxinline] as [mathjaxinline]{\tt sqrt(1-x)}[/mathjaxinline]; also, please make sure to explicitly denote multiplication operations, e.g. [mathjaxinline]x + 2y[/mathjaxinline] should be entered as [mathjaxinline]{\tt x + 2 * y}[/mathjaxinline], and [mathjaxinline]x(1-y)[/mathjaxinline] should be entered as [mathjaxinline]{\tt x * (1-y)}[/mathjaxinline]. </p>
<ol class="enumerate">
<li value="1">
<p>
[mathjaxinline]p_1=1[/mathjaxinline], [mathjaxinline]p_2=0[/mathjaxinline] and [mathjaxinline]\sigma _1,\sigma _2[/mathjaxinline] are arbitrary. </p>
<p>
<p style="display:inline">[mathjaxinline]P_{err, opt} =[/mathjaxinline]</p>
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<p>
<p style="display:inline">[mathjaxinline]{\rm Best\ state\ to\ guess} =[/mathjaxinline]</p>
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<p>
[mathjaxinline]p_1 \geq p_2 \geq 0[/mathjaxinline] are arbitrary (subject to [mathjaxinline]p_1+p_2=1[/mathjaxinline]) and [mathjaxinline]\sigma _1=\sigma _2[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]P_{err, opt} =[/mathjaxinline]</p>
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<li value="3">
<p>
[mathjaxinline]p_1 = p_2 = 1/2[/mathjaxinline] and [mathjaxinline]\sigma _1,\sigma _2[/mathjaxinline] are arbitrary. Express your answer in terms of [mathjaxinline]x \equiv \| \sigma _1 - \sigma _2\| _1[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]P_{err, opt} =[/mathjaxinline]</p>
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<li value="4">
<p>
[mathjaxinline]p_1=p_2=1/2[/mathjaxinline], [mathjaxinline]\sigma _1 = \left|\psi _1\right\rangle \left\langle \psi _1\right|[/mathjaxinline], [mathjaxinline]\sigma _2=\left|\psi _2\right\rangle \left\langle \psi _2\right|[/mathjaxinline]. Express your answer in terms of [mathjaxinline]y \equiv |\langle {\psi _1}|{\psi _2}\rangle |^2[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]P_{err, opt} =[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">TPCP maps are quantum operations</h2>
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TM1. TPCP maps are quantum operations (1 of 3)
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<p>
In this problem we will show that if [mathjaxinline]\mathcal{N}[/mathjaxinline] is a trace-preserving (TP), completely positive (CP) linear map on [mathjaxinline]L(A)[/mathjaxinline] then it is a valid quantum operation (i.e. has a Kraus decomposition). </p>
<p>
Let [mathjaxinline]d=\dim A[/mathjaxinline] and define another system [mathjaxinline]R[/mathjaxinline] (for &#8220;reference") also of dimension [mathjaxinline]d[/mathjaxinline]. Define [mathjaxinline]|{\Phi }\rangle := \frac{1}{\sqrt{d}}\sum _{i=1}^{d} |{i}\rangle _ A \otimes |{i}\rangle _ R[/mathjaxinline] and write the corresponding density matrix as [mathjaxinline]\Phi = \left|\Phi \right\rangle \left\langle \Phi \right|[/mathjaxinline]. Define the Choi-Jamio&#322;kowski state </p>
<table id="a0000000011" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]J(\mathcal{N}) := (\mathcal{N}_ A \otimes {\rm id}_ R)(\Phi ).[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.4)</td>
</tr>
</table>
<p>
We would like to show that if [mathjaxinline]\mathcal{N}[/mathjaxinline] is a TPCP linear map then [mathjaxinline]J(\mathcal{N})[/mathjaxinline] is a valid density matrix. The argument has two parts: first, [mathjaxinline]J(\mathcal{N})[/mathjaxinline] must have nonnegative eigenvalues. This argument is simple. </p>
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Second, the trace of [mathjaxinline]J(\mathcal{N})[/mathjaxinline] must be one. Complete the following calculation to show that fact: </p>
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<h3 class="hd hd-3 problem-header" id="ps1a_tcp_2-problem-title" aria-describedby="block-v1:MITx+8.371.1x+2T2018+type@problem+block@ps1a_tcp_2-problem-progress" tabindex="-1">
TM2. TPCP maps are quantum operations (2 of 3)
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<p>
Complete the following sequence of logic to show that [mathjaxinline]\mathcal{N}[/mathjaxinline] can be recovered from [mathjaxinline]J(\mathcal{N})[/mathjaxinline] via the equation </p>
<table id="a0000000012" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr id="eqn1p5">
<td class="equation" style="width:80%; border:none">[mathjax]\mathcal{N}(X) = d {\rm Tr}_ R[J(\mathcal{N})(I \otimes X^ T)].[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.5)</td>
</tr>
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<p>
This implies that the map [mathjaxinline]\mathcal{N}\rightarrow J(\mathcal{N})[/mathjaxinline] is injective. (In fact it is known as the Choi-Jamio&#322;kowski isomorphism.) </p>
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<h2 class="hd hd-2 unit-title">Quantum channels</h2>
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QC1. Quantum channels (1 of 4)
</h3>
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<p>
Complete the following line of reasoning to show that any linear operator [mathjaxinline]\mathcal{N}[/mathjaxinline] from [mathjaxinline]L(\mathbb {C}^{d_1})[/mathjaxinline] to [mathjaxinline]L(\mathbb {C}^{d_2})[/mathjaxinline] can be written in a form like [mathjaxinline]\mathcal{N}(X) = \sum _ a A_ a X B_ a^\dagger[/mathjaxinline] for some matrices [mathjaxinline]A_ a,B_ a[/mathjaxinline]. </p>
<p>
Let [mathjaxinline]\{ |{1}\rangle , \ldots , |{d_1}\rangle \}[/mathjaxinline] be an orthonormal basis for [mathjaxinline]\mathbb {C}^{d_1}[/mathjaxinline]. To understand the action of [mathjaxinline]\mathcal{N}[/mathjaxinline], it suffices to consider its action on a basis for [mathjaxinline]L(\mathbb {C}^{d_1})[/mathjaxinline]. A simple choice of basis is the matrices of the form [mathjaxinline]|{i}\rangle \langle {j}|[/mathjaxinline] for [mathjaxinline]i,j \in [d_1][/mathjaxinline]. </p>
<p>
Define the complex coefficients [mathjaxinline]N_{ijkl}[/mathjaxinline] according to [mathjaxinline]\mathcal{N}(|{i}\rangle \langle {j}|) = \sum _{kl} N_{ijkl} |{k}\rangle \langle {l}|[/mathjaxinline]. We can write </p>
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To describe the dimension of the matrices, use the convention that a linear map from [mathjaxinline]\mathbb {C}^{d_ A}[/mathjaxinline] to [mathjaxinline]\mathbb {C}^{d_ B}[/mathjaxinline] has dimension [mathjaxinline]d_ B \times d_ A[/mathjaxinline]. </p>
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<h3 class="hd hd-3 problem-header" id="ps1b_q_channels2-problem-title" aria-describedby="block-v1:MITx+8.371.1x+2T2018+type@problem+block@ps1b_q_channels2-problem-progress" tabindex="-1">
QC2. Quantum channels (2 of 4)
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When we write a channel in the Stinespring representation as [mathjaxinline]\mathcal{N}(\rho ) = {\rm Tr}_ E V\rho V^\dagger[/mathjaxinline], the outcome is the same if we perform a further isometry on system [mathjaxinline]E[/mathjaxinline] before tracing it out. What effect does this have on the Kraus operators? We will see that in general they change, and this will lead us to the conclusion that the Kraus representation of a channel is not unique. </p>
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Suppose [mathjaxinline]\dim E = d[/mathjaxinline]. Let [mathjaxinline]W[/mathjaxinline] be an isometry from [mathjaxinline]L(\mathbb {C}^{d})[/mathjaxinline] to [mathjaxinline]L(\mathbb {C}^{d'})[/mathjaxinline] mapping system [mathjaxinline]E[/mathjaxinline] to a [mathjaxinline]d'[/mathjaxinline] dimensional system [mathjaxinline]E'[/mathjaxinline] (with orthonormal basis [mathjaxinline]\{ |{1'}\rangle , \ldots , |{d'}\rangle \}[/mathjaxinline], for [mathjaxinline]d'\geq d[/mathjaxinline]). If we apply this after [mathjaxinline]V[/mathjaxinline] then we obtain the composed isometry [mathjaxinline](I_ B \otimes W)V[/mathjaxinline]. This yields a new set of Kraus operators [mathjaxinline]\{ Y_{j}\} _{j=1}^{d'}[/mathjaxinline] given by </p>
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<h3 class="hd hd-3 problem-header" id="ps1b_q_channels3-problem-title" aria-describedby="block-v1:MITx+8.371.1x+2T2018+type@problem+block@ps1b_q_channels3-problem-progress" tabindex="-1">
QC3. Quantum channels (3 of 4)
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Define the <em>Hilbert-Schmidt</em> inner product between two matrices to be </p>
<table id="a0000000013" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
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<td class="equation" style="width:80%; border:none">[mathjax]\langle X,Y\rangle := {\rm Tr}[X^\dagger Y].[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.6)</td>
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The adjoint of a superoperator [mathjaxinline]T \in L(L(A),L(B))[/mathjaxinline] with respect to this inner product is defined by the expression </p>
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<td class="equation" style="width:80%; border:none">[mathjax]\langle X,T(Y)\rangle = \langle T^\dagger (X), Y\rangle .[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.7)</td>
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This is also known as the Heisenberg picture for quantum operations. </p>
<p>
Let [mathjaxinline]T(\rho ) = \sum _{i \in [k]} A_ i \rho A_ i^\dagger[/mathjaxinline], where [mathjaxinline][k] = \{ 1,\ldots ,k\}[/mathjaxinline]. We would like to find the Kraus operators of [mathjaxinline]T^\dagger[/mathjaxinline]. <span><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps1b_q_channels3_2_1" class="capa_inputtype">
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QC4. Quantum channels (4 of 4)
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<p>
[mathjaxinline]{\rm Tr}_ C[/mathjaxinline] is a quantum channel from [mathjaxinline]B \otimes C[/mathjaxinline] to [mathjaxinline]B[/mathjaxinline]. Therefore [mathjaxinline]{\rm Tr}^\dagger _ C[/mathjaxinline] is a quantum channel from [mathjaxinline]V[/mathjaxinline] to [mathjaxinline]W[/mathjaxinline]. What spaces are [mathjaxinline]V[/mathjaxinline] and [mathjaxinline]W[/mathjaxinline]? </p>
<p>
Let [mathjaxinline]X_ B[/mathjaxinline] be an arbitrary matrix in [mathjaxinline]L(B)[/mathjaxinline]. Write an expression for [mathjaxinline]{\rm Tr}_ C^\dagger [X_ B][/mathjaxinline]. </p>
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<h2 class="hd hd-2 unit-title">Introduction to amplitude damping</h2>
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<h3 class="hd hd-3 problem-header" id="s12-wk1-adamp-problem-title" aria-describedby="block-v1:MITx+8.371.1x+2T2018+type@problem+block@s12-wk1-adamp-problem-progress" tabindex="-1">
Introduction to amplitude damping
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Amplitude damping is an important process in real physical systems; it models spontaneous emission, inelastic scattering, thermalization of spins to the lattice, and many other microscopic processes where energy is exchanged between the system and environment. In this problem, we study the properties of amplitude damping acting on a single qubit, using the operator sum representation of the process. </p>
<p>
The ampltiude damping channel for a single qubit is described by [mathjaxinline]{\cal E}(\rho ) = \sum _ k E_ k \rho E_ k^\dagger[/mathjaxinline], where the Kraus operators are </p>
<table id="a0000000015" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
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<td class="equation" style="width:80%; border:none">[mathjax]E_0 = \left[ \begin{array}{cc} {1} &amp; {0}\\ {0} &amp; \sqrt{1-g} \end{array} \right] ~ ~ ~ ~ ~ ~ ~ ~ E_1 = \left[ \begin{array}{cc} {0} &amp; {\sqrt{g}}\\ {0} &amp; {0} \end{array} \right] \, .[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.8)</td>
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<p>
Let [mathjaxinline]g= 1-e^{-t/T_1}[/mathjaxinline], where [mathjaxinline]t[/mathjaxinline] is time and [mathjaxinline]T_1[/mathjaxinline] is the amplitude-damping time constant. </p>
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<p>
Let [mathjaxinline]|\psi \rangle = (|0\rangle + |1\rangle )/\sqrt{2}[/mathjaxinline], and [mathjaxinline]\rho = {\cal E}(|\psi \rangle \langle \psi |) = \sum _ k E_ k |\psi \rangle \langle \psi | E_ k^\dagger[/mathjaxinline] be the density matrix obtained for the qubit after amplitude damping. Compute the fidelity of [mathjaxinline]\rho[/mathjaxinline] with respect to [mathjaxinline]|\psi \rangle[/mathjaxinline], [mathjaxinline]F(t) = F(|\psi \rangle ,\rho ) = \sqrt{\langle \psi |\rho |\psi \rangle }[/mathjaxinline]. </p>
<p>
You may wish to plot this as a function of [mathjaxinline]t[/mathjaxinline] or [mathjaxinline]g[/mathjaxinline], for your own appreciation of what amplitude damping can do to a qubit superposition state. </p>
<p>
Please give your answer as a function of [mathjaxinline]g[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]F(g) =[/mathjaxinline]</p>
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