# TraceDistanceCoherence

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TraceDistanceCoherence | |

Computes the trace distance of coherence of a quantum state | |

Other toolboxes required | CVX |
---|---|

Related functions | L1NormCoherence RelEntCoherence RobustnessCoherence |

Function category | Coherence and incoherence |

Usable within CVX? | no |

` TraceDistanceCoherence` is a function that computes the trace distance of coherence of a quantum state $\rho$, defined as follows:

\[C_{\text{tr}}(\rho) := \min_{D \in \mathcal{I}}\big\{ \|\rho - D\|_{\text{tr}}\big\},\]

where $\mathcal{I}$ is the set of incoherent quantum states (i.e., the set of density matrices that are diagonal in the standard basis).

## Syntax

`TDC = TraceDistanceCoherence(RHO)``[TDC,D] = TraceDistanceCoherence(RHO)`

## Argument descriptions

### Input arguments

`RHO`: A state (either pure or mixed) to have its trace distance of coherence computed.

### Output arguments

`TDC`: The trace distance of coherence of`RHO`.`D`: A vector such that`diag(D)`is the closest incoherent state to`RHO`.

## Examples

### Maximally coherent states

The largest possible value of the trace distance of coherence on $d$-dimensional states is $2 - 2/d$, 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
>> TraceDistanceCoherence(v)
ans =
1.6000
>> 2 - 2/d
ans =
1.6000
```

## Source code

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