We propose a measurement setup reaching a Heisenberg scaling precision for the estimation of any distributed parameter φ (not necessarily a phase) encoded into a generic M-port linear network composed only of passive elements. The scheme proposed can be easily implemented from an experimental point of view since it employs only Gaussian states and Gaussian measurements. Due to the complete generality of the estimation problem considered, it was predicted that one would need to carry out an adaptive procedure which involves both the input states employed and the measurement performed at the output; we show that this is not necessary: Heisenberg scaling precision is still achievable by only adapting a single stage. The non-adapted stage only affects the value of a pre-factor multiplying the Heisenberg scaling precision: we show that, for large values of M and a random choice of the non-adapted stage, this pre-factor takes a typical value which can be controlled through the encoding of the parameter φ into the linear network.
Abstract We consider the estimation of an arbitrary parameter φ, such as the temperature or a magnetic field, affecting in a distributed manner the components of an arbitrary linear optical passive network, such as an integrated chip. We demonstrate that Heisenberg scaling precision (i.e. of the order of 1/N, where N is the number of probe photons) can be achieved without any iterative adaptation of the interferometer hardware and by using only a simple, single, squeezed light source and well-established homodyne measurements techniques. Furthermore, no constraint on the possible values of the parameter is needed but only a preliminary shot-noise estimation (i.e. with a precision of N ) easily achievable without any quantum resources. Indeed, such a classical knowledge of the parameter is enough to prepare a single, suitable optical stage either at the input or the output of the network to monitor with Heisenberg-limited precision any variation of the parameter to the order of 1 / N without the need to iteratively modify such a stage.
The simultaneous estimation of multiple parameters is a central task in quantum metrology, distributed sensing, and the calibration of large photonic interferometers. A fundamental question is how many independent parameter combinations can inherit Heisenberg scaling from a given number of squeezed probes in a multimode Gaussian network. Here, we answer this question for arbitrary passive linear optical networks. For a
p-parameter,
M-channel interferometer probed by
k single-mode squeezed states and at least one coherent state in the remaining channels, we show that the rank of the Heisenberg-scaling coefficient of the quantum Fisher information matrix is bounded by
nHS≤min{p,k(k+3)/2}, which corresponds to the maximum number of independent combinations of parameters that can be estimated with Heisenberg-scaling sensitivity. The bound separates into two geometrically distinct contributions. The covariance contribution of the quantum Fisher information, which describes squeezing-enhanced fluctuations, provides at most
k(k+1)/2 parameter combinations estimable at Heisenberg-scaling sensitivity, while the first-moment contribution provides at most
k additional independent parameter combinations with Heisenberg-scaling sensitivity. We identify the conditions for saturating these bounds and construct a passive family of interferometers that saturates these bounds.
We propose an NN-photon Gaussian measurement scheme which allows the estimation of a parameter φφ encoded into a multi-port interferometer with a Heisenberg scaling precision (i.e. of order 1/N1/N). In this protocol, no restrictions on the structure of the interferometer are imposed other than linearity and passivity, allowing the parameter φφ to be distributed over several components. In all previous proposals Heisenberg scaling has been obtained provided that both the input state and the measurement at the output are suitably adapted to the unknown parameter φφ. This is a serious drawback which would require in practice the use of iterative procedures with a sequence of trial input states and measurements, which involve an unquantified use of additional resources. Remarkably, we find that only one stage has to be adapted, which leaves the choice of the other stage completely arbitrary. We also show that our scheme is robust against imperfections in the optimized stage. Moreover, we show that the adaptive procedure only requires a preliminary classical knowledge (i.e to a precision 1/N−−√1/N) on the parameter, and no further additional resources. As a consequence, the same adapted stage can be employed to monitor with Heisenberg-limited precision any variation of the parameter of the order of 1/N−−√1/N without any further adaptation.
We address the issue of precisely estimating small parameters encoded in a general linear transformation of the modes of a bosonic quantum field. Such Bogoliubov transformations frequently appear in the context of quantum optics. We provide a set of instructions for computing the quantum Fisher information for arbitrary pure initial states. We show that the maximally achievable precision of estimation is inversely proportional to the squared average particle number and that such Heisenberg scaling requires nonclassical but not necessarily entangled states. Our method further allows us to quantify losses in precision arising from being able to monitor only finitely many modes, for which we identify a lower bound.