# L1NormCoherence

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 Other toolboxes required L1NormCoherence Computes the ℓ1-norm of coherence of a quantum state none RelEntCoherenceRobustnessCoherenceTraceDistanceCoherence Coherence and incoherence yes (convex)

L1NormCoherence is a function that computes the ℓ1-norm of coherence of a quantum state $\rho$, defined as follows:

$C_{\ell_1}(\rho) := \sum_{i \neq j} |\rho_{ij}|,$

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

## Syntax

• L1C = L1NormCoherence(RHO)

## Argument descriptions

• RHO: 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 $|v\rangle\langle v|$:

>> 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

### 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.

>> v = ones(3,1)/sqrt(3); % this is a maximally coherent state
>> L1NormCoherence(v)

ans =

2.0000

### 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 $|v\rangle = (1,1,1,1,1)/\sqrt{5}$:

>> 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

## Source code

Click on "expand" to the right to view the MATLAB source code for this function.

1. %%  L1NormCoherence    Computes the l1-norm of coherence of a quantum state
2. %   This function has one required argument:
3. %     RHO: a density matrix or pure state vector
4. %
5. %   L1C = L1NormCoherence(RHO) is the l1-norm of coherence, which is the
6. %   sum of the absolute values of the off-diagonal entries of the density
7. %   matrix RHO (in the standard basis).
8. %
9. %   URL: http://www.qetlab.com/L1NormCoherence
10. 
11. %   requires: pure_to_mixed.m
12. %   author: Nathaniel Johnston (nathaniel@njohnston.ca)
13. %   package: QETLAB
14. %   last updated: January 12, 2016
15. 
16. function L1C = L1NormCoherence(rho)
17. 
18. rho = pure_to_mixed(rho); % Let the user specify rho as either a density matrix or a pure state vector
19. 
20. L1C = sum(sum(abs(rho))) - trace(rho);