Questions tagged [fidelity]
For questions about the fidelity between quantum states.
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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 ...
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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, \...
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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
$$
F_{|\psi\rangle\!\langle \psi|} = \mathrm{Tr}\bigg(U | \psi \rangle\!\langle \psi | U^\dagger \mathcal M\big(| \psi \rangle\!\langle \psi |\...
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Inner product in terms of Hadamard and controlled SWAP gates
On p. 7 of this paper (https://arxiv.org/abs/2112.04958) it is claimed that:
"For any two states $|φ\rangle$ and $|ψ\rangle$ with the same dimensions, the
fidelity $F(|φ\rangle, |ψ\rangle)$ can ...
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Does the quantum state fidelity satisfy $F(\rho ,\sigma) \le F(\mathcal{A}(\rho), \mathcal{A}(\sigma))$ if ${\cal A}$ is a process involving ancillae?
It is a well known fact that the fidelity is preserved by unitary evolution, i.e.
$$
F(\rho ,\sigma) = F(U\rho U^\dagger, U\sigma U^\dagger),
$$
for any unitary operator $U$.
However in most quantum ...
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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 \...
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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} | \...
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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{\...
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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 ...
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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, ...
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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 ...
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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 ...
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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♦
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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 ...
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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 ...
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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 ...
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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 ...
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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:
...
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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 ...
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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 ...
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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
$$\mathsf{Pr}\big[|\langle \Phi|\psi \rangle|^2 \...
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Error Correction over time
I am in year 12 and am a Science Extension student. My research question was "To what extent does an error-correcting algorithm reduce the inaccuracy of a quantum computation over time?".
To ...
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Qiskit state_fidelity not accepting my Density Matrices
I'm attempting to use qiskit's state_fidelity(state1, state2, validate=True) but keep getting the following error: QiskitError: 'Input quantum state is not a valid'
...
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Bounding inner product of states with distance
Suppose we are given two states quantum states $|{\psi}\rangle$ and $|{\phi}\rangle$ over $n$ qubits. We know that the distance between the states is bounded by $\epsilon$:
$$|| |{\psi}\rangle- |{\phi}...
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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 ...
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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\...
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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 ...
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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 ...
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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♦
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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 ...
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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 (...
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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 ...
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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 $\...
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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 ...
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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 ...
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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 $p_x,p_y,p_z$ ...
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Commutative operators
I have got a 2-qubit circuit with the following instructions:
...
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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.
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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♦
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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 ...
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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♦
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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 ...
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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 ...
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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 ...
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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:
...
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What use cases are there for 127 qubit QPUs?
IBM have recently announced their 127 qubit Eagle processor. Other approaches, such as Rydberg arrays, have now 256 qubits, as for example in QuEra's QPU QPU.
While these are without a doubt ...
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Is it overestimating noise in a QPU if we use infidelities and also quantum channel such as depolarising or amplitude damping?
I have mainly seen two ways of studying noise in quantum algorithm simulation.
The first one is to suppose that your quantum gate can be implemented with a probability of success of $\mathcal{F}$ (...
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Qubit fidelity of DWAVE device
Since DWAVE quantum device is constructed using superconducting flux qubits, each qubit cannot be produced identically so that the fidelity of the qubit must be different. DWAVE only provides the ...
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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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