Questions tagged [fidelity]
For questions about the fidelity between quantum states.
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Can a quantum state be certified using self-test?
I am reading the paper Certifying almost all quantum states with
few single-qubit measurements. The main result of the paper (Theorem 1) is that
given an $n$-qubit target pure state $|\psi\rangle$ and ...
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How do I efficiently compute the fidelity between two stabilizer tableau states?
I have two stabilizer tableaus $T_1$ and $T_2$. How do I efficiently compute the fidelity of their stabilizer 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 ...
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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
$$ F(N,U) = \int \langle \psi| U \, N(| \psi \rangle \langle \psi |) \, U^\dagger| \psi \rangle d\mu_H(\psi)$...
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Can different density matrices have 100% fidelity with a given pure state?
I am trying to understand fidelity a bit better, to do so consider the bell state:
$$|\Psi\rangle=\frac{1}{\sqrt{2}}\left(|01\rangle-|10\rangle\right),$$
the density matrix associated with this state ...
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What is the definition of physical gate error rate?
The fidelity of two quantum states $\rho$ and $\sigma$ is a well-defined (up to discussions about a square):
$$
F(\rho, \sigma) = \text{Tr}\left( \sqrt{ \sqrt{\rho} \sigma \sqrt{\rho}}\right)^2.
$$
...
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Doubt in understanding the use of Uhlmann's Theorem in BB84 security proof
I was going through the security proof of the BB84 protocol by Dr. Ramona Wolf. I am having trouble following the equation (used at 8:53) for the fidelity,
\begin{align*}
F(\rho_{ABE},\left|\phi\right\...
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Infidelity as distance measure
Let $\mathcal{X} \in {\rm CP}(\mathcal{H}, \mathcal{K})$ and unital (compositive positive and unital maps). Let $\mathcal{Y} \in {\rm CPT}(\mathcal{H}, \mathcal{K})$(complete positive and trace ...
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The lower bound on the probability of error in quantum hypothesis testing
Prove the following lower bound on the probability of error $P_e$ in a quantum hypothesis test to distinguish $\rho$ from $\sigma$:
\begin{align}
P_e \geq \frac{1}{2} \left(1-\sqrt{1-F(\rho, \...
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Can you compute the average fidelity of two qubits by averaging over a finite number of states?
For a single qubit and a quantum operation, the fidelity
$$
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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