Quality assessment of quantum teleportation through the distribution of fidelity

TL;DR

The paper addresses the limitation of single-number fidelity benchmarks by deriving the full fidelity probability distribution for single-qubit teleportation under two noise models, then introduces a universal certification framework based on prior importance functions (e.g., Beta distributions) that unifies moment-based and threshold-based criteria. It provides closed-form fidelity PDFs for classical measure-and-prepare protocols and for standard quantum teleportation with Bell-diagonal resources or local amplitude-damping noises, revealing nontrivial statistical features and asymmetries hidden by average fidelity. The results show that high-fidelity certification requires stronger entanglement or nonlocality and demonstrate that the so‑called fighting-noise-with-noise effect can be an artifact of the chosen prior. Overall, the framework enables tailored, application-specific teleportation benchmarks and clarifies when average fidelity is an adequate or misleading success criterion.

Abstract

In this work, we introduce a comprehensive statistical framework for assessing single-qubit quantum teleportation performance beyond the conventional average-fidelity benchmark. At first, we derive a closed-form expression for the full probability density function of actual teleportation fidelities and apply it to both classical measure-and-prepare schemes and standard quantum teleportation, considering two relevant noise models: Bell-diagonal resource states and local amplitude-damping channels. These results reveal that protocols with identical average fidelities can exhibit markedly different statistical behaviors, and that relying solely on average fidelity can mask inherent asymmetries introduced by local noise, potentially leading to spurious conclusions of symmetry. Secondly, we introduce a certification method based on prior importance functions (e.g., Beta distributions), which unifies moment-based criteria and threshold-based success probabilities into a single figure of merit. Applying this framework, we show that certifying high-fidelity teleportation requires increasingly stronger entanglement or non-locality, and we clarify that the so-called ``fighting noise with noise'' effect arises from the chosen prior importance function rather than representing a genuine advantage. Our approach thus provides versatile tools for tailored, application-specific teleportation benchmarks.

Figures (7)

• Figure 1: Plots of the probability density function $f^{\text{pf}}(F)$, Eq. \ref{['eq:distrProbBellDiagonal_phase_flip']}, for different values of $c_{\text{pf}}$, Eq. \ref{['eq:phaseflipfunc']}. This case corresponds to the standard quantum teleportation protocol for an ideal resource (Bell state $\ket{\Phi_1}$) affected by two local phase flip noises with parameters $p_A$ and $p_B$ related to $c_{\text{pf}}$ through Eq. \ref{['eq:phaseflipfunc2']}.
• Figure 2: Standard quantum teleportation protocol: Contour lines representing maximal $F^{\max}$ and minimal $F^{\min}$ fidelity values obtained for local amplitude-damping noise. These functions of noise parameters $p^A$ and $p^B$, do not depend on the outcome $k$ of the measurement, $F^{\min}(p^A,p^B) = F_{k}^{\min}(p^A,p^B)$ and $F^{\max}(p^A,p^B) = F_{k}^{\max}(p^A,p^B)$. Notably, these optimal values showcase the non-symmetry (with respect to $p^A\leftrightarrow p^B$) inherent to this noise model.
• Figure 3: Fidelity probability density function, $f^{\text{ad}}(F)$, Eq. \ref{['eq:distrprobtwoadc']}, for different local amplitude-damping noises $p^A$ and $p^B$: In the upper (lower) figures, noise is fixed in the system $B$ ($A$), $p^B=0.85$ ($p^A=0.85$), and increased in system $A$ ($B$). Plots vertically aligned (i.e., in the same column) display probability density functions that imply equal average fidelities despite having highly different statistical behaviors. $\sigma^{\text{ad}}$ is the fidelity standard deviation in each corresponding case.
• Figure 4: The importance prior function $\text{W}_{ \alpha , 1 } (F)$, Eq. \ref{['eq:importanceprior']}, for different values of $\alpha$ and $\beta=1$. Higher values of $\alpha$ assign greater importance to higher fidelity values.
• Figure 5: Tetrahedron of Bell-diagonal states: Set of diagonal correlation matrices $\mathbb{w}$ [Eq. \ref{['eq:BellDiagonalStates']}] corresponding to physical states. The green region stands for Bell-diagonal states satisfying the CHSH inequality. Black dots represent Bell-diagonal states leading to a non-successful certification ($\gamma_{\text{W}_{ \alpha , \beta } }(f^{\text{Bd}},f^\mathrm{mp}_{\mathrm{opt}})<0$, see Eq. \ref{['eq:criteriaPriorImportance']}), according to different importance prior functions $\{\text{W}_{ \alpha , 1 } \}_{\alpha=1}^6$, Eq. \ref{['eq:importanceprior']}. As the parameter $\alpha$ is increased, the teleportation assessment becomes more rigorous by assigning greater priotiry to higher fidelity values, thereby improving the teleportation quality (see Fig. \ref{['fig:importanceprior']}). We generated $20000$ points, uniformly distributed in the tetrahedron, and black colored only if they were non-successful. We see that as we assign more importance to higher fidelity values, more entangled states lead to a non-successful protocol. Non-local states may seem to hold a successful certification even for $\text{W}_{ 6 , 1 }$.

Theorems & Definitions (5)

• Proposition 1
• Proposition 2
• Corollary 1
• Proposition 3
• Proposition 4