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# Questions tagged [fidelity]

For questions about the fidelity between quantum states.

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### Polar decomposition of $\sqrt{\sqrt{\rho} \sigma \sqrt{\rho}}$

By polar decomposition of a square invertible matrix $A$, I understand $A = |A| U$ for some unitary matrix $U$, where $|A| = \sqrt{A^\dagger A}$ with $\dagger$ denoting the conjugate-transpose ...
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1 vote
56 views

### 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?
• 40.1k
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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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### 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 ...
• 121
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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 ...
• 319
90 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 \...
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