Recovery of optical losses with the Petz recovery map

TL;DR

This work analyzes approximate recovery of information lost in a single optical mode due to Gaussian losses, using the Petz recovery map for a forward channel modeled as a beam splitter with transmissivity $\eta$. It derives the Petz map parameters for Gaussian lossy channels with thermal environment and thermal reference, showing the map is Gaussian and can be implemented as a beam splitter or an amplifier depending on the scenario, with explicit formulas for $X_P$, $Y_P$, and $\eta'$. The authors compare Petz recovery against naive benchmarks using fidelity, showing Petz often outperforms replacing the state by a prior and, for thermal inputs, outperforms keeping the noisy state, though there are regimes where preserving the noisy state is preferable. They demonstrate near-optimality of the Petz map among the considered recovery protocols, quantified via relative transmissivity and fidelity metrics, and provide physical intuition for when Petz acts as a splitter vs an amplifier. Overall, the results advance Gaussian error correction by offering a concrete, near-optimal recovery strategy for optical losses with thermal priors and illuminate when Petz recovery is advantageous in quantum optical information tasks.

Abstract

Optical systems are a main platform for quantum information processing, while a hidden challenge in these systems is information loss due to scattering into unmonitored modes, typically modeled as state-independent beam-splitter interactions. While such losses simply erase information encoded across modes, they directly degrade information encoded in the quantum state of a mode. Perfect correction of these Gaussian lossy channels with Gaussian operations alone is known to be impossible. In this work, we investigate the Petz recovery map as an approximate recovery. We construct the Petz recovery of single mode losses and its implementations. In particular, we show that the recovery performance of Petz recovery map is better than the recovery protocol that replaces the noisy state with the belief state. Also, when the reference state is far from the true state, it is better not to use the Petz recovery map but to leave the noisy state instead. We discuss the physical intuition of Petz recovery map and finally shows that it is near-optimal among the considered recovery protocols.

Figures (4)

• Figure 1: Setup of the noise model. In the forward channel, a beam splitter with transmissivity $\eta$ is used and the environment in the state $\xi$ will be traced out after the beam splitter.
• Figure 2: Comparison between the Petz recovery $\mathcal{P}_{\mathcal{N},\sigma}$ and the trivial recovery protocols $\mathcal{R}^{(0)},\mathcal{R}^{(1)}_{\sigma}$. A forward noise model considered here is a beam splitter of transmissivity $\eta=0.5$. Subfigures (a)–(c) correspond to the case where the environment of the forward channel is a thermal state with mean photon number $\overline{n}_\xi = 10$. In this regime, the Petz recovery map operates as a beam splitter over $0 \leq \overline{n}_\sigma \leq 10$. Subfigures (d)–(f) correspond to the case where the environment of the forward channel is the vacuum state, i.e., $\overline{n}_\xi = 0$. In this case, the Petz recovery map operates as a phase insensitive amplifier over $0 \leq \overline{n}_\sigma \leq 10$. We consider a reference state $\sigma$ that is also a thermal state with its photon number $\overline{n}_\sigma$. The figure shows the plots of the fidelity against $\overline{n}_\sigma$ between a input state and a state recovered by $\mathcal{P}_{\mathcal{N},\sigma}$ (green line), $\mathcal{R}^{(1)}_\sigma$ (purple line), and $\mathcal{R}^{(0)}$ (dashed blue line), while considering three input states: (a) and (d) thermal state $V_{th}=(2\overline{n}_{th}+1)I$ with mean photon number $\overline{n}_{th}=2$, (b) and (e) a squeezed state $V_{sq} = \rm{diag}(2.5,10)$ with $\overline{\textbf{r}}_{sq} =0$, (c) and (f) a coherent state $\ket{\alpha}$ with amplitude $\alpha = 1/2\sqrt{2}(1+i)$. In all three plots, the dotted line indicates when respective recovery protocols achieves the optimal fidelity.
• Figure 3: Comparison between the Petz recovery $\mathcal{P}_{\mathcal{N},\sigma}$ and and other recovery protocols $\mathcal{R}\in \mathfrak{R}_{\mathcal{N},\sigma}$. The transmissivity of the forward channel is set to be $\eta=0.5$. The dotted line is a guide for the eye to show where the transmissivity of Petz recovery map lies. The mean photon numbers of the environment and the mean photon numbers of the thermal reference states are (a) $\overline{n}_\xi = 0$ and $\overline{n}_\sigma = 4$, (b) $\overline{n}_\xi = 2$ and $\overline{n}_\sigma = 6$, and (c) $\overline{n}_\xi = 10$ and $\overline{n}_\sigma = 4$, respectively. Notably, (a) and (b) are the cases where the Petz recovery map is a phase-insensitive amplifier while in (c), it is a beam splitter. The three input state are considered in all of these cases; thermal state $V_{th}=(2\overline{n}_{th}+1)I$ with mean photon number $\overline{n}_{th}=2$ (green line); a squeezed state $V_{sq} = \rm{diag}(2.5,10)$ with $\overline{\textbf{r}}_{sq} =0$ (blue line); and a coherent state $\ket{\alpha}$ with amplitude $\alpha = 1/2\sqrt{2}(1+i)$ (purple line).
• Figure 4: The relative transmissivity difference $\eta_{\text{rel}}(\mathcal{P})$ and the relative fidelity difference $F_{\text{rel}}(\rho, \mathcal{P})$ against the transmissivity $\eta$ of the forward channel $\mathcal{N}$. The purple line is the case where the environment of the forward channel is a thermal state with $\overline{n}_\xi = 10$ and the reference state is a thermal state with $\overline{n}_\sigma = 4$, such that $\eta_\mathcal{P} \leq 1$. The green line is the case where the environment of the forward channel is a thermal state with $\overline{n}_\xi = 2$ and the reference state is another thermal state with $\overline{n}_\sigma = 6$, such that $\eta_\mathcal{P} >1$. Here, we consider three input states; thermal state $V_{th}=(2\overline{n}_{th}+1)I$ with mean photon number $\overline{n}_{th}=2$ (solid line); a squeezed state $V_{sq} = \rm{diag}(2.5,10)$ with $\overline{\textbf{r}}_{sq} =0$ (dotted line); and a coherent state $\ket{\alpha}$ with amplitude $\alpha = 1/2\sqrt{2}(1+i)$ (dashed line).

Theorems & Definitions (2)

• proof
• proof