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Computes the maximum probability of distinguishing quantum states

Other toolboxes required CVX
Related functions ChannelDistinguishability
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.


  • 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.


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 =


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 =


>> 1/2 + TraceNorm(rho - sigma)/4

ans =


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);
>> Distinguishability(rho)

ans =


Source code

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