# Questions tagged [fidelity]

In quantum information theory, fidelity is a measure of the "closeness" of two quantum states. It expresses the probability that one state will pass a test to identify as the other. The fidelity is not a metric on the space of density matrices, but it can be used to define the Bures metric on this space. (Wikipedia)

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### What can be said about the closeness of two states if the difference of their fidelity measured with respect to a fixed state is close to 0?

Suppose I have two states $\rho$ and $\sigma$. We are given that, $$Tr((\rho - \sigma)|\psi\rangle\langle\psi|) \geq \epsilon$$ where $|\psi\rangle$ is a fixed state and $\epsilon \rightarrow 0$, ...
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### What is a difference between error rates and qubit/gate fidelity?

What is a difference between error rates and qubit/gate fidelity? A bit of maths in the explanation is fine but I am an A Level student doing a research project so definitions would be preferred.
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### How to calculate state fidelity in Qiskit?

I have a circuit with different structures, now I'm trying to calculate the fidelity between those with the original one. How do I calculate the fidelity? I want also to initialize the state vector by ...
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### How can I find the fidelity of the preparation operation $|0\rangle$ of IBMQ?

I want to know the fidelity (or error rate) of the preparation of $|0\rangle$. How can I obtain it?
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### Can we combine the square roots inside the definition of the fidelity?

The (Uhlmann-Jozsa) fidelity of quantum states $\rho$ and $\sigma$ is defined to be $$F(\rho, \sigma) := \left(\mathrm{tr} \left[\sqrt{\sqrt{\rho} \sigma \sqrt{\rho}} \right]\right)^2.$$ However, as ...
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### Which of the Jozsa axioms does the Hilbert-Schmidt inner product violate?

The paper Quantum fidelity measures for mixed states considers various differently-normalized variants of the Hilbert-Schmidt inner product $\mathrm{Tr}(A^\dagger B)$ on linear operators as candidate ...
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### Can I compute the fidelity between two states without having to diagonalise them?

So I have been given the following quantum states: $$\rho = \frac{I}{2} + \frac{\bar{s}.\bar{\sigma}}{2}$$ $$\pi = \frac{I}{2} + \frac{\bar{r}.\bar{\sigma}}{2}$$ How do I calculate the fidelity ...
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### What is the difference between the "Fubini-Study distances" $\arccos|\langle\psi|\phi\rangle|$ and $\sqrt{1-|\langle\psi|\phi\rangle|}$?

I sometimes see the "Fubini-Study distance" between two (pure) states $|\psi\rangle,|\phi\rangle$ written as $$d(\psi,\phi)_1=\arccos(|\langle\psi|\phi\rangle|),$$ for example in the Wikipedia page. ...
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### What are the "nice" properties of the diamond norm and why is it used?

I have heard about the diamond norm, and from what I understood it is a "nice" tool to quantify quality of quantum gates in the NISQ era. I would like to know a little more before going in detail in ...
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### Intuitive role of the polar decomposition in proof of Uhlmann's theorem for fidelity

I have read the Wikipedia article which relates the polar decomposition to a complex number being split into its modulus and phase but this analogy isn't very intuitive to me. In Nielsen and Chuang, ...
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### How can I calculate the inner product of two quantum registers of different sizes?

I found an algorithm that can compute the distance of two quantum states. It is based on a subroutine known as swap test (a fidelity estimator or inner product of two state, btw I don't understand ...
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### How to find the fidelity between two state when one is an operator?

I am going through Nielsen and Chuang and am finding the chapter on error-correction particularly confusing. At the moment I am stuck on exercise 10.12 which states Show that the fidelity between the ...
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### Do avoided crossings / CTs /ZEFOZs optimize quantum fidelity in practice?

CTs / ZEFOZs: Energy level structures that include avoided crossings at accessible energies tend to be resilient to noise and therefore present high coherence times, at least in the case of spin ...
Often, when comparing two density matrices, $\rho$ and $\sigma$ (such as when $\rho$ is an experimental implementation of an ideal $\sigma$), the closeness of these two states is given by the quantum ...