# LocalDistinguishability

LocalDistinguishability | |

Computes the maximum probability of distinguishing quantum states by symmetric-extendible measurements | |

Other toolboxes required | CVX |
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Related functions | ChannelDistinguishability Distinguishability |

Function category | Distinguishing objects |

` LocalDistinguishability` is a function that computes the maximum probability of distinguishing two or more quantum states via PPT or symmetric-extendible measurements, which approximate separable measurements. 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. However, the only measurements that are available to you are "almost" local in the sense that they are PPT and/or have a symmetric extension.

## Syntax

`DIST = LocalDistinguishability(X)``[DIST,MEAS,DUAL_SOL] = LocalDistinguishability(X)``[DIST,MEAS,DUAL_SOL] = LocalDistinguishability(X,...)`

## 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. *`DIM`(optional, default has both subsystems of equal dimension): A 1-by-2 vector containing the dimensions of the subsystems on which the states described by`X`act.`COPIES`(optional, default 2): The number of copies of the second subsystem (i.e., the measurements will have a`COPIES`-level symmetric extension).`PPT`(optonal, default`true`): A flag (either`true`or`false`) indicating whether the symmetric extensions of the measurement operators must have positive partial transpose.`BOS`(optonal, default`true`): A flag (either`true`or`false`) indicating whether the symmetric extensions of the measurement operators must be Bosonic (i.e., be supported on the symmetric subspace).`TOL`(optonal, default`eps^(1/4)`): The numerical tolerance used when determining whether or not a symmetric extension exists.

**Important note:** The input arguments are mutually optional (this is different from other QETLAB functions!). For example, if you want to specify the number of `COPIES` to 3 and the dimension of the two subsystems to `[2 3]`, you can call the function as follows: `LocalDistinguishability(X, 'COPIES', 3, 'DIM', [2 3])`

### 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`.`DUAL_SOL`(optional): A dual solution that verifies that the maximum distinguishability probability is indeed no larger than`DIST`.

## Examples

Some examples should be added.

## Source code

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

## Notes

The methods used by this function to compute the local distinguishability of quantum states was developed in ^{[1]}.

## References

- ↑ A. Cosentino.
*Quantum State Local Distinguishability via Convex Optimization*. PhD thesis, University of Waterloo, 2015.