Quantum Beam Splitter as a Deterministic Entanglement Source: Exact Photon-Number Statistics, Hong-Ou-Mandel Interference, and von Neumann Entropy Scaling from Balanced Inputs
Abstract
In this work, we present a comprehensive analytical study of a lossless quantum beam splitter (BS) with transmission coefficient T and reflection coefficient R. Using the unitary operator representation and binomial expansion, we derive the exact probability distribution P_k for detecting k photons in one output port when the input ports are prepared in Fock states |N>_0 |M>_1. The balanced configuration R = T = 1/2 and N = M is examined in detail: we prove that destructive quantum interference suppresses all odd photon numbers at the outputs, leaving only even photon counts. The special case N = M = 1 reproduces the famous Hong-Ou-Mandel (HOM) dip with P_1 = 0. Beyond photon statistics, we demonstrate that the beam splitter functions as a deterministic, tunable source of photon-number entanglement. By tracing over one output mode, we construct the reduced density matrix rho_2 and compute the von Neumann entropy S = - sum_k P_k log2 P_k, a direct measure of entanglement between the two output ports. For the balanced case with N = M = 1, we obtain S = 1 ebit, i.e., maximal Bell-state entanglement. For larger equal photon numbers, the entropy increases monotonically. Our results transform the passive beam splitter into a deterministic source of photon-number entanglement, with direct applications in optical quantum computing, interferometry, and quantum information processing. Numerical three-dimensional plots of the probability distribution are provided, and all derivations are supported by rigorous algebra and a full appendix.
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BibTeX
@misc{khansari2026quantum,
title = {{Quantum Beam Splitter as a Deterministic Entanglement Source: Exact Photon-Number Statistics, Hong-Ou-Mandel Interference, and von Neumann Entropy Scaling from Balanced Inputs}},
author = {Seyed Mohsen Moosavi Khansari},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.02151},
eprint = {2605.02151},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.02151},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Quantum Beam Splitter as a Deterministic Entanglement Source: Exact Photon-Number Statistics, Hong-Ou-Mandel Interference, and von Neumann Entropy Scaling from Balanced Inputs
AU - Seyed Mohsen Moosavi Khansari
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2605.02151
UR - https://arxiv.org/abs/2605.02151
AB - In this work, we present a comprehensive analytical study of a lossless quantum beam splitter (BS) with transmission coefficient T and reflection coefficient R. Using the unitary operator representation and binomial expansion, we derive the exact probability distribution P_k for detecting k photons in one output port when the input ports are prepared in Fock states |N>_0 |M>_1. The balanced configuration R = T = 1/2 and N = M is examined in detail: we prove that destructive quantum interference suppresses all odd photon numbers at the outputs, leaving only even photon counts. The special case N = M = 1 reproduces the famous Hong-Ou-Mandel (HOM) dip with P_1 = 0. Beyond photon statistics, we demonstrate that the beam splitter functions as a deterministic, tunable source of photon-number entanglement. By tracing over one output mode, we construct the reduced density matrix rho_2 and compute the von Neumann entropy S = - sum_k P_k log2 P_k, a direct measure of entanglement between the two output ports. For the balanced case with N = M = 1, we obtain S = 1 ebit, i.e., maximal Bell-state entanglement. For larger equal photon numbers, the entropy increases monotonically. Our results transform the passive beam splitter into a deterministic source of photon-number entanglement, with direct applications in optical quantum computing, interferometry, and quantum information processing. Numerical three-dimensional plots of the probability distribution are provided, and all derivations are supported by rigorous algebra and a full appendix.
ER -