Nathaniel: requires CVX

2016-01-12T19:05:32Z

requires CVX

Nathaniel: Created page with "{{Function |name=RobustnessCoherence |desc=Computes the robustness of coherence of a quantum state |rel=L1NormCoherence
RelEntCoherence
TraceDistanceCohere..."

2016-01-12T15:51:15Z

Created page with "{{Function |name=RobustnessCoherence |desc=Computes the robustness of coherence of a quantum state |rel=L1NormCoherence RelEntCoherence TraceDistanceCohere..."

New page

{{Function
|name=RobustnessCoherence
|desc=Computes the robustness of coherence of a quantum state
|rel=[[L1NormCoherence]] [[RelEntCoherence]] [[TraceDistanceCoherence]]
|cat=[[List of functions#Coherence_and_incoherence|Coherence and incoherence]]
|upd=January 12, 2016
|cvx=yes (convex)}}
<tt>'''RobustnessCoherence'''</tt> is a [[List of functions|function]] that computes the robustness of coherence of a quantum state $\rho$, defined as follows <ref>C. Napoli, T. R. Bromley, M. Cianciaruso, M. Piani, N. Johnston, G. Adesso. ''Robustness of coherence: An operational and observable measure of quantum coherence''. Preprint, 2016.</ref><ref>C. Napoli, T. R. Bromley, M. Cianciaruso, M. Piani, N. Johnston, G. Adesso. ''Robustness of asymmetry and coherence of quantum states''. Preprint, 2016.</ref>:

: <math>C_{R}(\rho) := \min_{\tau}\left\{s \geq 0 \, \Big| \, \frac{\rho + s\tau}{1+s} \in \mathcal{I}\right\},</math>

where the minimization is over all density matrices $\tau$ and $\mathcal{I}$ is the set of incoherent density matrices (i.e., the set of density matrices that are diagonal in the computational basis).

==Syntax==
* <tt>ROC = RobustnessCoherence(RHO)</tt>

==Argument descriptions==
* <tt>RHO</tt>: A state (either pure or mixed) to have its robustness of coherence computed.

==Examples==
===Pure states===
If $|v\rangle$ is a pure state then its robustness of coherence and ℓ1-norm of coherence coincide:
<syntaxhighlight>
>> v = RandomStateVector(4);
>> L1NormCoherence(v)

ans =

2.5954

>> RobustnessCoherence(v)

ans =

2.5954
</syntaxhighlight>

===Can be used within CVX===
The robustness 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 robustness of coherence or use the robustness of coherence in constraints of CVX optimization problems. For example, the following code minimizes the robustness of coherence over all density matrices that are within a trace distance of $1/2$ from the maximally coherent state $|v\rangle = (1,1,1,1,1)/\sqrt{5}$:
<syntaxhighlight>
>> d = 5;
>> v = ones(d,1)/sqrt(d); % this is a maximally coherent state
>> cvx_begin sdp quiet
variable rho(5,5) hermitian;

minimize RobustnessCoherence(rho)

subject to
TraceNorm(rho - v*v') <= 0.5;
% the next two constraints force rho to be a density matrix
rho >= 0;
trace(rho) == 1;
cvx_end
cvx_optval

cvx_optval =

2.7500

>> rho

rho =

0.2000 0.1375 0.1375 0.1375 0.1375
0.1375 0.2000 0.1375 0.1375 0.1375
0.1375 0.1375 0.2000 0.1375 0.1375
0.1375 0.1375 0.1375 0.2000 0.1375
0.1375 0.1375 0.1375 0.1375 0.2000
</syntaxhighlight>

{{SourceCode|name=RobustnessCoherence}}

==References==
<references />

@@ Line 2: / Line 2: @@
 |name=RobustnessCoherence
 |desc=Computes the robustness of coherence of a quantum state
 |rel=[[L1NormCoherence]]<br />[[RelEntCoherence]]<br />[[TraceDistanceCoherence]]
 |cat=[[List of functions#Coherence_and_incoherence|Coherence and incoherence]]

RobustnessCoherence - Revision history

Nathaniel: requires CVX

Nathaniel: Created page with "{{Function |name=RobustnessCoherence |desc=Computes the robustness of coherence of a quantum state |rel=L1NormCoherenceRelEntCoherenceTraceDistanceCohere..."

Nathaniel: Created page with "{{Function |name=RobustnessCoherence |desc=Computes the robustness of coherence of a quantum state |rel=L1NormCoherence
RelEntCoherence
TraceDistanceCohere..."