Robustness and optimization of N00N-state interferometry

Abstract

Quantum-enhanced interferometry is often discussed in terms of ideal resources and asymptotic scalings, whereas in practice its performance is set by a delicate interplay between losses, state imbalance, and photon number. We address this interplay in a folded Franson interferometer fed with partially entangled N00N states, treating asymmetric losses and tunable input imbalance on equal footing. From exact detection probabilities we obtain closed-form expressions for the fringe visibility and the Fisher information, and show that these two figures of merit respond very differently to imperfections. In particular, we demonstrate that perfect interference contrast can always be recovered by compensating loss asymmetry with an appropriate input imbalance, while the Fisher information generally peaks at a distinct operating point, reflecting the irreducible trade-off between coherence restoration and signal attenuation. By determining the exact optima and benchmarking against single-photon strategies, we identify the critical loss and minimum entanglement required to maintain a genuine quantum advantage over optimized single-photon strategies under identical loss conditions, and establish their scaling with the photon number N . Beyond delineating the fundamental trade-offs between loss, entanglement, and sensitivity, this work establishes a comprehensive theoretical framework that both underpins and extends the experimental demonstration of quantum advantage reported in [1], providing a unified description of the relevant operating regimes.

Figures (6)

• Figure 1: General scheme of the interferometric configuration considered in this work, based on a N00N state propagating in a Mach--Zehnder--like folded Franson interferometer. The two arrows represent the lower and upper arms of the interferometer. A relative phase shift and relative losses are introduced in the upper arm and are modeled by the unitary transformations $\hat{U}_{\phi}$ and $\hat{U}_{L}$, respectively.
• Figure 2: (a) Fringe visibility [Eq. (\ref{['eq: visibility']})] as a function of the entanglement parameter $\alpha$ for fixed relative losses ($p=0.3$), for $N=1,2,$ and $5$ photons (blue, red, and green curves, respectively). Dotted lines indicate the global maxima, showing that the visibility can be fully restored by an appropriate tuning of $\alpha$. (b) Optimal entanglement parameter $\alpha_{v}$ given by Eq. (\ref{['eq: Vis opt t']}) as a function of the relative losses, yielding the maximum visibility; in the lossless limit the optimum corresponds to a maximally entangled state. (c) Visibility as a function of the relative losses $p$ for a fixed degree of entanglement ($\alpha=0.3$). Dotted lines mark the local maxima. Increasing the photon number makes the visibility increasingly sensitive to losses. (d) Optimal relative loss $p_{v}$ given by Eq. (\ref{['eq: Vis opt p']}) as a function of the entanglement parameter, maximizing the visibility.
• Figure 3: (a) Output detection probabilities for $N=1$ (blue) and $N=2$ (red) photons, for relative loss $p=0.1$ and entanglement parameter $\alpha=0.4$. The dotted curves correspond to the total probability given by Eq. (\ref{['eq: energy conservation']}). (b) Corresponding Fisher information computed from Eq. (\ref{['eq: Fisher info 2']}). The lower and upper dotted horizontal lines indicate the standard quantum limit and the Heisenberg limit, respectively.
• Figure 4: (a) Fisher information [Eq. (\ref{['eq: Fisher info max']})] as a function of the relative loss $p$ for a fixed entanglement parameter $\alpha=0.2$ and $N=1,2,$ and $5$ photons (blue, green, and red curves, respectively). At low loss, increasing $N$ enhances the sensitivity, but also makes it more fragile to attenuation. (b) Optimal relative loss maximizing the Fisher information [Eq. (\ref{['eq: FI opt p']})], showing that for any $\alpha$ and $N$ the maximum is always reached in the lossless limit. (c) Maximum Fisher information at optimal loss [Eq. (\ref{['eq: FI max opt p']})] as a function of $\alpha$, exhibiting a peak at $\alpha=1/2$ and a linear scaling with $N$. (d) Fisher information as a function of $\alpha$ for $p=0.2$; dotted lines indicate the optimal values. (e) Optimal entanglement parameter $\alpha_{\mathrm{opt}}$ maximizing the Fisher information [Eq. (\ref{['eq: FI opt t']})] as a function of the relative loss. The optimum shifts toward increasingly unbalanced states as $p$ increases. (f) Maximum Fisher information at optimal entanglement [Eq. (\ref{['eq: FI max opt t']})] as a function of $p$. The sensitivity decreases monotonically with loss, with a faster degradation for larger photon numbers.
• Figure 5: Boundaries for surpassing the SQL with optimized parameters as a function of $N$. Purple curve shows the minimum degree of entanglement from Eq. \ref{['eq: sup opt p']} at zero relative losses. The orange curve shows the maximum tolerable relative loss from Eq. \ref{['eq: sup opt t']} when $\alpha$ is optimized. As $N$ increases, the tolerance to imperfect entanglement increase while the maximum acceptable relative losses decreases.