RobustnessCoherence

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RobustnessCoherence
Computes the robustness of coherence of a quantum state

Other toolboxes required CVX
Related functions L1NormCoherence
RelEntCoherence
TraceDistanceCoherence
Function category Coherence and incoherence
Usable within CVX? yes (convex)

RobustnessCoherence is a function that computes the robustness of coherence of a quantum state $\rho$, defined as follows [1][2]:

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

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

  • ROC = RobustnessCoherence(RHO)

Argument descriptions

  • RHO: 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:

>> v = RandomStateVector(4);
>> L1NormCoherence(v)

ans =

    2.5954

>> RobustnessCoherence(v)

ans =

    2.5954

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

>> 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

Source code

Click here to view this function's source code on github.

References

  1. 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.
  2. C. Napoli, T. R. Bromley, M. Cianciaruso, M. Piani, N. Johnston, G. Adesso. Robustness of asymmetry and coherence of quantum states. Preprint, 2016.