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Questions tagged [fidelity]

For questions about the fidelity between quantum states.

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How to derive the higher terms in the Taylor expansion of the Bures fidelity?

The Wikipedia article for the quantum Fisher Information mentions that one can expand the Bures fidelity and the quantum Fisher Information will appear as the second-order correction term. However in ...
I1ussion's user avatar
5 votes
1 answer
113 views

Is there a concentration inequality for the quantum gate fidelity $F(C,U)$ for a channel $C$ such that $\int dU F(C,U)=X$?

For a fixed quantum channel $N$ and a unitary channel $U$, we define $N$'s gate fidelity as $$ F(N,U) = \int \langle \psi| U \, N(| \psi \rangle \langle \psi |) \, U^\dagger| \psi \rangle d\mu_H(\psi)$...
Davide Li Calsi's user avatar
2 votes
1 answer
69 views

Can different density matrices have 100% fidelity with a given pure state?

I am trying to understand fidelity a bit better, to do so consider the bell state: $$|\Psi\rangle=\frac{1}{\sqrt{2}}\left(|01\rangle-|10\rangle\right),$$ the density matrix associated with this state ...
Adrien Amour's user avatar
8 votes
2 answers
97 views

What is the definition of physical gate error rate?

The fidelity of two quantum states $\rho$ and $\sigma$ is a well-defined (up to discussions about a square): $$ F(\rho, \sigma) = \text{Tr}\left( \sqrt{ \sqrt{\rho} \sigma \sqrt{\rho}}\right)^2. $$ ...
Frederik Ravn Klausen's user avatar
2 votes
1 answer
145 views

Doubt in understanding the use of Uhlmann's Theorem in BB84 security proof

I was going through the security proof of the BB84 protocol by Dr. Ramona Wolf. I am having trouble following the equation (used at 8:53) for the fidelity, \begin{align*} F(\rho_{ABE},\left|\phi\right\...
HypnoticZebra's user avatar
3 votes
0 answers
40 views

Infidelity as distance measure

Let $\mathcal{X} \in {\rm CP}(\mathcal{H}, \mathcal{K})$ and unital (compositive positive and unital maps). Let $\mathcal{Y} \in {\rm CPT}(\mathcal{H}, \mathcal{K})$(complete positive and trace ...
Michael.Andy's user avatar
1 vote
0 answers
19 views

The lower bound on the probability of error in quantum hypothesis testing

Prove the following lower bound on the probability of error $P_e$ in a quantum hypothesis test to distinguish $\rho$ from $\sigma$: \begin{align} P_e \geq \frac{1}{2} \left(1-\sqrt{1-F(\rho, \...
Michael.Andy's user avatar
4 votes
1 answer
197 views

Can you compute the average fidelity of two qubits by averaging over a finite number of states?

For a single qubit and a quantum operation, the fidelity $$ F_{|\psi\rangle\!\langle \psi|} = \mathrm{Tr}\bigg(U | \psi \rangle\!\langle \psi | U^\dagger \mathcal M\big(| \psi \rangle\!\langle \psi |\...
Omar Nagib's user avatar
2 votes
1 answer
43 views

Inner product in terms of Hadamard and controlled SWAP gates

On p. 7 of this paper (https://arxiv.org/abs/2112.04958) it is claimed that: "For any two states $|φ\rangle$ and $|ψ\rangle$ with the same dimensions, the fidelity $F(|φ\rangle, |ψ\rangle)$ can ...
pll04's user avatar
  • 21
2 votes
1 answer
96 views

Does the quantum state fidelity satisfy $F(\rho ,\sigma) \le F(\mathcal{A}(\rho), \mathcal{A}(\sigma))$ if ${\cal A}$ is a process involving ancillae?

It is a well known fact that the fidelity is preserved by unitary evolution, i.e. $$ F(\rho ,\sigma) = F(U\rho U^\dagger, U\sigma U^\dagger), $$ for any unitary operator $U$. However in most quantum ...
SRichoux's user avatar
  • 319
0 votes
0 answers
77 views

What is the expectation value of the overlap of two uniformly random pure states? [duplicate]

Let $\psi$ and $\phi$ be two uniformly random pure state $\psi, \phi \sim\mathbb{C}^d$. The the following equality holds \begin{align} \mathbb{E}_{\psi, \phi \sim \mathbb{C}^d} {\rm Tr}[\vert \phi \...
Michael.Andy's user avatar
3 votes
2 answers
206 views

Prove the fidelity equals $F( \rho , \sigma) = |\langle \psi_{\rho} | \psi_{\sigma}\rangle|^2$ for pure states

I am trying to learn by myself quantum computing and information and I have a very simple question concerning the demonstration of the following equality: $F( \rho , \sigma) = |\langle \psi_{\rho} | \...
X0-user-0X's user avatar
3 votes
1 answer
74 views

Is fidelity of mixed $\sigma$ and pure $|\psi\rangle$ equal to $\||\psi\rangle\langle\psi|\sigma\|_1$?

The quantum state fidelity between a pure quantum state $\rho:= \vert \psi \rangle \langle \psi \vert$ and a state $\sigma$ is \begin{align} F(\rho, \sigma):= {\rm Tr}[\sqrt{\sqrt{\rho}\sigma\sqrt{\...
Michael.Andy's user avatar
1 vote
0 answers
63 views

What are the similarity and difference between quantum fidelity estimation and parameter estimation problem?

Quantum fidelity (1) estimation is to estimate the similarity between two quantum states or process. Could quantum fidelity be viewd as a parameter? And what are the similarity and difference between ...
Michael.Andy's user avatar
2 votes
0 answers
54 views

Is the quantum state fidelity defined as $|\langle\psi|\phi\rangle|$ or $|\langle\psi|\phi\rangle|^2$? [duplicate]

I am reading Preskill's notes on quantum information as well as Chuang's textbook. I saw that fidelity is defined in two different ways in Preskill's notes and Chuang's book. In Preskill's notes, ...
Anindita Sarkar's user avatar
2 votes
1 answer
60 views

Global vs local: global density matrix and all the reduced density matrix

I prepare a $n$-qubit quantum state $\sigma$ whose ideal state is $\rho$, then perform state tomography on all the $m$- qubit reduced states. Ideally, I find that all the $m$- qubit reduced states are ...
Michael.Andy's user avatar
0 votes
2 answers
129 views

How to improve the state fidelity of state tomography in qiskit

I want to simulate state tomography on a 8 qubit state. I use the example https://qiskit.org/ecosystem/experiments/manuals/verification/state_tomography.html as a guide. My problem is that I get a ...
Luis Andres Colmenarez's user avatar
1 vote
0 answers
43 views

Do the eigenvalues of $|\sqrt\rho\sqrt\sigma|$ have a physical interpretation?

The (square-root) fidelity between a pair of states $\rho,\sigma$ is the quantity $$\sqrt F(\rho,\sigma) = \|\sqrt\rho\sqrt\sigma\|_1 = \operatorname{tr}|\sqrt\rho\sqrt\sigma| =\operatorname{tr}\left[\...
glS's user avatar
  • 25.3k
1 vote
0 answers
28 views

Pure Product State Problem Clarification about $\alpha$-closeness and $\beta$-farness

I am reading a paper on entanglement, specifically, determining if a state is close to being entangled or not. The problem first introduces the $(\alpha, \beta, l)$-Pure product state problem. This ...
Loic Stoic's user avatar
1 vote
0 answers
33 views

What are the estimate-estimate and estimate-project algorithms for quantum overlap?

In this paper on improvements to the traditional SWAP test to measure quantum state overlap (or fidelity), they mention two methods called estimate-estimate and estimate-project. I googled about these ...
Loic Stoic's user avatar
1 vote
1 answer
86 views

An intuitive definition of "One-way LOCC distance"

I am reading up on this paper which covers many aspects about fidelity, separability, and came across the "One way LOCC-distance" in section 2.4. Below is an excerpt: I am having trouble ...
Loic Stoic's user avatar
5 votes
0 answers
181 views

Explanation of the generalized SWAP test: the permutation test?

I am reading this paper on the state separability problem, and came across the term “Permutation Test.” This is on page 7 section 2.3. Apparently, the more famous SWAP test is a special case of ...
TwentyCents's user avatar
0 votes
1 answer
57 views

How are mixed states given to a quantum algorithm?

I've been reading this paper about quantum fidelity estimation, but really have no idea what's going on when it comes to density matrix notation. In the abstract, they have the following quote: ...
Loic Stoic's user avatar
1 vote
1 answer
315 views

pennylane:fidelity calculation after swap test between entagled states. Swap test issue

What I am trying to do is first take an image and encode it into quantum states, for this I have taken an image from the MNIST dataset and then reshaped it to (4,4) and now I wrote the following ...
Pratyush's user avatar
1 vote
1 answer
192 views

Why can't Quantum Fisher Information be negative?

Quantum Fisher Information is proportional to Fidelity susceptibility. Mathematically the equation is: $QFI=-\frac{\partial^2 d_B(\epsilon) }{\partial \epsilon^2}$ where above equation shows QFI is ...
Chetan Waghela's user avatar
7 votes
1 answer
210 views

Fidelity concentration bound for random stabilizer states

Let $|\Phi\rangle$ be a normalized vector in $\mathbb{C}^d$ and let $|\psi\rangle$ be a random stabilizer state. I am trying to compute the quantity $$\mathsf{Pr}\big[|\langle \Phi|\psi \rangle|^2 \...
BlackHat18's user avatar
  • 1,335
1 vote
1 answer
98 views

Error Correction over time

I am in year 12 and am a Science Extension student. My research question was "To what extent does an error-correcting algorithm reduce the inaccuracy of a quantum computation over time?". To ...
Liam Mitchell's user avatar
0 votes
2 answers
322 views

Qiskit state_fidelity not accepting my Density Matrices

I'm attempting to use qiskit's state_fidelity(state1, state2, validate=True) but keep getting the following error: QiskitError: 'Input quantum state is not a valid' ...
PGibbon's user avatar
  • 462
3 votes
1 answer
259 views

Bounding inner product of states with distance

Suppose we are given two states quantum states $|{\psi}\rangle$ and $|{\phi}\rangle$ over $n$ qubits. We know that the distance between the states is bounded by $\epsilon$: $$|| |{\psi}\rangle- |{\phi}...
Apo's user avatar
  • 545
1 vote
1 answer
308 views

In what sense is $\langle\psi|\rho|\psi\rangle$ the overlap between $|\psi\rangle$ and $\rho$?

The fidelity between a pure state $|\psi\rangle$ and an arbitrary mixed state $\rho$ is given by, $F(|\psi\rangle,\rho)=\sqrt{\langle\psi|\rho|\psi\rangle}$, which is stated to be equal to the square ...
Sooraj S's user avatar
  • 821
3 votes
1 answer
113 views

Strange inequivalence between superdense coding and teleportation

Let the fidelity between two quantum states be defined as $$F(\rho, \sigma) = \|\sqrt{\rho}\sqrt{\sigma}\|_1.$$ If $\rho = \vert\psi\rangle\langle\psi\vert$, then $F(\rho, \sigma) = \sqrt{\langle\psi\...
user1936752's user avatar
  • 2,997
2 votes
1 answer
190 views

What is the quantum relative entropy between pure states?

Given two pure quantum state $\rho=|\psi_\rho\rangle\langle\psi_\rho\mid$ and $\sigma=\mid\psi_\sigma\rangle\langle\psi_\sigma\mid$ ($\rho\neq\sigma$). We know that the fidelity between quantum ...
m1rohit's user avatar
  • 93
4 votes
0 answers
41 views

Bounds relating min-fidelity and induced one-norm

Consider two CPTP maps $M_{A\rightarrow B}$ and $N_{A\rightarrow B}$. Let $\Phi = M - N$. To distinguish between the two maps, there are several measures but here I want to compare two of them. The ...
user1936752's user avatar
  • 2,997
2 votes
1 answer
122 views

Why is state discrimination possible to infidelity $\delta$ using $n=\Theta(1/\delta)$ states?

In (Haah et al. 2015), in the first section, the authors study the asymptotic behaviours of fidelity and trace distance between $\rho^{\otimes n}$ and $\sigma^{\otimes n}$ for some given pair of ...
glS's user avatar
  • 25.3k
2 votes
0 answers
69 views

Fidelity and Uhlmann's theorem in the context of source coding

In quantum source coding, we have an encoder $\mathcal{E}$ and a decoder $\mathcal{D}$ which are some quantum channels. Given a state $\rho_A$ on Hilbert space $\mathcal{H}_A$, we wish to encode and ...
user1936752's user avatar
  • 2,997
1 vote
1 answer
132 views

Calculating state fidelity and space complexity of Minimum Eigen Optimizers (VQE, QAOA and Grover Optimizer) in qiskit

I'm a beginner in using Qiskit and my Computer Science background is not extensive. But I understand the Quantum Physics aspects of it relatively well. I solved a QUBO problem in Qiskit using VQE (...
Janani Ananthanarayanan's user avatar
3 votes
1 answer
147 views

Does a fidelity of $\mathcal{F}(U_1|0\rangle, U_2|0\rangle)=1$ imply that $U_1=U_2$?

I'm now studying quantum ML and now studying about fidelity ($\mathcal{F}$). To my knowledge, fidelity means the distance between two quantum states, $\textit{i.e.,}$ if $\mathcal{F} ==1$, then the ...
JERMY's user avatar
  • 111
8 votes
2 answers
466 views

Prove that a channel is close to acting on only one system

Background Suppose I have a quantum channel $\Phi:B(\mathcal{H}_1)\rightarrow B(\mathcal{H}_1)\otimes B(\mathcal{H}_2)$, such that there is some small $\epsilon$ such that for any two input states $\...
Sam Jaques's user avatar
  • 2,066
4 votes
0 answers
111 views

Are there any systematic tools for estimating expected error rate?

Like we all probably know, today’s NISQ computers are, as their name implies - very noisy. Hence, if we desire to obtain valuable results then we should come up with circuit designs that minimizes the ...
Ohad's user avatar
  • 1,759
6 votes
2 answers
1k views

What is fidelity in quantum computing?

As I studied quantum computing, I saw term 'Fidelity' in many papers that related to quantum algorithm. So, I really wonder about following two things. What is the real meaning of 'Fidelity' (As I ...
김시윤's user avatar
4 votes
1 answer
764 views

How to calculate the average fidelity of Pauli error channel

This question is related to this and this. I am working on Qiskit to design QEC schemes. My model works with Pauli errors. I would like to give to my Pauli error channel probabilities $p_x,p_y,p_z$ ...
Daniele Cuomo's user avatar
0 votes
0 answers
82 views

Commutative operators

I have got a 2-qubit circuit with the following instructions: ...
Марина Лисниченко's user avatar
1 vote
2 answers
120 views

which hardware platform is best for single qubits?

Please share a paper most recent data on single qubit hardware comparisons. Mainly gate single qubit gate fidelities and coherence times.
Hafiz muhammad Ibrahim jaffar's user avatar
4 votes
0 answers
499 views

Why is the fidelity, rather than the trace distance, the standard choice to compare quantum states?

I don't think it's particularly controversial to say that the "standard" way people use to compare quantum states is via the fidelity. Yes, sometimes the trace distance is used as well, but ...
glS's user avatar
  • 25.3k
2 votes
1 answer
180 views

What makes dynamical decoupling a good method since the fidelity after using it can only reach 0.85 or so?

From this paper, I see that the fidelity of single qubit gate after using dynamical decoupling only reach around 0.85 while I normally saw experiment papers state their fidelity can reach around 0.99 ...
Sherlock's user avatar
  • 695
5 votes
0 answers
90 views

Interpretation of the PVM $\mu$ whose Bhattacharyya coefficient equals the fidelity: $\|\sqrt P\sqrt Q\|_1={\rm B}(P,Q|\mu)$

(Bhattacharyya coefficient vs state fidelity) Given two vectors $u,v$ with nonnegative entries, their Bhattacharyya coefficient is $$\mathrm B(u,v)\equiv \sum_a \sqrt{u_a v_a}.$$ Given two positive ...
glS's user avatar
  • 25.3k
4 votes
0 answers
240 views

Is (square root) fidelity strictly concave?

We know that (square root) Fidelity which is defined as $\text{F}(\rho,\sigma) = \| \sqrt{\rho} \sqrt{\sigma} \|_1 = \text{Tr}(\sqrt{\sqrt{\sigma} \rho \sqrt{\sigma}})$ is satisfies the property of ...
Afham's user avatar
  • 41
1 vote
1 answer
77 views

Scaling of magic state distillation with single & two-qubit gate error

How does the resulting error in magic state distillation scale with the fidelity of single qubit gates and two qubit gates? In most experimental systems, two-qubit gate errors are much larger than ...
shixian105's user avatar
3 votes
1 answer
109 views

Game formulation of Quantum GAN

Quantum Generative Adversarial Network (QuGAN) generates a desired quantum state via a minimax game between generator and discriminator (equivalently, it's optimizing a trace distance between ...
userflux9674's user avatar
1 vote
0 answers
260 views

Problem with fidelity value range obtained from this python code

I'm calculating for instance the fidelity between two pure two qubit state, using the following python code: ...
Farhad's user avatar
  • 105