# ChannelDistinguishability

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ChannelDistinguishability | |

Computes the maximum probability of distinguishing two quantum channels | |

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

Related functions | Distinguishability LocalDistinguishability |

Function category | Distinguishing objects |

` ChannelDistinguishability` is a function that computes the maximum probability of distinguishing two quantum channels. That is, this function computes the maximum probability of winning the following game: You are given a complete description of two quantum channels $\Phi$ and $\Psi$, and then are given one of those two channels, and asked to determine which channel was given to you (by supplying some input state to the channel and then measuring the output).

## Syntax

`DIST = ChannelDistinguishability(PHI,PSI)``DIST = ChannelDistinguishability(PHI,PSI,P)``DIST = ChannelDistinguishability(PHI,PSI,P,DIM)`

## Argument descriptions

`PHI,PSI`: The quantum channels to be distinguished. They can either be input as Choi matrices or as cells of Kraus operators.`P`(optional, default`[1/2, 1/2]`): A vector that specifies that`PHI`and`PSI`are chosen with probability`P(1)`and`P(2)`, respectively.`DIM`(optional, by default tries to guess the input and output dimensions): A 1-by-2 vector containing the input and output dimensions of`PHI`and`PSI`.`DIM`is required if and only if both`PHI`and`PSI`are provided as Choi matrices and the input and output dimensions are different.

## Examples

### Perfectly distinguishable channels

The following code demonstrates that the simple example of two channels that can be perfectly distinguished from ^{[1]} can indeed be perfectly distinguished:

```
>> n = 4;
>> Phi = SymmetricProjection(n)*2/(n+1);
>> Psi = AntisymmetricProjection(n)*2/(n-1);
>> ChannelDistinguishability(Phi,Psi)
ans =
1.0000
```

## Source code

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

## References

- ↑ John Watrous. Lecture 20: Channel distinguishability and the completely bounded trace norm,
*Theory of Quantum Information Lecture Notes*, 2011.