# 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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### Purpose of using Fidelity in Randomised Benchmarking

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 ...
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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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### What is the longest time a qubit has survived with 0.9999 fidelity?

I am pretty intrigued by the record time that a qubit has survived.
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### How do the extra energy levels of a transmon qubit affect computation/fidelity?

I was reading about transmon qubits, and I know that they are not true two-level systems. Are there any math/papers which talk about how those extra energy levels affect the computation? I'm assuming ...
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### What is a stabilizer state?

I am reading through the paper "Direct Fidelity Estimation from Few Pauli Measurements" (arXiv:1104.4695) and it mentions 'stabilizer state'. "The number of repetitions depends on the ...
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### How to calculate the average fidelity of an amplitude damping channel

An answer to this question shows how to calculate the average fidelity of a depolarizing channel. How would one go about calculating this for an amplitude dampening channel? I tried working out the ...
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### What would be an ideal fidelity measure to determine the closeness between two non unitary matrices?

The Hibert Schmidt norm $\mathrm {tr}(A^{\dagger}B)$ works well for unitaries. It has a value of one when the matrices are equal and less than one otherwise. But this norm is absolutely unsuitable for ...
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### How are eavesdroppers detected when using BB84 in the presence of noise?

I would like to expand upon this question: What is the probability of detecting Eve's tampering, in BB84? Let's say that when the receiver (colloquially referred to as Bob) receives a qubit and ...
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### Clarification on Watrous' proof of Alberti's theorem on the fidelity function

I am reading John Watrous' quantum information theory book. In the proof of Theorem 3.19 (practically the Alberti's theorem on the characterization of the fidelity function) he claims the following ...
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### What is the connection between Bures metric and (finite) Bures distance?

The Wikipedia page discussing the Bures metric introduces it as the Hermitian 1-form operator $G$ defined implicitly by $\rho G+G\rho = \mathrm d\rho$, and which induces the corresponding Bures ...
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### How to Aggregate Multiple Gate Fidelities

The fidelity of a qubit is nicely defined here and gate fidelity as "the average fidelity of the output state over pure input states" (defined here). How can one combine the fidelies of two (...
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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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### Saturating the Fuchs-van de Graaf inequality

It is well-known that one side of the Fuchs-van de Graaf inequality is saturated for pure states, i.e. $F(\rho,\sigma)^2 = 1-d(\rho,\sigma)^2$ when $\rho$ and $\sigma$ are pure (here we are using 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 ...
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### How to limit the error probability in large scale quantum computers

I am quite stumped by the fact that the roadmaps for quantum computers as given by IBM, Google, IonQ, etc. seem to imply a linear/exponential growth in the size of their quantum computers. Naively, I ...
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### Prove the fidelity can be written in terms of Pauli expectation values as ${\rm tr}(\rho\sigma)=\sum_k \chi_\rho(k)\chi_\sigma(\rho)$

I am reading through "Direct Fidelity Estimation from Few Pauli Measurements" and it states that the measure of fidelity between a desired pure state $\rho$ and an arbitrary state $\sigma$ ...
Given two states $\rho_A, \sigma_A$, Uhlmann's theorem states that the fidelity between them is achieved in the following way $$F(\rho_A, \sigma_A) = \max_{U_{R'}}F(\rho_{AR'}, (I\otimes U_{R'})\... 1answer 194 views ### Proving the inequality |\mathrm{tr}(AU)|\le \mathrm{tr}|A| in Uhlmann's theorem In Nielsen and Chuang, in the Fidelity section, (Lemma 9.5, page 410 in the 2002 edition), they prove the following.$$ \mathrm{tr}(AU) = |\mathrm{tr}(|A|VU)| = |\mathrm{tr}(|A|^{1/2}|A|^{1/2}VU)| $$... 1answer 143 views ### The proof of monotonicity of fidelity for channels and its meaning I have two questions regarding the exercise 9.2.8 of Quantum information by Wilde, which is as follows: Let \rho,\sigma \in \mathcal{D}(\mathcal{H}_A) and let \mathcal{N: L(H}_A)\rightarrow \... 1answer 546 views ### Is there any way we get the state vector/density matrix of a noisy simulation in qiskit? In Qiskit we can't use noise models in the 'state vector_simulator' or the 'unitary simulator', hence making it impossible to compute fidelity of the output of the noisy circuit and the noiseless ... 1answer 616 views ### How to calculate state fidelity in Qiskit? I have a circuit with different structures, now I'm trying to calculate the fidelity between those with the original one. How do I calculate the fidelity? I want also to initialize the state vector by ... 0answers 48 views ### Given \rho,\sigma, bound \min F(\bar\rho,\sigma) over \bar\rho such that F(\bar\rho,\rho)\ge1-\epsilon Let \rho, \sigma be states such that$$F(\rho,\sigma) = \delta >0,$$where F(\rho,\sigma) = \|\sqrt{\rho}\sqrt{\sigma}\|_1. Now consider all possible states \bar{\rho} such that F(\bar{\rho},... 2answers 62 views ### What can be said about the closeness of two states if the difference of their fidelity measured with respect to a fixed state is close to 0? Suppose I have two states \rho and \sigma. We are given that,$$Tr((\rho - \sigma)|\psi\rangle\langle\psi|) \geq \epsilon where $|\psi\rangle$ is a fixed state and $\epsilon \rightarrow 0$, ...
Suppose $\vert\Phi\rangle_{AR} = \frac{1}{\sqrt{|D|}}\sum_{i\in D} \vert ii\rangle_{AR}$ is the maximally entangled state. Let $V_{A\rightarrow BE}$ and $\tilde{V}_{A\rightarrow BE}$ be two isometries ...