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)
Argument descriptions
- 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.
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 here to view this function's source code on github.
References
- ↑ John Watrous. Theory of Quantum Information lecture notes, Fall 2011.