Saturation of the quantum Cramér-Rao bound for distributed sensing via error sensitivity in SU(1,1)-SU(<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi></mml:math>) interferometry

Breaking the standard quantum limit in the sensing of parameters at different spatial locations, such as in a quantum network, is of great importance. Using the framework of quantum Fisher information, many strategies based on squeezed quantum probes and multipath multiphoton or multiqubit entangled states have been considered. In this context there is always the question of what the simplest measurement is that would saturate the quantum Cramér-Rao bound (QCRB). The simplest quantity to measure would be characteristics of photon flux or population distribution in the case of qubits. Previous studies have shown that the error sensitivity in SU(1,1) interferometry, also known by several other terms such as nonlinear interferometry and time-reversed measurements, does saturate the QCRB for single parameters such as phase, displacement, and loss. In this work we reveal the great utility of generalized SU(1,1) interferometry in distributed sensing. Generalized SU(1,1) interferometry is a combination of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>SU</a:mi><a:mo>(</a:mo><a:mi>m</a:mi><a:mo>)</a:mo></a:mrow></a:math> and SU(1,1) elements, where <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>m</b:mi></b:math> is the number of nodes in the network. The <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mi>SU</c:mi><c:mo>(</c:mo><c:mi>m</c:mi><c:mo>)</c:mo></c:mrow></c:math> element is used to produce distributed entanglement starting from a squeezed photonic or matter probe. We demonstrate how error sensitivity measurement or more precisely the method of moment sensitivity at just one output port can saturate or nearly saturate the QCRB and thus results in Heisenberg sensitivity of network sensing.