# Distinguishing density operators with the same diagonal elements

If I have two sources of qubits and one source produces the density matrix:

$$\rho_1 = \begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix}$$

and the other source produces:

$$\rho_2 = \begin{pmatrix} 1 & 1 \\ 1 & 1\end{pmatrix}$$

Is it possible to perform a measurement to determine which source the qubit is coming from? I understand that the diagonal elements tell us the probability of finding the qubit in that state - and so just measuring the state of the qubit in this case will not be enough to distinguish them. I also understand that the non-diagonal elements tell us the extent to which the state is a mixed state or a pure state - so in the first case we have a statistical mixture and in the second we have a pure state, but I'm unsure how we could this fact to distinguish them?

• the trace of a density matrix is equal to one. Commented Oct 31, 2020 at 22:42

## 2 Answers

The second density matrix is actually a rank-1 projection (if normalised) hence a dyadic product and therefore a pure state. It is enough then to measure against a state which is perpendicular to this vector (i.e. $(1/\sqrt 2,1/\sqrt 2)$) to say whether the qubit is not coming from the second source.

• Thanks for your answer. Do you know a good reference for why rank-1 projection implies pure state? I can't seem to find this anywhere. So measuring against a state which is perpendicular means projecting onto the span of that state? Commented Mar 5, 2015 at 16:40
• every rank-1 operator is of the form $x\mapsto (y,x)z$ for a certain pair of vectors $y,z$. Using, e.g., Dirac notation you can easily show that this is a dyadic product. Yes, you can think you are evaluating your state $\rho_2$ on the dyadic product of the orthogonal state, which is simply the projection onto it. This will give you the modulus squared of the inner product between the two vectors, which in this case is 0 by orthogonality. Commented Mar 5, 2015 at 16:44

It is not possible to distinguish them with a single measurement because the first density matrix is fully mixed and can produce either result at any angle of measurement. If you can repeat the process, there is one detector angle at which a measurement of the second density matrix will always produce the same result. So with many measurements, you can be almost certain the source is the second one, or you can be sure the source is the first.