# What is the probability of error in quantum fingerprinting using cswap test?

In quantum fingerprinting, the states are sent to a 3rd party in order to verify if both qubits match or not. To do so, a swap test is needed by the 3rd party.

I read some information on swap test including this link here and I noticed that the probability of cswap test is:

If $$|\phi _{x}\rangle =\ |\phi _{y}\rangle$$ then we observe a 0 with $$prob(0) = 1$$ according to wikipedia.

So I was wondering what are the probabilities of a false positive and false negative in quantum fingerprinting?

According to what I saw I'm guessing:

• Probability to get a false positive (a 0 even if both states don't match) is 1 - P(getting 0) so it's 0?
• Probability of getting a false negative (a 1 even if both states match) is 1/2?

But I'm not sure if my conclusion is correct, could someone please confirm this and give some explanation?

## 1 Answer

Conclusion first. The first one is correct, but the scenario is not practical, since it is challenging(or more directly, impossible) to build a perfect channel, so even if the two states are actually the same at the very beginning, what the third party gets are two different states(but highly likely is the channel is a good one).

Besides, sometimes we do not hope to get two states that are exactly the same, so the value of the fidelity(for example, 0.5623) is what really needed.

This paper gives a detailed reduction using density matrix at SUPPLEMENTARY NOTE 4.

I'll just sketch the conclusion. The probability that the measurement of the control qubit is in the $$|0\rangle$$ state is $$p_0=\frac{1}{2}+\frac{1}{2}F(|\phi\rangle,rho)$$, where $$|\phi\rangle$$ is the state vector of one state, and $$\rho$$ is the density matrix. Thus the fidelity of the two states is $$F(|\phi\rangle, rho)=2*p_0-1$$.

Since by its definition, fidelity can be a number varting continuously from $$0$$ to $$1$$, just a few measurements(say three or four times) is never sufficient. If sufficiently many measurements are performed, then we'll have confidence in estimating its value, e.g., if we performed 100 measurements then we know that the decile of the reduced value of fidelity should be reasonable.