I have a general question regarding the interpretation of a enganglement distillation protocol.

In general you have a set of entanglet qubit pairs in a Werner-state. Point of matter of this is that I can define the fidelity for one particular state (a Bell-state in my case) on this mixed Werner-state. Alice and Bob have each one qubit of each pair. Both apply some unitaries (quanten gates) on their qubits and measure some of them in some basis. In some cases, for example both measure the qubit to be "1", they accept the protocol and in some not.

Now comes the crucial part for my question: After the measurement I renormalize the after-measurement-state with the probability that Alice and Bob accept the protocol and can calculate the fidelity for this renormalized state which should be bigger than the fidelity before application of the protocol. This is clear.

But how can I interpret the following situation: I do my measurement as normal but do not renormalize the state and calculate instead the "fidelity". What is the interpretation of this "fidelity"? Does it mean that for a particular input fidelity the qubit pair is in a particular (Bell-)state? I mean if I talk of the fidelity and the success-probabilty as "dimensions" I have in one case the "dimension" fidelity and in the other case "probability * fidelity*.

Or is it somewhat of "I apply the Protocol X times to Y qubit pairs and this is the overall probability that I had success and my qubit has my desired state"?

I hope I could clearly formulate my question.


I don't think there is a good answer to your question. For me, the fidelity is a distance measure (not a metric though). This means, it just tells you how "far" two states are from each other. The further they are away (e.g. the smaller the fidelity) the less they behave like each other, the closer the fidelity is to one, the better the states match. In order for this interpretation to work at all, you need normalized states - otherwise, the fidelity will just not be normalized (e.g. same state: Fidelity = 1), but arbitrary.

Therefore and since the fidelity doesn't have a clear operational meaning anyway, I doubt that an interpretation like you want really exists. This does not exclude the possibility that in this particular case, the number has an interpretation like you want (the success probability of course does enter: this will probably be the norm of the resulting state that nees to be renormalized, so the success probability could be the ratio of the unnormalized and normalized probability - careful though, I didn't check this).


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