# Distinguishability

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 Other toolboxes required Distinguishability Computes the maximum probability of distinguishing quantum states CVX ChannelDistinguishabilityLocalDistinguishability 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 $\frac{1}{2} + \frac{1}{4}\|\rho - \sigma\|_1$, 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.