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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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},...
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How to find the distance between a given $\rho$ and the nearest pure state(s)?

I have a $d$-dimensional state $\rho$. Is there any way to find the (possibly not unique) trace distance to the nearest pure state: $$ \min_{|\psi\rangle} \,\,\lVert \rho - |\psi\rangle\langle \psi| \...
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What is the best quantum process tomography method?

This question is somewhat related to this question. What is currently the best method for quantum process tomography? By best I mean, the one that can achieve the best accuracy of estimation per qubit ...
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What is the best method for estimating average channel fidelity?

This thesis shows an efficient way to estimate average channel fidelity (in chapter 4). However, it is somewhat old (from 2005). Are there any better methods out there? By better I mean: are there ...
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Can Gate Set Tomography work on Quantum Channels?

I stumbled across a new paper on gate set tomography. Can gate set tomography be applied to a quantum channel or multiple quantum channels? Will the same advantages still apply of not having to 'rely ...
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Can quantum error correction work on any type of channel?

It says on wikipedia that quantum error correction can (at best) correct phase flips and bit flips. A popular form of representing a quantum channel is in its Kraus representation (scroll down to ...
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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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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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How is the connection between Bures fidelity and quantum Fisher information derived?

I recently came to know that there is a connection between Bures Fidelity $(F_B)$ and Quantum Fisher Information $(F_Q)$ given by $$F_{B}(\rho, \rho_\theta) = 1 - \frac{\theta^2}{4} F_Q[\rho, A] + \...
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What is the relation between fidelity and concurrence for a two qubit maximally mixed state?

I am trying to understand the relation between Fidelity and Concurrence for a two qubit maximally mixed state. When I calculate the Fidelity and Concurrence, I observe that Concurrence is zero whereas ...
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Is un-computing $U$ a good proxy for circuit fidelity?

I'm trying to estimate the fidelity of some family of unitaries $U(\theta)$ implemented on a noisy quantum computer. To do so, I start from an uninitialized state $|0\rangle$ and run the circuit $U(\...
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Simulate a quantum channel with a certain fidelity

I am looking for an easy-to-use framework for simulating a quantum channel that can accept the desired average fidelity of the channel as input. For example, if I want a channel with 98% average ...
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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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Fidelity of extensions of states

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'})\...
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In Uhlmann's theorem, should the polar decomposition be written as $A=|A|V$ or $A=V|A|$?

In the proof of Uhlmann's theorem, the book writes the polar decomposition: $A = |A|V$, with $|A| = \sqrt{A^\dagger A}$. Shouldn't it be $V|A|$ instead? The former case is $A^\dagger A = V^\dagger|A||...
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Closeness of purifications of states

Uhlmann's theorem states that if two states $\rho_A, \sigma_A$ satisfy $F(\rho_A, \sigma_A)\geq 1 - \varepsilon$, then there for any purification $\Psi_{AR}$ of $\rho_A$, one can find a purification $\...
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Properties of composition of isometry and a perturbed adjoint

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 ...
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Non-ideal coin tossing

Can someone please check if the following makes sense? We have a non-ideal coin-tossing scheme as follows. Alice and Bob know what $|0\rangle,|1\rangle$ are. Bob wins when the coin is 1. Honest Alice ...
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Non-ideal bit commitment and coin tossing

Can someone please explain rigorously the reasoning behing non-ideal ($\epsilon$-concealing and $\delta$-binding) bit commitment scheme and the impossibility of coin tossing. What is bias in coin ...
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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 \...
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How do I compute the fidelity on the IBM Q using qiskit without the statevector simulator?

I want to compare the fidelity of a circuit implemented on IBMQ Santiago with the ideal circuit simulated using the statevector backend. Is there a way to do this? So far I have only seen examples ...
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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 ...
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Given $\rho,\sigma$ such that $F(\rho,\sigma)=0$, what can we say about $F(\text{Tr}_A(\rho),\text{Tr}_B(\sigma))$?

Lets say $\rho,\sigma$ satisfy $F(\rho,\sigma)=0$, i.e., they are quantum states living on orthogonal supports. What can we say about $F(\text{Tr}_A(\rho),\text{Tr}_B(\sigma))$? I am looking for upper ...
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What is the intuition behind Bures and angle metrics?

I am reading Distance measures to compare real and ideal quantum processes and it is explained the motivation behind Bures metric and angle metric. Bures metric is defined as: $$B(\rho,\sigma)=\sqrt{2-...
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Calculating fidelity for mixed states: are there tricks or it is “brute force” calculations?

The fidelity between two density matrix $\rho$ and $\sigma$ is the following: $$F(\rho,\sigma)=\operatorname{Tr}\left[\sqrt{\sqrt{\rho} \sigma \sqrt{\rho}} \right]^2$$ If one of the two state is a ...
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What does quantum gate fidelity mean?

The formal definition states that it's the distance between two quantum states. What does that mean experimentally? Does distance here mean the distance between two states on the Bloch Sphere? I am a ...
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Is the quantum min-relative entropy $D_{\min}(\rho\|\sigma)=-\log(F(\rho, \sigma)^2)$ or $D_{\min}(\rho\|\sigma)=-\log(tr(\Pi_\rho\sigma))$?

In John Watrous' lectures, he defines the quantum min-relative entropy as $$D_{\min}(\rho\|\sigma) = -\log(F(\rho, \sigma)^2),$$ where $F(\rho,\sigma) = tr(\sqrt{\rho\sigma})$. Here, I use this ...
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What can be inferred about the closeness of reduced qubit states from the closeness of the bipartite quantum state?

Given a qubit state $|\psi\rangle \in \mathcal{H}$, and two bipartite general mixed states $\rho$ and $\sigma$, such that, $$\langle \psi|\otimes \langle \psi|\rho - \sigma |\psi\rangle \otimes |\psi \...
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Can the fidelity $F(\rho,\sigma)$ be computed knowing only $\rho - \sigma$?

The motivation for this question comes from trace distance. For any two states $\rho, \sigma$, the trace distance $T(\rho, \sigma)$ is given by $$T(\rho, \sigma) = |\rho - \sigma|_1,$$ where $|\cdot|...
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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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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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Why does the state fidelity satisfy $\operatorname{tr}|\sqrt{\rho}\sqrt{\sigma}|=\operatorname{tr}\sqrt{\sigma^{1/2}\rho\sigma^{1/2}}$?

Given the the two states $\rho$ and $\sigma$ of a quantum system, with $|\psi\rangle$ and $|\varphi\rangle$ as their purification respectively, the fidelity is defined as: $$F(\rho,\sigma)=\max_{|\...
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Joint Concavity of (Root) Fidelity

I have some problem in understanding the proof of the concavity of root fidelity given in Chapter 9.2 of Mark M. Wilde's "Quantum Information Theory". Here, the fidelity is defined by $F(\...
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Fidelity With Bell State Calculation

Let's say I have the following state: $$ |\psi\rangle = \sqrt{\frac{2}{3}} |0000\rangle_{a_1b_1a_2b_2} + \sqrt{\frac{1}{6}} \big( |0011\rangle_{a_1b_1a_2b_2} + |1100\rangle_{a_1b_1a_2b_2} \big). $$ I ...
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Is the quantum state fidelity defined as $F(\rho, \sigma)=\text{tr}\sqrt{\rho^{1/2}\sigma\rho^{1/2}}$ or its square?

I have seen two different definition of Fidelity in different sources. For example, Nielsen & Chuang QCQI, 10th edition, page 409 defines Fidelity like the following: $$ F(\rho, \sigma) := \...
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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 preparation?

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-Josza) 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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Does higher channel fidelity imply higher entanglement fidelity?

Consider two noisy quantum channels (CPTP maps), $\Phi_1^A$ and $\Phi_2^A$, acting on a system $A$. Suppose that for any pure state $\left|\psi\right>\in \mathcal H_A$, $$ F\big(\psi, \Phi_1^A(\psi)...
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On the distribution of the fidelity of a random product state with an arbitrary many-qubit state

Consider an arbitrary $n$-qubit state $\lvert \psi \rangle$. How much do we understand about the probability distribution of the fidelity of $\lvert \psi \rangle$ with a tensor product $\lvert \alpha \...
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144 views

Is phase factor negligible in fidelity of quantum states?

One well-known fidelity is defined as $(Tr\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}})^2$. And for pure states, fidelity is always in the form $|\langle\psi|\phi\rangle|^2$. As we know, in the context of two-...
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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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What does fidelity mean?

I am learning qiskit software and this term keeps popping up and I am unable to get a grasp on the technical definition given by wikipedia. For example, the functions state fidelity and process ...
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635 views

Quantum fidelity simplified formula while both of the density matrices are single qubit states

I have a question while reading the quantum fidelity definition in Wikipedia Fidelity of quantum states, at the end of the Definition section of quantum fidelity formula, it says Explicit expression ...
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How to calculate the fidelity of a certain gate of a IBMQ device in Qiskit using randomized benchmarking/tomography?

For example, I want to calculate the fidelity of a 1-qubit and 2-qubit gates (similar to the result shown in figure 2 in this paper). Is there any way to do that in Qiskit? I've gone through the ...