# Distinguishability

Distinguishability | |

Computes the maximum probability of distinguishing quantum states | |

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

Related functions | ChannelDistinguishability LocalDistinguishability |

Function category | Distinguishing objects |

` Distinguishability` is a function that computes the maximum probability of distinguishing two or more quantum states. That is, this function computes the maximum probability of winning the following game: You are given a complete description of a set of $k$ quantum states $\rho_1, \ldots, \rho_k$, and then are given one of those $k$ states, and asked to determine (via quantum measurement) which state was given to you.

## Syntax

`DIST = Distinguishability(X)``DIST = Distinguishability(X,P)``[DIST,MEAS] = Distinguishability(X,P)`

## Argument descriptions

### Input arguments

`X`: The quantum states to be distinguished.`X`can either be a cell containing 2 or more density matrices, or`X`can be a matrix whose columns are pure vector states.`P`(optional, default`[1/k, 1/k, ..., 1/k]`, where`k`is the number of quantum states): A vector whose j-th entry is the probability that the state $\rho_j$ is given to you in the game described above. All entries must be non-negative, and the entries of this vector must sum to 1.

### Output arguments

`DIST`: The maximum probability of distinguishing the states specified by`X`.`MEAS`(optional): A cell containing optimal measurement operators that distinguish the states specified by`X`with probability`DIST`.

## Examples

### Orthogonal states can be perfectly distinguished

Any number of quantum states can be perfectly distinguished (i.e., distinguished with probability 1) if they are mutually orthogonal. The following code generates a random $6\times 6$ unitary matrix (i.e., a matrix with orthogonal pure states as columns) and verifies that those pure states are perfectly distinguishable:

```
>> Distinguishability(RandomUnitary(6))
ans =
1
```

### Two states

The maximum probability of distinguishing two quantum states $\rho$ and $\sigma$ is exactly $\frac{1}{2} + \frac{1}{4}\|\rho - \sigma\|_1$^{[1]}, where $\|\cdot\|_1$ is the trace norm. We can verify this in a special case as follows:

```
>> rho = RandomDensityMatrix(4);
>> sigma = RandomDensityMatrix(4);
>> Distinguishability({rho, sigma})
ans =
0.7762
>> 1/2 + TraceNorm(rho - sigma)/4
ans =
0.7762
```

### Three or more states

We can also compute the maximum probability of distinguishing three or more states, but no simple formula is known in this case.

```
>> for j = 1:6
rho{j} = RandomDensityMatrix(4);
end
>> Distinguishability(rho)
ans =
0.4156
```

## Source code

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

## References

- ↑ John Watrous. Theory of Quantum Information lecture notes, Fall 2011.