Two-detector reconstruction of multiphoton states in linear optical networks

We propose a method for partial state reconstruction of multiphoton states in multimode ( <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>N</a:mi> </a:math> -photon <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>M</b:mi> </b:math> -mode) linear optical networks (LONs) employing only two bucket photon-number-resolving detectors. The reconstructed Heisenberg-Weyl-reduced density matrix captures quantum coherence and symmetry with respect to Heisenberg-Weyl operators. Employing deterministic quantum computing with one qubit (DQC1) circuits, we reduce the detector requirement from <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi>M</c:mi> </c:math> to 2, while the requirement on measurement configurations is retained <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mn>2</d:mn> <d:msup> <d:mi>M</d:mi> <d:mn>3</d:mn> </d:msup> <d:mo>−</d:mo> <d:mn>2</d:mn> <d:mi>M</d:mi> </d:mrow> </d:math> . To ensure physicality, maximum likelihood estimation is incorporated into the DQC1 reconstruction process, with numerical simulations demonstrating the efficiency of our approach and its robustness against interferometer noises. This method offers a resource-efficient solution for state characterization in large-scale LONs to advance photonic quantum technologies.