Transfer of entanglement from nonlocal photon to non-Gaussian CV states

This paper proposes a mechanism to transfer quantum entanglement from a nonlocal photon to non-Gaussian continuous variable states, demonstrating that while the process is probabilistic with standard squeezed vacuum states, it can be made nearly deterministic (>98% probability) by utilizing single-photon-subtracted states, thereby offering an optimal trade-off between success probability and output brightness for quantum applications.

The Gist

Imagine you have two isolated islands (let's call them Island A and Island B). On each island, there is a quiet, calm lake representing a specific type of quantum state called a "squeezed vacuum." These lakes are beautiful, but they are completely separate; they don't know about each other, and they aren't connected.

Now, imagine you have a special, magical messenger bird (a nonlocal photon) that can fly to both islands at the exact same time. The goal of this research is to use this bird to make the two lakes "entangled"—meaning they become so deeply connected that what happens to one instantly affects the other, even though they never touched.

Here is the story of how the scientists achieved this, broken down into simple steps:

1. The Problem: The "Ghost" Connection

Usually, to connect two things, you have to bring them together or send a signal between them. But in the quantum world, sometimes you want to connect things that are far apart without them ever directly interacting.

The scientists wanted to take a single photon (the bird) that is already in a "superposition" state (it's flying over both islands simultaneously) and use it to transfer that connection to the two lakes.

2. The First Attempt: The "Quiet Lakes" (SMSV States)

First, they tried to connect the lakes using their natural, calm state (Single-Mode Squeezed Vacuum).

• The Setup: They sent the magical bird to interact with both lakes using a special mirror system (Beam Splitters).
• The Catch: To make the connection work perfectly, the mirrors had to be set up in a very specific way.
  • Scenario A (The "Super Reflective" Mirror): If they used mirrors that mostly bounced the light back, the connection happened almost 100% of the time! BUT, the price was high: the lakes became almost empty. The "brightness" (energy) of the connection dropped to near zero. It was like connecting two empty rooms; they were linked, but there was nothing in them to do anything useful.
  • Scenario B (The "Balanced" Mirror): If they used mirrors that let light pass through equally, the lakes stayed bright and useful. BUT, the connection only worked about 23% of the time. It was like trying to catch a specific type of fish; you might get lucky, but most of the time, you come home empty-handed.

The Lesson: You can't have both a perfect success rate and a bright, energetic connection using these "quiet lakes."

3. The Breakthrough: The "Active Lakes" (Odd CV States)

The scientists realized the problem was that the "quiet lakes" were too dominated by "vacuum" (empty space). So, they decided to change the lakes before the bird arrived.

They took the lakes and subtracted one photon from each. In quantum terms, this turned them into "Odd CV states." Think of this as adding a tiny, specific ripple to the water before the bird arrives.

• The Result: When they sent the magical bird to these "rippled" lakes, something amazing happened.
  • The connection became nearly 100% successful (probability > 98%).
  • Crucially, they could tune the mirrors to keep the lakes bright and energetic.

The Analogy: The "Herald" Technique

Imagine you are trying to get two people to shake hands across a room, but they are shy.

• Method 1 (Quiet Lakes): You throw a ball at them. Sometimes they catch it and shake hands (23% chance). If you make the room huge so they always catch it, the ball is so small they can't feel it (low brightness).
• Method 2 (Odd States): You first teach them a special handshake dance (subtracting a photon). Now, when you throw the ball, they are ready. They catch it almost every time, and the handshake is strong and visible.

Why This Matters

This research is a big deal for the future of the Quantum Internet.

• Reliability: We need quantum networks that work every time, not just when we get lucky. This new method makes the "entanglement transfer" nearly deterministic (guaranteed).
• Efficiency: It solves the trade-off. Before, you had to choose between "it works often" or "it works well." Now, you can have a system that works almost always and has enough energy to be useful for computing or sensing.
• No Direct Contact: The two quantum states (the lakes) never touched each other. The connection was purely mediated by the single photon and the measurement. This is like two people becoming best friends because they both talked to the same third person, without ever meeting each other.

In a Nutshell

The scientists found a way to use a single "magic bird" to tie two distant quantum systems together. By slightly tweaking the systems beforehand (subtracting a photon), they turned a gamble (23% success) into a near-certainty (98% success) without losing the power of the connection. This is a major step toward building a reliable, high-speed quantum internet.

Technical Summary

Here is a detailed technical summary of the paper "Transfer of entanglement from nonlocal photon to non-Gaussian quantum states" by Mikhail S. Podoshvedov and Sergey A. Podoshvedov.

1. Problem Statement

The paper addresses the challenge of generating Continuous Variable (CV) entanglement between two spatially separated quantum nodes without direct interaction between them.

• Context: While Discrete Variable (DV) entanglement (e.g., single photons) is well-established, generating high-quality CV entanglement (involving quadratures of light) is crucial for quantum communication, sensing, and computing.
• Limitations of Existing Methods:
  • Standard CV entanglement generation often relies on Gaussian operations (like beam splitters acting on squeezed vacuum), which are limited by vacuum noise dominance, resulting in low photon numbers (low brightness).
  • Entanglement swapping protocols typically require two initial entangled states and Bell State Measurements (BSM), which are resource-intensive and difficult to implement deterministically in optical systems.
  • Probabilistic protocols often suffer from exponentially decreasing success probabilities as networks scale.
• Goal: The authors propose a mechanism to transfer entanglement from a single nonlocal photon (a DV resource) to two initially separate Continuous Variable (CV) states in a manner that is either deterministic or nearly deterministic, while preserving high brightness and avoiding direct CV-CV interaction.

2. Methodology

The authors propose a Transfer of Quantum Entanglement (TQE) protocol. The core mechanism involves mixing two separated CV states with a single nonlocal photon using beam splitters (BS), followed by photon-number-resolving (PNR) detection.

A. The Setup

1. Input Resources:
   • Nonlocal Photon: A single photon in a maximally entangled state propagating in two spatial modes (3 and 4): $|\xi\rangle = \frac{1}{\sqrt{2}}(|01\rangle_{34} + |10\rangle_{34})$. This is generated by passing a single photon through a balanced beam splitter.
   • Target States: Two identical Single-Mode Squeezed Vacuum (SMSV) states in modes 1 and 2.
2. Interaction:
   • Mode 1 interacts with Mode 3 via Beam Splitter 1 ($BS_{13}$).
   • Mode 2 interacts with Mode 4 via Beam Splitter 2 ($BS_{24}$).
   • The beam splitters are identical with transmittance $t$ and reflectance $r$, parameterized by $B = r^2/t^2$.
3. Measurement:
   • PNR detectors in modes 3 and 4 register photon counts $k_1$ and $k_2$.
   • The detection of specific photon numbers projects the remaining modes (1 and 2) into an entangled CV state.

B. Two Scenarios Analyzed

The paper investigates two distinct input configurations:

1. Scenario 1: SMSV Inputs: Two standard squeezed vacuum states.
2. Scenario 2: Odd CV Inputs: Two "odd" CV states (generated by subtracting one photon from an SMSV state). These are non-Gaussian states with definite parity (odd photon number).

C. Theoretical Framework

• The protocol relies on measurement-induced entanglement. The joint detection of photons $k_1$ and $k_2$ conditions the output state.
• The output state is a superposition of two non-Gaussian states with different parities: $|\Psi\rangle \propto |\Psi^{(0)}\rangle_1 |\Psi^{(1)}\rangle_2 + b |\Psi^{(1)}\rangle_1 |\Psi^{(0)}\rangle_2$.
• Amplitude Distortion Factor ($b$): The degree of entanglement depends on a coefficient $b_{k_1 k_2}$. For maximal entanglement, $b$ must equal 1. If $b \neq 1$, the state is partially entangled.
• Optimization: The authors optimize the beam splitter parameter ($B$) and the initial squeezing ($S$) to maximize the probability of achieving $b=1$ (perfect TQE) while maintaining high output brightness (average photon number).

3. Key Results

A. TQE with SMSV Inputs (Gaussian to Non-Gaussian)

• Probabilistic Nature: When using standard SMSV states, the protocol is inherently probabilistic.
• Vacuum Dominance: The "no-click-no-click" ($k_1=k_2=0$) event has a very high probability if highly reflective beam splitters (HRBS) are used. However, this results in a final state with very low brightness (few photons), rendering it practically useless.
• Optimal Strategy: The authors found that the best trade-off between success probability and brightness occurs for the two-photon subtraction events ($k_1=0, k_2=2$ or $k_1=2, k_2=0$).
  • Success Probability: The maximum probability for generating maximally entangled states is $P \approx 0.2344$.
  • Parameters: Achieved with initial squeezing $S \approx 13.3$ dB and a near-balanced beam splitter ($B \approx 0.98$).
  • Limitation: While this preserves brightness, the protocol remains probabilistic (success rate < 25%).

B. TQE with Odd CV Inputs (Non-Gaussian to Non-Gaussian)

• Heralded Preparation: The input states are prepared by subtracting one photon from SMSV states, creating "odd" parity states.
• Deterministic Performance: This configuration drastically improves the protocol's efficiency.
  • Success Probability: The probability of generating maximal entanglement exceeds $P > 0.98$.
  • Mechanism: The "no-click" probability is significantly suppressed in odd states compared to SMSV states. The protocol effectively becomes nearly deterministic.
  • Brightness Trade-off: Initially, achieving this high probability required highly reflective beam splitters ($B \approx 276$), which reduced brightness. However, the authors demonstrated that by reducing $B$ (e.g., to $B=10$), the probability remains high ($P > 0.968$) while significantly increasing the average photon number (brightness) of the output state.

C. Robustness

• The paper analyzes the impact of detector inefficiency (quantum efficiency $\eta < 1$).
• With high-efficiency detectors ($\eta = 0.98$), the fidelity of the generated entanglement remains high ($F \approx 0.976$), and the success probability is largely unaffected.

4. Key Contributions

1. Novel Entanglement Transfer Mechanism: The paper proposes a method to transfer entanglement from a single nonlocal photon (DV) to two separated CV states without requiring a second entangled resource or Bell State Measurement (BSM).
2. Parity-Based Entanglement: It demonstrates how to generate parity-entangled non-Gaussian CV states deterministically.
3. Optimization of Non-Gaussian Resources: The authors show that using odd CV states (photon-subtracted squeezed states) as inputs transforms a probabilistic protocol into a nearly deterministic one ($>98\%$ success), solving the scalability issue of probabilistic entanglement distribution.
4. Brightness-Probability Trade-off: The study provides a detailed analysis of the trade-off between the beam splitter parameter ($B$), success probability, and output brightness, offering practical parameter sets for experimental implementation.

5. Significance

• Quantum Networking: This protocol offers a scalable solution for distributing entanglement in quantum networks. By achieving near-deterministic transfer, it overcomes the exponential decay of success probability that plagues standard probabilistic swapping.
• Hybrid Quantum Systems: It bridges the gap between Discrete Variable (photon-based) and Continuous Variable (field-based) quantum information, facilitating the conversion of qubits to continuous-variable qubits.
• Practical Implementation: The results suggest that high-brightness, maximally entangled CV states can be generated on-demand using current technology (squeezed light sources and transition-edge sensors), making it highly relevant for quantum metrology, sensing, and computation.
• Resource Efficiency: The protocol reduces resource requirements by needing only one entangled resource (the nonlocal photon) instead of two, simplifying the experimental setup.

In conclusion, the paper establishes that transferring entanglement from a nonlocal photon to odd non-Gaussian CV states is a superior strategy, enabling high-probability, high-brightness entanglement generation suitable for future quantum networks.