RelEntCoherence

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RelEntCoherence
Computes the relative entropy of coherence of a quantum state

Other toolboxes required none
Related functions L1NormCoherence
RobustnessCoherence
TraceDistanceCoherence
Function category Coherence and incoherence
Usable within CVX? no

RelEntCoherence is a function that computes the relative entropy of coherence of a quantum state $\rho$, defined as follows:

\[C_{r}(\rho) := -\mathrm{Tr}\big(\rho(\log_2(\rho) - \log_2(\rho_{\text{diag}}))\big),\]

where $\rho_{\text{diag}}$ is the matrix that has the same diagonal as $\rho$ but is zero elsewhere.

Syntax

  • REC = RelEntCoherence(RHO)

Argument descriptions

  • RHO: A state (either pure or mixed) to have its relative entropy of coherence computed.

Examples

Maximally coherent states

The largest possible value of the relative entropy of coherence on $d$-dimensional states is $\log_2(d)$, and is attained exactly by the "maximally coherent states": pure states whose entries all have the same absolute value.

>> d = 5;
>> v = ones(d,1)/sqrt(d); % this is a maximally coherent state
>> RelEntCoherence(v)
 
ans =
 
    2.3219
 
>> log(5)/log(2)
 
ans =
 
    2.3219

Source code

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

  1. %%  RelEntCoherence    Computes the relative entropy of coherence of a quantum state
  2. %   This function has one required argument:
  3. %     RHO: a density matrix or pure state vector
  4. %
  5. %   REC = RelEntCoherence(RHO) is the relative entropy of coherence, which
  6. %   equals S(diag(RHO)) - S(RHO), where S(.) is the von Neumann entropy.
  7. %
  8. %   URL: http://www.qetlab.com/RelEntCoherence
  9.  
  10. %   requires: Entropy.m, pure_to_mixed.m
  11. %   author: Nathaniel Johnston (nathaniel@njohnston.ca)
  12. %   package: QETLAB
  13. %   last updated: January 12, 2016
  14.  
  15. function REC = RelEntCoherence(rho)
  16.  
  17. rho = pure_to_mixed(rho); % Let the user specify rho as either a density matrix or a pure state vector
  18.  
  19. REC = Entropy(diag(diag(rho))) - Entropy(rho);