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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Closest quantum state with a fixed marginal: Analytical solution?

Let $\rho_{AB}$ be a bipartite state and let $\sigma_{B}$ be another state. What state $\tilde{\rho}_{AB}$ is closest to $\rho_{AB}$ and satisfies $\tilde{\rho}_B = \sigma_B$? We can define closeness ...
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Properties of the generalized fidelity for subnormalized states

The generalized fidelity for quantum states that may be sub-normalized is given by (Defn 3.12) $$F_{*}(\rho, \tau):=\left(\operatorname{Tr}|\sqrt{\rho} \sqrt{\tau}|+\sqrt{(1-\operatorname{Tr} \rho)(1-\...
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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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What scheme should be used in case of applying non-Cliffords to estimate probability of success?

For Clifford gates (when performing randomized benchmarking and starting from ground state) the final state is always ground. It is acquired by applying at the end recovery gate, which transfers the ...
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97 views

How to calculate the threshold for gate fidelity?

I've been interested in gate fidelity lately. In the meantime, I came up with a specific example. Let's think about the encoding of situation $\alpha_0 |0\rangle + \alpha_1 |1\rangle \rightarrow \...
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How can I get fidelity of a gate from randomized benchmarking?

I found (page 33) the method of finding fidelity from fit by "interleaved and reference decay" according to the sequence fidelity formula: $$F_{ref}=Ap_{ref}^{m}+B,$$ where $p_{ref}^{m}$ is ...
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Amplitude encoding: distinction between negative and positive real values

I want to encode two vectors into qubits and compute a distance between them that corresponds/is proportional to the euclidian distance of the real vectors. The encoding I use is A) qiskit ...
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How large can we make the fidelity between mixed states by allowing unitaries?

For pure states, it is known that one can always find a unitary that relates the two i.e. for any choice of states $\vert\psi\rangle$ and $\vert\phi\rangle$, there exists a unitary $U$ such that $U\...
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What are the "higher moments" of the gate fidelity?

Reading the paper Gate fidelity fluctuations and quantum process invariants I came across the concept of higher moments of the gate fidelity, for example in the following excerpt from the introduction:...
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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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What are min and max overlaps of a maximally entangled state with a separable state?

Let $A,B$ be Hilbert spaces of dimension $d$. Let $\rho$ be some separable quantum state of the composite system $AB$. Given a maximally entangled state: $$\vert\phi\rangle = \frac{1}{\sqrt{d}}\sum_{i=...
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Approximate Cloning

Question Consider two single qubit states $\left\{|\alpha_0\rangle,|\alpha_1\rangle\right\}$ which are not orthogonal or parallel, i.e. $\left|\langle\alpha_0|\alpha_1\rangle\right|\ne0,1$. ...
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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 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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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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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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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 $\sigma$ ...
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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)]^2 = 1 - \frac{\theta^2}{4} F_Q[\rho, A] + ...
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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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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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If $\rho,\sigma$ are classical-quantum states, can the fidelity $F(\rho,\sigma)$ be expressed in terms of $F(\rho_i,\sigma_i)$?

Let $\rho = \sum_i \vert i\rangle\langle i\vert \otimes \rho_i$ and $\sigma = \sum_i\vert i\rangle\langle i\vert\otimes\sigma_i$ where we are using the same orthonormal basis indexed by $\vert i\...
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How to prove generalized Uhlmann's theorem?

I think the Uhlmann theorem should be in general of this form: Let $\rho$ and $\sigma$ be density operators acting on $A$, with Schmidt degrees at most $r$, and let $B$ be another Hilbert space with ...
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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. $$ \mathrm{tr}(AU) = |\mathrm{tr}(|A|VU)| = |\mathrm{tr}(|A|^{1/2}|A|^{1/2}VU)| $$ ...
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Trace distance between mixed state and pure state vs trace distance between their purifications

Let $\rho$ be a mixed state and $\vert\psi\rangle\langle\psi\vert$ be a pure state on some Hilbert space $H_A$ such that $$\|\rho - \vert\psi\rangle\langle\psi\vert \|_1 \leq \varepsilon,$$ where $\|A\...
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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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987 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 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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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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How does $F(\psi, \phi) = [\sum_{x}\sqrt{p(x)q(x)}]^{2}$ [duplicate]

From Quantum Information Theory by Mark Wilde, pg 243 asks to show that $F(\psi, \phi) = [\sum_{x}\sqrt{p(x)q(x)}]^{2}$, which is described as the Bhattacharyya overlap, or classical fidelity, between ...
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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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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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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 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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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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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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Quantum circuit for computing fidelity

Suppose we use Uhlmann-Jozsa fidelity $$ F(\rho, \sigma):=\left(\mathrm{tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}\right)^2. $$ Can we construct a quantum circuit that helps us calculate the fidelity of ...
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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 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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Are there disadvantages in using the inner product between states instead of the fidelity?

Would there be any disadvantages of using inner product, that is, $\mathrm{Tr}(A^{\dagger}B)$ (say making it, $\mathrm{Tr}(\sqrt A \sqrt B)$ to normalise) to quantify how far two quantum states are ...
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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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1answer
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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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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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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 ...