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<g id="a_node12"><a xlink:title="Extreme Points and Even Perturbations">
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<g id="a_node16"><a xlink:title="General Characterization">
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<text text-anchor="middle" x="283" y="-691.1" font-family="Arial" font-size="8.00">General Characterization</text>
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<g id="a_node20"><a xlink:title="Finite Number of Constraints">
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<text text-anchor="middle" x="283" y="-556.1" font-family="Arial" font-size="8.00">Finite Number of Constraints</text>
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<text text-anchor="middle" x="283" y="-421.1" font-family="Arial" font-size="8.00">Correlation Matrices and Hadamard Rank Inequality</text>
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<text text-anchor="middle" x="283" y="-178.1" font-family="Arial" font-size="8.00">Quantum Mechanical States</text>
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<text text-anchor="middle" x="283" y="-70.1" font-family="Arial" font-size="8.00">Conclusion</text>
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<g id="a_node3"><a xlink:title="Characterize extreme points of convex set C.">
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<text text-anchor="middle" x="549" y="-1366.1" font-family="Arial" font-size="8.00">Characterize extreme points of convex set C.</text>
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<text text-anchor="middle" x="549" y="-1312.1" font-family="Arial" font-size="8.00">C is set of positive linear operators on Hilbert space.</text>
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<text text-anchor="middle" x="549" y="-1258.1" font-family="Arial" font-size="8.00">Linear constraints are given by TrPA = c for every α.</text>
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<text text-anchor="middle" x="549" y="-1204.1" font-family="Arial" font-size="8.00">P is set of positive linear operators on Hilbert space.</text>
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<text text-anchor="middle" x="549" y="-1150.1" font-family="Arial" font-size="8.00">S is set of bounded linear operators on Hilbert space.</text>
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<g id="a_node9"><a xlink:title="TrA is trace of operator A.">
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<text text-anchor="middle" x="549" y="-1096.1" font-family="Arial" font-size="8.00">TrA is trace of operator A.</text>
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<text text-anchor="middle" x="549" y="-1042.1" font-family="Arial" font-size="8.00">R(A) is range of operator A.</text>
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<g id="a_node11"><a xlink:title="N(A) is null space of operator A.">
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<text text-anchor="middle" x="549" y="-988.1" font-family="Arial" font-size="8.00">N(A) is null space of operator A.</text>
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<g id="a_node13"><a xlink:title="P is extreme point if no even perturbation exists.">
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<title>139861107210240</title>
<g id="a_node14"><a xlink:title="Even perturbation is H such that P + H and P &#45; H are in C.">
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<text text-anchor="middle" x="549" y="-880.1" font-family="Arial" font-size="8.00">Even perturbation is H such that P + H and P &#45; H are in C.</text>
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<!-- 139861107209920&#45;&gt;139861107210240 -->
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<title>139861107209920&#45;&gt;139861107210240</title>
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<title>139861107201408</title>
<g id="a_node15"><a xlink:title="Theorem 3.1: Characterization of even perturbations.">
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<text text-anchor="middle" x="549" y="-826.1" font-family="Arial" font-size="8.00">Theorem 3.1: Characterization of even perturbations.</text>
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<!-- 139861107209920&#45;&gt;139861107201408 -->
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<title>139861107209920&#45;&gt;139861107201408</title>
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<title>139861107202688</title>
<g id="a_node17"><a xlink:title="Theorem 3.3: Three equivalent conditions for P to be extreme point.">
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<text text-anchor="middle" x="549" y="-772.1" font-family="Arial" font-size="8.00">Theorem 3.3: Three equivalent conditions for P to be extreme point.</text>
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<title>139861107209984&#45;&gt;139861107202688</title>
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<title>139861108090816</title>
<g id="a_node18"><a xlink:title="Conditions involve existence of nonzero self&#45;adjoint X.">
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<text text-anchor="middle" x="549" y="-718.1" font-family="Arial" font-size="8.00">Conditions involve existence of nonzero self&#45;adjoint X.</text>
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<!-- 139861107209984&#45;&gt;139861108090816 -->
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<title>139861107209984&#45;&gt;139861108090816</title>
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<title>139861108245632</title>
<g id="a_node19"><a xlink:title="X must satisfy certain properties.">
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<text text-anchor="middle" x="549" y="-664.1" font-family="Arial" font-size="8.00">X must satisfy certain properties.</text>
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<!-- 139861107209984&#45;&gt;139861108245632 -->
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<title>139861107209984&#45;&gt;139861108245632</title>
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<!-- 139861106515136 -->
<g id="node21" class="node">
<title>139861106515136</title>
<g id="a_node21"><a xlink:title="Theorem 3.5: Rank characterization of extreme points.">
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<text text-anchor="middle" x="549" y="-610.1" font-family="Arial" font-size="8.00">Theorem 3.5: Rank characterization of extreme points.</text>
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<!-- 139861106997632&#45;&gt;139861106515136 -->
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<title>139861106997632&#45;&gt;139861106515136</title>
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<!-- 139861106727552 -->
<g id="node22" class="node">
<title>139861106727552</title>
<g id="a_node22"><a xlink:title="Rank of P is finite if and only if P is extreme point.">
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<text text-anchor="middle" x="549" y="-556.1" font-family="Arial" font-size="8.00">Rank of P is finite if and only if P is extreme point.</text>
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<!-- 139861106997632&#45;&gt;139861106727552 -->
<g id="edge21" class="edge">
<title>139861106997632&#45;&gt;139861106727552</title>
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<g id="node23" class="node">
<title>139861107205056</title>
<g id="a_node23"><a xlink:title="Upper bound on rank of extreme points.">
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<text text-anchor="middle" x="549" y="-502.1" font-family="Arial" font-size="8.00">Upper bound on rank of extreme points.</text>
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<!-- 139861106997632&#45;&gt;139861107205056 -->
<g id="edge22" class="edge">
<title>139861106997632&#45;&gt;139861107205056</title>
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<!-- 139861106521856 -->
<g id="node25" class="node">
<title>139861106521856</title>
<g id="a_node25"><a xlink:title="Correlation matrix is P such that Pii = 1 for all i.">
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<text text-anchor="middle" x="549" y="-448.1" font-family="Arial" font-size="8.00">Correlation matrix is P such that Pii = 1 for all i.</text>
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<!-- 139861107206272&#45;&gt;139861106521856 -->
<g id="edge24" class="edge">
<title>139861107206272&#45;&gt;139861106521856</title>
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<!-- 139861107198976 -->
<g id="node26" class="node">
<title>139861107198976</title>
<g id="a_node26"><a xlink:title="Theorem 4.1: Characterization of extreme correlation matrices.">
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<text text-anchor="middle" x="549" y="-394.1" font-family="Arial" font-size="8.00">Theorem 4.1: Characterization of extreme correlation matrices.</text>
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<!-- 139861107206272&#45;&gt;139861107198976 -->
<g id="edge25" class="edge">
<title>139861107206272&#45;&gt;139861107198976</title>
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<!-- 139861108880384 -->
<g id="node27" class="node">
<title>139861108880384</title>
<g id="a_node27"><a xlink:title="Hadamard rank inequality is equality if and only if P is extreme.">
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<text text-anchor="middle" x="549" y="-340.1" font-family="Arial" font-size="8.00">Hadamard rank inequality is equality if and only if P is extreme.</text>
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<!-- 139861107206272&#45;&gt;139861108880384 -->
<g id="edge26" class="edge">
<title>139861107206272&#45;&gt;139861108880384</title>
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<!-- 139861108084864 -->
<g id="node29" class="node">
<title>139861108084864</title>
<g id="a_node29"><a xlink:title="Sensitivity analysis of PCA with incomplete data.">
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<text text-anchor="middle" x="549" y="-286.1" font-family="Arial" font-size="8.00">Sensitivity analysis of PCA with incomplete data.</text>
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<!-- 139861107805120&#45;&gt;139861108084864 -->
<g id="edge28" class="edge">
<title>139861107805120&#45;&gt;139861108084864</title>
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<!-- 139861107809664 -->
<g id="node30" class="node">
<title>139861107809664</title>
<g id="a_node30"><a xlink:title="Optimization on quantum mechanical states.">
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<text text-anchor="middle" x="549" y="-232.1" font-family="Arial" font-size="8.00">Optimization on quantum mechanical states.</text>
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<!-- 139861107805120&#45;&gt;139861107809664 -->
<g id="edge29" class="edge">
<title>139861107805120&#45;&gt;139861107809664</title>
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<!-- 139861107582080 -->
<g id="node32" class="node">
<title>139861107582080</title>
<g id="a_node32"><a xlink:title="State of particle is represented by density operator P.">
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<text text-anchor="middle" x="549" y="-178.1" font-family="Arial" font-size="8.00">State of particle is represented by density operator P.</text>
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<!-- 139861107806272&#45;&gt;139861107582080 -->
<g id="edge31" class="edge">
<title>139861107806272&#45;&gt;139861107582080</title>
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<!-- 139861112237120 -->
<g id="node33" class="node">
<title>139861112237120</title>
<g id="a_node33"><a xlink:title="P must satisfy certain properties.">
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<text text-anchor="middle" x="549" y="-124.1" font-family="Arial" font-size="8.00">P must satisfy certain properties.</text>
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<!-- 139861107806272&#45;&gt;139861112237120 -->
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<title>139861107806272&#45;&gt;139861112237120</title>
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<!-- 139861108095808 -->
<g id="node35" class="node">
<title>139861108095808</title>
<g id="a_node35"><a xlink:title="Summary of main results.">
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<text text-anchor="middle" x="549" y="-70.1" font-family="Arial" font-size="8.00">Summary of main results.</text>
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<!-- 139861107815808&#45;&gt;139861108095808 -->
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<title>139861107815808&#45;&gt;139861108095808</title>
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<!-- 139861110545792 -->
<g id="node36" class="node">
<title>139861110545792</title>
<g id="a_node36"><a xlink:title="Future research directions.">
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<text text-anchor="middle" x="549" y="-16.1" font-family="Arial" font-size="8.00">Future research directions.</text>
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<!-- 139861107815808&#45;&gt;139861110545792 -->
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<title>139861107815808&#45;&gt;139861110545792</title>
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