bibkey: mayer2021zerofidelity authors: K. Mayer year: 2021 title: “A short note on the 0-fidelity” doi: 10.48550/arXiv.2109.09629 url: https://arxiv.org/abs/2109.09629v1 claim: “For every n-qubit process, the process fidelity and the 0-fidelity defined by product single-qubit SIC states satisfy 1 − (3/2)(1 − F_0) ≤ F ≤ F_0; an SDP for n ≤ 4 suggests that the lower bound is tight for all n, which is conjectured, and a description of the worst-case process is stated as an open problem.” strata_touched:

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Mayer, a short note on the 0-fidelity

K. Mayer, A short note on the 0-fidelity, arXiv:2109.09629v1 (20 September 2021; quant-ph; the only version).

Verified locator

DOI: 10.48550/arXiv.2109.09629. Primary version: https://arxiv.org/abs/2109.09629v1. The TeX source main.tex of v1 supplies §I (Definitions 1 and 2), §II (Theorem 1 and its proof through the operator $\Gamma$), §III (the semidefinite programs and the conjecture) and §IV (the open problem).

Source statements

0-fidelity (Definition 1): “F_0(\cE)=\frac{1}{d^2}\sum_{k=1}^{d^2}\bra{\psi_k}\cE\big(\ket{\psi_k}\bra{\psi_k}\big)\ket{\psi_k}, for states $|{\psi }_k⟩=|{\psi }_{k_i}⟩\otimes \cdots \otimes |{\psi }_{k_n}⟩$ such that for all $i$, $\{|{\psi }_{k_i}⟩{\}}_{k_i=1}^4$ is a single-qubit SIC-POVM”, where a single-qubit SIC-POVM satisfies “$|⟨{\psi }_i⟩{\psi }_j{|}^2=\frac{1}{3}i\ne j$” and $d=2^n$.

Process fidelity (Definition 2): “F(\cE) = \bra{\phi}(\cI\otimes\cE)\big(\ket{\phi}\bra{\phi}\big)\ket{\phi}”. The maximally entangled state is displayed in §II as “$|\varphi ⟩=\frac{1}{\sqrt{d}}{\sum }_{x=1}^{d^2}|x⟩\otimes |x⟩$” for an orthonormal basis $\{|x⟩{\}}_{x=1}^d$ of ${\mathbb{C}}^d$; the upper summation index $d^2$ is inconsistent with that basis and with the normalization $1/\sqrt{d}$. The repository reads the sum over the $d$ basis labels, which gives the normalized maximally entangled vector (D5/S3/QuantumBounds/PeritoTsirelson.maxEntangledVector).

Theorem 1: “Let \cE be an $n$-qubit process with $0$-fidelity $F_0$ and process fidelity $F$. Then $1-\frac{3}{2}(1-F_0)\le F\le F_0.$”

Conjecture (§III): “we … find that the solution is $1-\frac{3}{2}(1-F_0)$, independent of $n$ for $n\le 4$. This leads us to conjecture that the lower bound in Th.~1 is tight for all $n$.”

Open problem (§IV): “A description of the worst-case process is an open problem.”

Scope

The lower bound of Theorem 1 is proved in the note through the spectrum of the operator $\Gamma$; its tightness for every $n$ and the worst-case processes are left open. For $n=1$ the product states form a 2-design and the lower bound holds with equality for every process.