Determining the phase in one arm of a quantum interferometer is discussed taking into account the three non-ideal aspects in real experiments: non-deterministic state preparation, non-unitary state evolution due to losses during state propagation, and imperfect state detection. A general expression is written for the probability of a measurement outcome taking into account these three non-ideal aspects. As an example of applying the formalism, the classical Fisher information and fidelity (Shannon mutual information between phase and measurements) are computed for few-photon Fock and N00N states input into a lossy Mach-Zehnder interferometer. These three non-ideal aspects lead to qualitative differences in phase estimation, such as a decrease in fidelity and Fisher information that depends on the true value of the phase.
Atomic Interferometer has two quantum unitary gates that must be realized for quantum sensing purposes from atomic gravimeter and atomic interferometer gyroscope. An optimal cost function which define the distance between two unitary operator is defines and based on it a pulse based stochastic gradient descent algorithm is derived for implementation of atomic mirror and atomic beam-splitter gates. By using numerical optimal control for those gate realization, we achieved to gate fidelity of 0.995 and 0.998 for mirror and beam-splitter gates, respectively. For this research, a two-level atomic …
Interferences in multi-path systems for single and multiple particles are theoretically analyzed. A holistic method is presented, which allows to construct the unitary transition matrix describing interferometers for any port number d and particle number n. This work has two centerpieces. First, a solution algorithm for realizing any desired unitary transformation in an interferometer for the single particle case is demonstrated. Secondly, a simple scheme for calculating the rotation matrices of beam splitters and phase shifters for multiple particles is shown. Such linear optical devices are …
Photon counting measurements are analyzed for obtaining a classical phase parameter in linear Mach Zehnder interferometer (MZI), by the use of phase estimation theories. The detailed analysis is made for four cases: a) Coherent states inserted into the interferometer. b) Fock number state inserted in one input port of the interferometer and the vacuum into the other input port. c) Coherent state inserted into one input port of the interferometer and squeezed-vacuum state in the other input port. d) Exchanging the first beam-splitter (BS1) of a MZI by a non-linear system which inserts a NOON st…
In this work we investigate the problem of simultaneous estimation of phases using generalised three- and four-mode Mach-Zehnder interferometer. In our setup, we assume that the phases are placed in each of the modes in the interferometer, which introduces correlations between estimators of the phases. These correlations prevent simultaneous estimation of all these phases, however we show that we can still obtain the Heisenberg-like scaling of precision of joint estimation of any subset of $d-1$ phases, $d$ being the number of modes, within completely fixed experimental setup, namely with the …
We develop the learning algorithm to build the architecture agnostic model of the reconfigurable optical interferometer. Programming the unitary transformation on the optical modes of the interferometer either follows the analytical expression yielding the unitary matrix given the set of phaseshifts or requires the optimization routine if the analytic decomposition does not exist. Our algorithm adopts the supervised learning strategy which matches the model of the interferometer to the training set populated by the samples produced by the device under study. The simple optimization routine use…
The quantum stochastic phase estimation has many applications in the precise measurement of various physical parameters. Similar to the estimation of a constant phase, there is a standard quantum limit for stochastic phase estimation, which can be obtained with the Mach-Zehnder interferometer and coherent input state. Recently, it has been shown that the stochastic standard quantum limit can be surpassed with non-classical resources such as the squeezed light. However, practical methods to achieve the quantum enhancement in the stochastic phase estimation remains largely unexplored. Here we pr…
We describe a direct method to experimentally determine local two-qubit invariants by performing interferometric measurements on multiple copies of a given two-qubit state. We use this framework to analyze two different kinds of two-qubit invariants of Makhlin and Jing et. al. These invariants allow to fully reconstruct any two-qubit state up to local unitaries. We demonstrate that measuring 3 invariants is sufficient to find, e.g., the optimal Bell inequality violation. These invariants can be measured with local or nonlocal measurements. We show that the nonlocal strategy that follows from M…
We theoretically study the phase sensitivity of an SU(1,1) interferometer with a thermal state and squeezed vacuum state as inputs and parity detection as measurement. We find that phase sensitivity can beat the shot-noise limit and approaches the Heisenberg limit with increasing input photon number.
By performing two local displacement operations (LDOs) inside an SU(1,1) interferometer, called as the displacement-assisted SU(1,1) [DSU(1,1)], both the phase sensitivity based on homodyne detection and quantum Fisher information (QFI) with and without photon losses are investigated in this paper. In this DSU(1,1) interferometer, we focus our attention on the extent to which the introduced LDO affects the phase sensitivity and the QFI, even in the realistic scenario. Our analyses show that the estimation performance of DSU(1,1) interferometer is always better than that of SU(1,1) interferomet…
The precise estimation of the gravitational acceleration is important for various disciplines. We consider making such an estimation using quantum optics. A Mach-Zehnder interferometer in an "optical fountain" type arrangement is considered and used to define a standard quantum limit for estimating the gravitational acceleration. We use an approach based on quantum field theory on a curved, Schwarzschild metric background to calculate the coupling between the gravitational field and the optical signal. The analysis is extended to include the injection of a squeezed vacuum to the Mach-Zehnder a…
Quantum metrology is the state-of-the-art measurement technology. It uses quantum resources to enhance the sensitivity of phase estimation beyond what reachable within classical physics. While single parameter estimation theory has been widely investigated, much less is known about the simultaneous estimation of multiple phases, which finds key applications in imaging and sensing. In this manuscript we provide conditions of useful entanglement (among multimode particles, qudits) for multiphase estimation and adapt them to multiarm Mach-Zehnder interferometry. We discuss benchmark multimode Foc…
In this paper, we propose a protocol for angular displacement estimation based upon orbital angular momentum coherent state and a SU(1,1)-SU(2) hybrid interferometer. This interferometer consists of an optical parametric amplifier, a beam splitter and reflection mirrors, hereon we use a quantum detection strategy $\---$ balanced homodyne detection. The results indicate that super-resolution and super-sensitivity can be realized with ideal condition. Additionally, we study the impact of photon loss on the resolution and the sensitivity, and the robustness of our protocol is also discussed. Fina…
We theoretically study the quantum Fisher information (QFI) of the SU(1,1) interferometer with phase shifts in two arms taking account of realistic noise effects. A generalized phase transform including the phase diffusion effect is presented by the purification process. Based on this transform, the analytical QFI and the bound to the quantum precision are derived when considering the effects of phase diffusion and photon losses simultaneously. To beat the standard quantum limit with the reduced precision of phase estimation due to noisy, the upper bounds of decoherence coefficients as a funct…
We explored the general characteristics of a Sagnac interferometer in a multiparameter estimation process. We find that in the two-parameter estimation scenario, one cannot make both parameter measurement results reach the Heisenberg limit (HL) simultaneously when the input resources are maximally entangled. Only one of the parameters' uncertainty can approach the HL while the other is only scaled by the standard quantum limit (SQL). We also discussed the constraint conditions that make the quantum Cramer-Rao bound saturable. These constraint conditions would prompt one to choose proper evolut…
With the rapid development of quantum technologies in recent years, the need for high sensitivity measuring techniques has become a key issue. In particular, optical sensors based on quantum states of light have proven to be optimal resources for high precision interferometry. Nevertheless, their performance may be severely affected by the presence of noise or imperfections. In this work we derive the quantum Fisher information matrix associated to the simultaneous estimation of an interferometric phase and the indistinguishability characterizing the probe state consisting of an even number of…
Photon counting measurement has been regarded as the optimal measurement scheme for phase estimation in the squeezed-state interferometry, since the classical Fisher information equals to the quantum Fisher information and scales as $\bar{n}^2$ for given input number of photons $\bar{n}$. However, it requires photon-number-resolving detectors with a large enough resolution threshold. Here we show that a collection of $N$-photon detection events for $N$ up to the resolution threshold $\sim \bar{n}$ can result in the ultimate estimation precision beyond the shot-noise limit. An analytical formul…
We give a detailed discussion of optimal quantum states for optical two-mode interferometry in the presence of photon losses. We derive analytical formulae for the precision of phase estimation obtainable using quantum states of light with a definite photon number and prove that maximization of the precision is a convex optimization problem. The corresponding optimal precision, i.e. the lowest possible uncertainty, is shown to beat the standard quantum limit thus outperforming classical interferometry. Furthermore, we discuss more general inputs: states with indefinite photon number and states…
The Cramér-Rao bound plays a central role in both classical and quantum parameter estimation, but finding the observable and the resulting inversion estimator that saturates this bound remains an open issue for general multi-outcome measurements. Here we consider multi-outcome homodyne detection in a coherent-light Mach-Zehnder interferometer and construct a family of inversion estimators that almost saturate the Cramér-Rao bound over the whole range of phase interval. This provides a clue on constructing optimal inversion estimators for phase estimation and other parameter estimation in any m…
We provide a general framework for handling the effects of a unitary disturbance on the estimation of the amplitude $λ$ associated to a unitary dynamics. By computing an analytical and general expression for the quantum Fisher information, we prove that the optimal estimation precision for $λ$ cannot be outperformed through the addition of such a unitary disturbance. However, if the dynamics of the system is already affected by an external field, increasing its strength does not necessary imply a loss in the optimal estimation precision.
In this paper, we revisit the quantum Fisher information (QFI) calculation in SU(1,1) interferometer considering different phase configurations. When one of the input modes is a vacuum state, we show by using phase averaging, different phase configurations give same QFI. In addition, by casting the phase estimation as a two-parameter estimation problem, we show that the calculation of the quantum Fisher information matrix (QFIM) is necessary in general. Particularly, within this setup, the phase averaging method is equivalent to a two parameter estimation problem. We also calculate the phase s…
Super-resolved angular displacement estimation is of crucial significances for quantum information process and optical lithography. Here we report on and experimentally demonstrate a protocol for angular displacement estimation based on a coherent state containing orbital angular momentum. In the lossless scenario, with using parity measurement, this protocol can theoretically achieve 4$\ell$-fold super-resolution with quantum number $\ell$, and shot-noise-limited sensitivity saturating the quantum Cramér-Rao bound. Several realistic factors and their effects are considered, including nonideal…
Recently, quantum Fourier transform interferometers have been demonstrated to allow a quantum metrological enhancement in phase sensitivity for a small number n of identical input single photons [1-3]. However, multiphoton distinguishability at the detectors can play an important role from an experimental point of view [4]. This raises a fundamental question: How is the phase sensitivity affected when the photons are completely distinguishable at the detectors and therefore do not interfere? In other words, which role does multiphoton interference play in these schemes? Here, we show that for …
We theoretically present the quantum Cramér-Rao bounds (QCRB) of an SU(1,1) interferometer for Gaussian states input with and without the internal photonic losses. The phase shifts in the single arm and in the double arms are studied and the corresponding analytical expressions of quantum Fisher information with Gaussian input states are presented. Different from the traditional Mach-Zehnder interferometer, the QCRB of single arm case is slightly higher or lower than that of double arms case depending on the input states. With a fixed mean photon number and for pure Gaussian state input, the o…
A nonlinear phase shift is introduced to a Mach-Zehnder interferometer (MZI), and we present a scheme for enhancing the phase sensitivity. In our scheme, one input port of a standard MZI is injected with a coherent state and the other input port is injected with one mode of a two-mode squeezed-vacuum state. The final interference output of the MZI is detected with the method of active correlation output readout. Based on the optimal splitting ratio of beam splitters, the phase sensitivity can beat the standard quantum limit and approach the quantum Cramér-Rao bound. The effects of photon loss …
We propose a quantum algorithm for finding eigenvalues of non-unitary matrices. We show how to construct, through interactions in a quantum system and projective measurements, a non-Hermitian or non-unitary matrix and obtain its eigenvalues and eigenvectors. This proposal combines ideas of frequent measurement, measured quantum Fourier transform, and quantum state tomography. It provides a generalization of the conventional phase estimation algorithm, which is limited to Hermitian or unitary matrices.
We provide general bounds of phase estimation sensitivity in linear two-mode interferometers. We consider probe states with a fluctuating total number of particles. With incoherent mixtures of state with different total number of particles, particle entanglement is necessary but not sufficient to overcome the shot noise limit. The highest possible phase estimation sensitivity, the Heisenberg limit, is established under general unbiased properties of the estimator. When coherences can be created, manipulated and detected, a phase sensitivity bound can only be set in the central limit, with a su…
Randomized benchmarking is a useful scheme for evaluation the average fidelity of a noisy quantum circuit. However, it is insensitive to the unitary error. Here, we propose a method of randomized benchmarking in which a unitary t-design is applied and by which the unitary error estimation can be converted to analysis of pseudo-randomness on a set of unitary operators. We give a bound on the number of randomized benchmarking sequences, when performing a unitary t-design on n-qubit d-dimensional system. By applying local random unitary operators, a decomposition of a unitary t-design, the bound …
The problem of optimally estimating an unknown unitary quantum operation with the aid of entanglement is addressed. The idea is to prepare an entangled pair, apply the unknown unitary to one of the two parts and then measure the joint output state. This measurement could be an entangled one or it could be separable (e.g., LOCC). A comparison is made between these possibilities and it is shown that by using non-separable measurements one can improve the accuracy of the estimation by a factor of $2(d+1)/d$ where $d$ is the dimension of the Hilbert space on which $U$ acts.
We investigate the simultaneous estimation of $n$ not necessarily identical unitary channels using multi-partite entanglement. We examine whether it is possible for the rate at which the mean square error decreases to be greater than that using the channels individually. For a reasonably general situation, in which there is no functional dependence between the channels, we show that this is not possible. We look at a case in which the channels are not necessarily identical but depend on a common variable. In this case, the mean square error decreases more rapidly using multi-partite entangleme…
We address the problem of the information-disturbance trade-off associated to the estimation of a quantum transformation, and show how the extraction of information about the a black box causes a perturbation of the corresponding input-output evolution. In the case of a black box performing a unitary transformation, randomly distributed according to the invariant measure, we give a complete solution of the problem, deriving the optimal trade-off curve and presenting an explicit construction of the optimal quantum network.
We study a sensor network of distributed Mach-Zehnder interferometers (MZIs) for the parallel (simultaneous) estimation of an arbitrary number $d \geq 1$ of phase shifts. The scheme uses a squeezed-vacuum state that is split between $d$ modes by a quantum circuit (QC). Each output mode of the QC is the input of one of $d$ MZIs, the other input of each MZI being a coherent state. In particular, ${\it i}$) we identify the optimal configuration of the sensor network that allows the estimation of any linear combination of the $d$ phases with maximal sensitivity. The protocol overcomes the shot-noi…
The quantum Fisher information (QFI) in SU(2) and SU(1,1) interferometers was considered, and the QFI-only calculation was overestimated. In general, the phase estimation as a two-parameter estimation problem, and the quantum Fisher information matrix (QFIM) is necessary. In this paper, we theoretically generalize the model developed by Escher et al [Nature Physics 7, 406 (2011)] to the QFIM case with noise and study the ultimate precision limits of SU(2) and SU(1,1) interferometers with photon losses because photon losses as a very usual noise may happen to the phase measurement process. Usin…
By exploiting the correlation properties of ultracold atoms in a multi-mode interferometer, we show how quantum enhanced measurement precision can be achieved with strong robustness to particle loss. While the potential for enhanced measurement precision is limited for even moderate loss in two-mode schemes, multi-mode schemes can be more robust. A ring interferometer for sensing rotational motion with non-interacting fermionic atoms can realize an uncertainty scaling of $1/(N\sqrtη)$ for $N$ particles with a fraction $η$ remaining after loss, which undercuts the shot noise limit of two mode i…
Enhancing Phase Estimation in a Hybrid Interferometer via Kerr Nonlinearity and Photon Subtraction
Jiangxi Normal University（江西师范大学）；Department of Physics, Jiangxi Normal University Science and Technology College（江西师范大学科学技术学院物理系）；School of Electronic and Information Engineering, Nanchang Institute of Technology（南昌工学院电子与信息工程学院）
We construct a practical method for finding optimal Gaussian probe states for the estimation of parameters encoded by Gaussian unitary channels. This method can be used for finding all optimal probe states, rather than focusing on the performance of specific states as shown in previous studies. As an example, we apply this method to find optimal probes for the channel that combines the phase-change and squeezing channels, and for generalized two-mode squeezing and mode-mixing channels. The method enables a comprehensive study of temperature effects in Gaussian parameter estimation. It has been…
We present a proposal of a set-up to measure the work distribution due to an arbitrary unitary process acting on the spatial transverse degrees of freedom of a light beam. Hermite-Gaussian optical modes representing a quantum harmonic oscillator are prepared in a thermal state and sent through an interferometer. We show that the Fourier transform of the work distribution, or the characteristic function, can be obtained by measuring the intensity at the output of the interferometer. The usefulness of the approach is illustrated by calculating the work distribution for a unitary operation that d…
We propose a multi-parameter quantum metrological protocol based on a Mach-Zehnder interferometer with a squeezed vacuum input state and an anti-squeezing operation at one of its output channels. A simple and intuitive geometrical picture of the state evolution is provided by the marginal Wigner functions of the state at each interferometer output channel. The protocol allows to detect the value of the sum $β=\frac{1}{2}(φ_1+φ_2)+θ_\mathrm{in}-θ_\mathrm{out}$, of the relative phase $θ_\mathrm{in}-θ_\mathrm{out}$ between the two squeezers, and the average of the phase delays $φ_1,φ_2$ in the tw…
