https://qetlab.com/wiki/index.php?title=L1NormCoherence&feed=atom&action=historyL1NormCoherence - Revision history2024-03-29T06:57:19ZRevision history for this page on the wikiMediaWiki 1.35.3https://qetlab.com/wiki/index.php?title=L1NormCoherence&diff=871&oldid=prevNathaniel: /* Can be used within CVX */ typo2016-02-15T14:11:56Z<p><span dir="auto"><span class="autocomment">Can be used within CVX: </span> typo</span></p>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Can be used within CVX===</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Can be used within CVX===</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The ℓ<sub>1</sub>-norm of coherence is a convex function and can be used in the same way as any other convex function within CVX. Thus you can minimize the ℓ<sub>1</sub>-norm of coherence or use the ℓ<sub>1</sub>-norm of coherence in constraints of CVX optimization problems. For example, the following code minimizes the ℓ<sub>1</sub>-norm of coherence over all density matrices that are within a trace distance of $1/2$ from the maximally coherent state $|v\rangle = (1,1,1,1,1)/sqrt{5}$:</div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The ℓ<sub>1</sub>-norm of coherence is a convex function and can be used in the same way as any other convex function within CVX. Thus you can minimize the ℓ<sub>1</sub>-norm of coherence or use the ℓ<sub>1</sub>-norm of coherence in constraints of CVX optimization problems. For example, the following code minimizes the ℓ<sub>1</sub>-norm of coherence over all density matrices that are within a trace distance of $1/2$ from the maximally coherent state $|v\rangle = (1,1,1,1,1)/<ins class="diffchange diffchange-inline">\</ins>sqrt{5}$:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><syntaxhighlight></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><syntaxhighlight></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>>> d = 5;</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>>> d = 5;</div></td></tr>
</table>Nathanielhttps://qetlab.com/wiki/index.php?title=L1NormCoherence&diff=852&oldid=prevNathaniel: Fixed category2016-01-12T15:43:21Z<p>Fixed category</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 15:43, 12 January 2016</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|rel=[[RelEntCoherence]]<br />[[RobustnessCoherence]]<br />[[TraceDistanceCoherence]]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|rel=[[RelEntCoherence]]<br />[[RobustnessCoherence]]<br />[[TraceDistanceCoherence]]</div></td></tr>
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</table>Nathanielhttps://qetlab.com/wiki/index.php?title=L1NormCoherence&diff=848&oldid=prevNathaniel: Created page with "{{Function |name=L1NormCoherence |desc=Computes the ℓ<sub>1</sub>-norm of coherence of a quantum state |rel=RelEntCoherence<br />RobustnessCoherence<br />TraceDist..."2016-01-12T13:43:41Z<p>Created page with "{{Function |name=L1NormCoherence |desc=Computes the ℓ<sub>1</sub>-norm of coherence of a quantum state |rel=<a href="/RelEntCoherence" title="RelEntCoherence">RelEntCoherence</a><br /><a href="/RobustnessCoherence" title="RobustnessCoherence">RobustnessCoherence</a><br />TraceDist..."</p>
<p><b>New page</b></p><div>{{Function<br />
|name=L1NormCoherence<br />
|desc=Computes the ℓ<sub>1</sub>-norm of coherence of a quantum state<br />
|rel=[[RelEntCoherence]]<br />[[RobustnessCoherence]]<br />[[TraceDistanceCoherence]]<br />
|cat=[[List of functions#Norms|Norms]]<br />
|upd=January 12, 2016<br />
|cvx=yes (convex)}}<br />
<tt>'''L1NormCoherence'''</tt> is a [[List of functions|function]] that computes the ℓ<sub>1</sub>-norm of coherence of a quantum state $\rho$, defined as follows:<br />
<br />
: <math>C_{\ell_1}(\rho) := \sum_{i \neq j} |\rho_{ij}|,</math><br />
<br />
where $\rho_{ij}$ is the $(i,j)$-entry of $\rho$ in the standard basis.<br />
<br />
==Syntax==<br />
* <tt>L1C = L1NormCoherence(RHO)</tt><br />
<br />
==Argument descriptions==<br />
* <tt>RHO</tt>: A state (either pure or mixed) to have its ℓ<sub>1</sub>-norm of coherence computed.<br />
<br />
==Examples==<br />
===Pure states or mixed states===<br />
If $|v\rangle$ is a pure state then its ℓ<sub>1</sub>-norm of coherence is computed from the density matrix $|v\rangle\langle v|$:<br />
<syntaxhighlight><br />
>> v = RandomStateVector(3)<br />
<br />
v =<br />
<br />
0.6233 + 0.1633i<br />
-0.3038 - 0.0142i<br />
0.6830 + 0.1609i<br />
<br />
>> L1NormCoherence(v)<br />
<br />
ans =<br />
<br />
1.7229<br />
<br />
>> L1NormCoherence(v*v')<br />
<br />
ans =<br />
<br />
1.7229<br />
</syntaxhighlight><br />
<br />
===Maximally coherent states===<br />
The largest possible value of the ℓ<sub>1</sub>-norm of coherence on $d$-dimensional states is $d-1$, and is attained exactly by the "maximally coherent states": pure states whose entries all have the same absolute value.<br />
<syntaxhighlight><br />
>> v = ones(3,1)/sqrt(3); % this is a maximally coherent state<br />
>> L1NormCoherence(v)<br />
<br />
ans =<br />
<br />
2.0000<br />
</syntaxhighlight><br />
<br />
===Can be used within CVX===<br />
The ℓ<sub>1</sub>-norm of coherence is a convex function and can be used in the same way as any other convex function within CVX. Thus you can minimize the ℓ<sub>1</sub>-norm of coherence or use the ℓ<sub>1</sub>-norm of coherence in constraints of CVX optimization problems. For example, the following code minimizes the ℓ<sub>1</sub>-norm of coherence over all density matrices that are within a trace distance of $1/2$ from the maximally coherent state $|v\rangle = (1,1,1,1,1)/sqrt{5}$:<br />
<syntaxhighlight><br />
>> d = 5;<br />
>> v = ones(d,1)/sqrt(d); % this is a maximally coherent state<br />
>> cvx_begin sdp quiet<br />
variable rho(5,5) hermitian;<br />
<br />
minimize L1NormCoherence(rho)<br />
<br />
subject to<br />
TraceNorm(rho - v*v') <= 0.5;<br />
% the next two constraints force rho to be a density matrix<br />
rho >= 0;<br />
trace(rho) == 1;<br />
cvx_end<br />
cvx_optval<br />
<br />
cvx_optval =<br />
<br />
2.7500<br />
<br />
>> rho<br />
<br />
rho =<br />
<br />
0.2000 0.1375 0.1375 0.1375 0.1375<br />
0.1375 0.2000 0.1375 0.1375 0.1375<br />
0.1375 0.1375 0.2000 0.1375 0.1375<br />
0.1375 0.1375 0.1375 0.2000 0.1375<br />
0.1375 0.1375 0.1375 0.1375 0.2000<br />
</syntaxhighlight><br />
<br />
{{SourceCode|name=L1NormCoherence}}</div>Nathaniel