L1NormCoherence - Revision history

Nathaniel: /* Can be used within CVX */ typo

2016-02-15T14:11:56Z

Can be used within CVX: typo

Nathaniel: Fixed category

2016-01-12T15:43:21Z

Fixed category

Nathaniel: Created page with "{{Function |name=L1NormCoherence |desc=Computes the ℓ₁-norm of coherence of a quantum state |rel=RelEntCoherence
RobustnessCoherence
TraceDist..."

2016-01-12T13:43:41Z

Created page with "{{Function |name=L1NormCoherence |desc=Computes the ℓ1-norm of coherence of a quantum state |rel=RelEntCoherence RobustnessCoherence TraceDist..."

New page

{{Function
|name=L1NormCoherence
|desc=Computes the ℓ1-norm of coherence of a quantum state
|rel=[[RelEntCoherence]] [[RobustnessCoherence]] [[TraceDistanceCoherence]]
|cat=[[List of functions#Norms|Norms]]
|upd=January 12, 2016
|cvx=yes (convex)}}
<tt>'''L1NormCoherence'''</tt> is a [[List of functions|function]] that computes the ℓ1-norm of coherence of a quantum state $\rho$, defined as follows:

: <math>C_{\ell_1}(\rho) := \sum_{i \neq j} |\rho_{ij}|,</math>

where $\rho_{ij}$ is the $(i,j)$-entry of $\rho$ in the standard basis.

==Syntax==
* <tt>L1C = L1NormCoherence(RHO)</tt>

==Argument descriptions==
* <tt>RHO</tt>: A state (either pure or mixed) to have its ℓ1-norm of coherence computed.

==Examples==
===Pure states or mixed states===
If $|v\rangle$ is a pure state then its ℓ1-norm of coherence is computed from the density matrix $|v\rangle\langle v|$:
<syntaxhighlight>
>> v = RandomStateVector(3)

v =

0.6233 + 0.1633i
-0.3038 - 0.0142i
0.6830 + 0.1609i

>> L1NormCoherence(v)

ans =

1.7229

>> L1NormCoherence(v*v')

ans =

1.7229
</syntaxhighlight>

===Maximally coherent states===
The largest possible value of the ℓ1-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.
<syntaxhighlight>
>> v = ones(3,1)/sqrt(3); % this is a maximally coherent state
>> L1NormCoherence(v)

ans =

2.0000
</syntaxhighlight>

===Can be used within CVX===
The ℓ1-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 ℓ1-norm of coherence or use the ℓ1-norm of coherence in constraints of CVX optimization problems. For example, the following code minimizes the ℓ1-norm of coherence over all density matrices that are within a trace distance of $1/2$ from the maximally coherent state $|v\rangle = (1,1,1,1,1)/sqrt{5}$:
<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 L1NormCoherence(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=L1NormCoherence}}

@@ Line 3: / Line 3: @@
 |desc=Computes the ℓ<sub>1</sub>-norm of coherence of a quantum state
 |rel=[[RelEntCoherence]]<br />[[RobustnessCoherence]]<br />[[TraceDistanceCoherence]]
-|cat=[[List of functions#Norms|Norms]]
+|cat=[[List of functions#Coherence_and_incoherence|Coherence and incoherence]]
 |upd=January 12, 2016
 |cvx=yes (convex)}}

@@ Line 55: / Line 55: @@
 ===Can be used within CVX===
-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 $|v\rangle = (1,1,1,1,1)/sqrt{5}$:
+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 $|v\rangle = (1,1,1,1,1)/\sqrt{5}$:
 <syntaxhighlight>
 >> d = 5;

L1NormCoherence - Revision history

Nathaniel: /* Can be used within CVX */ typo

Nathaniel: Fixed category

Nathaniel: Created page with "{{Function |name=L1NormCoherence |desc=Computes the ℓ1-norm of coherence of a quantum state |rel=RelEntCoherenceRobustnessCoherenceTraceDist..."

Nathaniel: Created page with "{{Function |name=L1NormCoherence |desc=Computes the ℓ₁-norm of coherence of a quantum state |rel=RelEntCoherence
RobustnessCoherence
TraceDist..."