Adapting coherent-state superpositions in noisy channels

TL;DR

This work addresses preserving non-Gaussian resources, notably Wigner-function negativity, for coherent-state superpositions propagating through noisy bosonic channels. It introduces an adaptive pre-squeezing strategy that deterministically protects CS states by matching the channel’s loss and noise characteristics, deriving a negativity-preserving condition $f_X^2 f_P^2 - σ_X σ_P > 0$ and extending it to even-parity states. The authors extend the single-channel analysis to concatenated Gaussian channels, showing composite channels are effectively reducible to an equivalent single channel with parameters $η_e$ and $V_e$, and provide guidance on optimal pre- and mid-squeezing rates. Complementary to central negativity, they define the Hilbert-Schmidt distance $Δ_ξ$ between opposite-parity CS states as a measurable figure of merit, deriving a closed-form expression and showing adaptation can enhance this distance, with the optimal squeezing parameters exhibiting systematic dependence on transmittance. The results offer a practical, implementable path to protecting non-Gaussian quantum states in optical and other bosonic platforms, potentially benefiting quantum computation, communication, and hybrid DV-CV architectures.

Abstract

Quantum non-Gaussian states are crucial for the fundamental understanding of non-linear bosonic systems and simultaneously advanced applications in quantum technologies. In many bosonic experiments the important quantum non-Gaussian feature is the negativity of the Wigner function, a cornerstone for quantum computation with bosons. Unfortunately, the negativities present in complex quantum states are extremely vulnerable to the effects of decoherence, such as energy loss, noise and dephasing, caused by the coupling to the environment, which is an unavoidable part of any experimental implementation. An efficient way to mitigate its effects is by adapting quantum states into more resilient forms. We propose an optimal protection of superpositions of coherent states against a sequence of asymmetric thermal lossy channels by suitable squeezing operations.

Figures (6)

• Figure 1: The signal (solid line) is transmitted through a lossy channel represented by a beam splitter with transmittance $\eta$ where it interacts with the environment (dashed line). The environment is assumed to be in an axis-aligned asymmetric thermal state, which can be interpreted as a symmetric thermal state characterized by its variance $V$ and the associated squeezing rate $\gamma_{t}$. The signal state can be protected against decoherence with an optional pre-squeezing operation. Its squeezing rate $\gamma$ can be adapted to offer the best protection of the transmitted CS state.
• Figure 2: Adaptive pre-squeezing protects odd-parity CS state (with ${\sqrt{2}\xi = 3}$) from decoherence due to loss and thermal noise with varying asymmetry. (a) Dashed lines represent CN without protection. Asymmetries of the thermal noise with ${V = 1}$ (twice the variance of the vacuum state) are represented with different colors. We use black for the symmetric case with ${\gamma_{t} = 0}$, red for ${\gamma_{t} = 1 \mathrm{dB}}$, orange for ${\gamma_{t} = 3 \mathrm{dB}}$, blue for ${\gamma_{t} = 5 \mathrm{dB}}$, and purple for ${\gamma_{t} = 6 \mathrm{dB}}$ asymmetric cases. The solid black line represents the best attainable CN and does not depend on the $\gamma_{t}$ rate of the asymmetry. (b) Optimal pre-squeezing rates $\gamma$ follow the same color scheme. Colored lines represent optimal rates for different asymmetries. The lines appear constantly shifted by the value of $\gamma_{t}$ from the solid black line, which represents the optimal pre-squeezing $\gamma$ found for the symmetric thermal state (${\gamma_{t} = 0}$). The colored bullet points correspond to $\gamma_{t}$ added to its values at regular intervals to emphasise the constant shifts.
• Figure 3: Adaptive pre-squeezing protects odd-parity CS state (with ${\sqrt{2}\xi = 3}$) from decoherence due to loss and thermal noise. We consider symmetric thermal noise, characterized by its variance ${V \in \{ 0.5, 1.0, 1.5, 2.0 \}}$. Colors are used to distinguish between individual variances. (a) Dashed lines represent CN without protection. Solid lines correspond to CN attained by optimally pre-squeezed CS states. (b) Optimal pre-squeezing rate $\gamma$ does not exhibit trivial dependence on the variance $V$ of the thermal noise.
• Figure 4: Illustration of a composite channel comprising a pair of lossy channels with pre-squeezing ($\gamma$) and mid-squeezing ($\gamma'$). The first channel is parametrized by ($V, \gamma_{t}$) describing the asymmetric thermal state and $\eta$ determining its transmittance. Parameters of the second channel are distinguished by primes.
• Figure 5: Adaptive squeezing protects odd-parity CS state (${\sqrt{2} \xi = 3}$) from decoherence due to interaction with asymmetric thermal states with $V = 1$, ${\gamma_{t} = \{ -2, -1, 1, 2 \}\;\mathrm{dB}}$ in the first channel and ${V' = 2}$, ${\gamma_{t}' = 1 \mathrm{dB}}$ in the second channel. Solid black lines represent the cases where both thermal states are symmetric. (a) Attainable central negativity. Dashed lines represent the attainable negativity without any adaptation. After adaptation, a part of which effectively symmetrizes the environment, the best attainable negativities coincide. The solid black line determines this best attainable negativity. (b) Optimal pre-squeezing rate $\gamma_{t}$ where the solid black line represents the optimal adaptation in the fundamental case when both thermals states of the environment are symmetric. Colored lines correspond to optimal pre-squeezing rates for asymmetric environments. These lines are shifted by a constant offset equal to $\gamma_{t}$. This fact is emphasized by the colored bullets that are obtained by adding $\gamma_{t}$ to the fundamental pre-squeezing rate represented by the solid black line. (c) Optimal mid-squeezing rate $\gamma'_{t}$ depends only on the asymmetry of the adjacent thermal states. Its value is determined by the difference $\gamma'_{t} - \gamma_{t}$.