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arXiv:2502.00467 · Physical Review A

Extended analysis of distillation and purification of squeezed states of light

Jaromı́r Fiurášek, Stephan Grebien, Roman Schnabel

Published Apr 3, 2025• 8 citations
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Abstract

Previously, the authors experimentally used two-photon subtraction to achieve distillation of single-mode optical squeezed states. Here, they present an extended theoretical study of distillation and purification of such states by conditional photon subtraction. They find parallels between their squeezing distillation scheme and recent schemes for the generation of approximate GKP states in fault-tolerant quantum computing.

DOI
10.1103/physreva.111.043704

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Previously, the authors experimentally used two-photon subtraction to achieve distillation of single-mode optical squeezed states. Here, they present an extended theoretical study of distillation and purification of such states by conditional photon subtraction. They find parallels between their squeezing distillation scheme and recent schemes for the generation of approximate GKP states in fault-tolerant quantum computing.
Recently, the strong single-mode squeezed vacuum state of light with −15-15−15−15 dB of classic noise has been created [Phys. Rev. Lett.117, 110801 (2016)10.1103/PhysRevLett.117.110801]. This is an important resource in continuous variable (CV) quantum communication and fault-tolerant quantum computing. Nevertheless, strong squeezing means a non-negligible population in high-photon-number subspace and may pose a strong challenge to photon-based technology, for example, entanglement distillation. Entanglement distillation is an efficient method for retrieving a higher quality of entanglement from a weakly entangled state. Up till now, almost all the schemes for entanglement distillation are restricted to the low-squeezing regime. The distillation of strong-squeezing induced entanglement is an interesting but open topic in the near future. Here, we take the single-mode squeezed entangled state (SMSE) as an example and show that conventional photon subtraction based distillation fails to generate a higher quality of entanglement with experimentally feasible optical beam splitters. To this point, we show that the superposition of photon annihilation and creation could be an effective method. A practical scheme for implementing such a superposition with on-line squeezing is suggested. Our method is verified numerically by the calculation of a 25 dB SMSE state with an ideal photon detector and for distillation of 20.5 dB SMSE state with practical dichotic on-off detectors. The analysis of the distillation of the strong squeezing effect could be extended straightforwardly to other squeezing-based quantum information processing, such as quantum metrology and quantum illumination.
A theoretical framework is developed to analyze the generation of the optical GKP state using a single-photon-added squeezed vacuum. This state, defined by the squeezing parameter rrr, is injected into a 50:50 beam splitter, and the optical GKP state is obtained through conditional measurement at one output port. The single-photon-added squeezed vacuum is especially prominent in this context because it provides a simpler and more experimentally accessible ingredient than Schrodinger cat states, while conditional measurement ensures projection onto a state that closely approximates the finite-energy GKP form. Fidelity is employed to quantify this closeness, and the analysis demonstrates that the scheme achieves a maximum fidelity of 85% at a squeezing level of 3.76 dB3.76 \ \text{dB}3.76 dB. This performance surpasses approaches based on squeezed optical odd Schrodinger cat states, underscoring the single-photon-added squeezed vacuum as a practical and effective pathway toward fault-tolerant photonic quantum computing.
Photon subtraction from squeezed states is a powerful scheme to create good approximation of so-called Schr\"odinger cat states. However, conventional continuous-wave-based methods actually involve some impurity in squeezing of localized wave packets, even in the ideal case of no optical losses. Here, we theoretically discuss this impurity by introducing mode match of squeezing. Furthermore, here we propose a method to remove this impurity by filtering the photon-subtraction field. Our method in principle enables creation of pure photon-subtracted squeezed states, which was not possible with conventional methods.
Abstract Squeezed states of light have been improving the sensitivity of gravitational-wave observatories and are nonclassical resources of quantum cryptography and envisioned photonic quantum computers. The higher the squeeze factor is, the higher is the quantum advantage. Almost all applications of squeezed light require multi-path optical interference, whose unavoidable imperfections introduce optical loss, degrade the squeeze factor, as well as the quantum advantage. Here, for the first time, we experimentally demonstrate the distillation of Gaussian squeezed states that suffered from Gaussian photon loss. Our demonstration already involves two distillation steps. The first step improved the squeeze factor from 2.4 dB to 2.8 dB by the subtraction of two photons. The second step improved the value from 2.8 dB to 3.4 dB by a Gaussification protocol. It was realised on data measured at different times via an 8-port balanced homodyne detector and via data post-processing. The number of distillation steps can be increased by longer data sampling times, without additional hardware. We propose and discuss the application to quantum cryptography and photonic quantum computers.
A near-deterministic source of non-Gaussianity is the missing piece to reach universal and fault-tolerant quantum computing with continuous-variable optics. Due to a lack of strong non-linearities, the probabilistic photon-subtracted Gaussian states remain the most promising contender. In combination with Gaussian measurements and single-mode feed-forward operations, they have been shown to provide sufficient non-Gaussianity at high rates. However, these existing breeding protocols lack the necessary loss tolerance to be experimentally feasible. In this work, we introduce a mathematical formalism based on the application of a Gaussian blur and filter on the measured Fock Wigner function to describe general photon-subtracted Gaussian states. Within our representation, the initial squeezing, Gaussian measurements, and photon loss can all be understood as contributions to the total blurring and interchanged accordingly. We find that this intuitive approach can be used to find improvements to cat, cubic phase, and GKP state generation by compromising between success rates, feed-forward compatibility, and loss tolerance. In the case of multiple photon subtractions, this is achieved by introducing a hybrid setup variant bridging the gap between protocols based on breeding and post-selection. Finally, we establish a general lower bound on the resources needed to generate a given target state near-deterministically by introducing the expected Wigner logarithmic negativity as an additive monotone of non-Gaussianity.