[Paper Review] Generation of entangled states of two traveling modes for fixed number of photon

Xu‐Bo Zou, K. Pahlke+1 more|ArXiv.org|Oct 26, 2001

Quantum Information and Cryptography3 citations

TL;DR

This paper proposes a deterministic scheme to generate arbitrary two-mode photonic entangled states with a fixed total number of photons, using sequential beam splitter transformations and conditional zero-photon detection. The method enables preparation of any superposition state of the form $\Psi = \sum_{n=0}^{N} C_n |n\rangle_1 |N-n\rangle_2 $, achieving a success probability of 1/16 for generating a maximally entangled four-photon state, outperforming prior schemes.

ABSTRACT

We present an scheme to generate entangled state between two traveling modes for fixed number of photon, which is based on beam splitter transformation and conditional zero photon counters.

Motivation & Objective

• To develop a scalable, deterministic method for generating arbitrary two-mode photonic entangled states with a fixed total number of photons.
• To address the challenge of preparing non-Gaussian entangled states essential for quantum metrology and quantum information protocols.
• To improve the success probability of generating high-photon-number entangled states compared to existing schemes.
• To provide a theoretically feasible and experimentally realizable framework using linear optics and conditional measurements.

Proposed method

• The scheme uses a sequence of beam splitter transformations and conditional zero-photon detection to prepare entangled states of the form $\Psi = \sum_{n=0}^{N} C_n |n\rangle_1 |N-n\rangle_2 $.
• It leverages the algebraic structure of the state's coefficients to map the desired state to a product of displacement-like operators parameterized by roots of a characteristic polynomial.
• Each step involves injecting a single photon into a beam splitter, followed by conditional measurement of zero photons in one output port, which conditionally adds a photon to the other mode.
• The transformation sequence is derived from the Baker-Campbell-Hausdorff formula, enabling the mapping of the initial one-photon state to the target N-photon entangled state.
• The scheme is implemented via a cascade of beam splitters with tunable reflectance and transmittance, combined with on-off photon detectors to post-select vacuum outcomes.
• The success probability is calculated recursively based on the conditional probabilities of zero detection at each stage, with explicit formulas derived for the overall success rate.

Experimental results

Research questions

• RQ1 Can arbitrary two-mode entangled photonic states with a fixed total number of photons be generated using only linear optics and conditional measurements?
• RQ2 What is the optimal sequence of beam splitter operations and conditional detections to prepare a desired entangled state of fixed photon number?
• RQ3 How does the success probability scale with the number of photons in the target entangled state?
• RQ4 Can the scheme outperform existing protocols in terms of success probability for generating high-photon-number entangled states?
• RQ5 What are the experimental constraints, such as detector sensitivity and photon source requirements, that limit the feasibility of the scheme?

Key findings

• The scheme enables deterministic preparation of any two-mode entangled state of the form $\Psi = \sum_{n=0}^{N} C_n |n\rangle_1 |N-n\rangle_2 $ using only beam splitters and conditional zero-photon detection.
• The success probability for generating a maximally entangled four-photon state $\Psi = \frac{1}{\sqrt{2}}(|0\rangle_1|4\rangle_2 - |4\rangle_1|0\rangle_2 $ is 1/16.
• This success probability exceeds that of the scheme in Ref. [20], which achieves only 3/64 for the same state.
• The method is scalable in principle, as the required operations are defined recursively via the roots of a characteristic polynomial derived from the state coefficients.
• The scheme requires high-efficiency photon number-resolving detectors capable of distinguishing vacuum from one or more photons.
• The theoretical framework is validated by showing that the output state matches the target state under the correct choice of beam splitter parameters and detection conditions.