We investigate in detail a recently introduced “coherent averaging scheme” in terms of its usefulness for achieving Heisenberg limited sensitivity in the measurement of different parameters. In the scheme, N quantum probes in a product state interact with a quantum bus. Instead of measuring the probes directly and then averaging as in classical averaging, one measures the quantum bus or the entire system and tries to estimate the parameters from these measurement results. Combining analytical results from perturbation theory and an exactly solvable dephasing model with numerical simulations, we draw a detailed picture of the scaling of the best achievable sensitivity with N , the dependence on the initial state, the interaction strength, the part of the system measured, and the parameter under investigation. In particular, we identify the situations allowing one to reach Heisenberg‐limited scaling of the sensitivity.
We investigate in detail a recently introduced “coherent averaging scheme” in terms of its usefulness for achieving Heisenberg limited sensitivity in the measurement of different parameters. In the scheme, N quantum probes in a product state interact with a quantum bus. Instead of measuring the probes directly and then averaging as in classical averaging, one measures the quantum bus or the entire system and tries to estimate the parameters from these measurement results. Combining analytical results from perturbation theory and an exactly solvable dephasing model with numerical simulations, we draw a detailed picture of the scaling of the best achievable sensitivity with N , the dependence on the initial state, the interaction strength, the part of the system measured, and the parameter under investigation. In particular, we identify the situations allowing one to reach Heisenberg‐limited scaling of the sensitivity.
This thesis is dedicated to the understanding of the metrology of quantum systems by using the tools of quantum parameter estimation, in particular the quantum Fisher information (QFI). Our first project deals with a specific protocol of quantum enhanced measurement known as coherent averaging [Braun and Martin, 2011]. This protocol is based on a star topology, with one central object, the so-called quantum bus, connected to N extra subsystems, called probes. For the estimation of a parameter characteristic of the interaction between the quantum bus and the probes, coherent averaging leads to a Heisenberg limited (HL) scaling for the QFI (QFI proportional to N 2 ). Importantly this HL scaling can be obtained while starting with a separable state. This provides an advantage as generally one needs to use entangled states to achieve this scaling. Another important aspect in coherent averaging is the possibility to obtain the HL scaling by performing a measurement on the quantum bus only. These results were obtained using perturbation theory in the regime of weak interactions. In this thesis we go one step further in the study of the coherent averaging protocol. We extend the formalism of perturbation theory to encompass the possibility of estimating any parameter, in the regimes of strong and weak interactions. To illustrate the validity of our results, we introduce two models as examples for a coherent averaging scheme. In these models both the quantum bus and all the probes are qubits. In the ZZXX model, the free Hamiltonians do not commute with the interaction Hamiltonians and we have to rely on numerics to find non-perturbative solutions .In the ZZZZ model the free evolution Hamiltonians commute with the interaction Hamiltonians and we can find the exact solution analytically. Perturbation theory shows that in the strong interaction regime and starting with a separable state, we can estimate the parameter of the free evolution of the probes with a HL scaling if the free Hamiltonians do not commute with the interaction Hamiltonians. This is confirmed by the non-perturbative numerical results for the ZZXX model. In the weak interaction regime we only obtain a standard quantum limit (SQL) scaling for the parameter of the free evolution of the probes (QFI proportional to N ). When one has only access to the quantum bus, we show that the HL scaling found using the perturbation theory does not necessarily survive outside the regime of validity of the perturbation. This is especially the case as N becomes large. It is shown by comparing the exact analytical result to the perturbative result with the ZZZZ model. The same behaviour is observed with the ZZXX model using the non-perturbative numerical results. In our second project we investigate the estimation of the depolarizing channel and the phase-flip channel under non-ideal conditions. It is known that using an ancilla can lead to an improvement of the channel QFI (QFI maximized over input states feeding the channel) even if we act with the identity on the ancilla. This method is known as channel extension. In all generality the maximal channel QFI can be obtained using an ancilla whose Hilbert space has the same dimension as the dimension of the Hilbert space of the original system. In this ideal scenario using multiple ancillas — or one ancilla with a larger Hilbert space dimension — is useless. To go beyond this ideal result we take into account the possibility of loosing either the probe or a finite number of ancillas. The input states considered are GHZ and W states with n + 1 qubits (the probe plus n ancillas). We show that for any channel, when the probe is lost then all the information is lost, and the use of ancillas cannot help. For the phase-flip channel the introduction of ancillas never improves the channel QFI and ancillas are useless. For the depolarizing channel the maximal channel QFI can be reached using one ancilla and feeding the extended channel with a Bell state, but if the ancilla is lost then all the advantage is lost. We show that the GHZ states do not help to fight the loss of ancillas: If one ancilla or more are lost all the advantage provided by the use of ancillas is lost. More interestingly, we show that the W states with more than one ancilla are robust against loss. For a given number of lost ancillas, there always exists an initial number of ancillas for which a W state provides a higher QFI than the one obtained without ancillas. Our last project is about Hamiltonian parameter estimation for arbitrary Hamiltonians. It is known that channel extension does not help for unitary channels. Instead we apply the idea of extension to the Hamiltonian itself and not to the channel. This is done by adding to the Hamiltonian an extra term, which is independent of the parameter and which possibly encompasses interactions with an ancilla. We call this technique Hamiltonian extension. We show that for arbitrary Hamiltonians there exists an upper bound to the channel QFI that is in general not saturated. This result is known in the context of non-linear metrology. Here we show explicitly the conditions to saturate the bound. We provide two methods for Hamiltonian extensions, called signal flooding and Hamiltonian subtraction, that allow one to saturate the upper bound for any Hamiltonian. We also introduce a third method which does not saturate the upper bound but provides the possibility to restore the quadratic time scaling in the channel QFI when the original Hamiltonian leads only to a periodic time scaling of the channel QFI. We finally show how these methods work using two different examples. We study the estimation of the strength of a magnetic field using a NV center, and show how using signal flooding we saturate the channel QFI. We also consider the estimation of a direction of a magnetic field using a spin-1. We show how using signal flooding or Hamiltonian subtraction we saturate the channel QFI. We also show how by adding an arbitrary magnetic field we restore the quadratic time scaling in the channel QFI. Eventually we explain how coherent averaging can be scrutinized in the formalism of Hamiltonian extensions.
Abstract In all quantitative sciences, it is common practice to increase the signal-to-noise ratio of noisy measurements by measuring identically prepared systems N times and averaging the measurement results. This leads to a scaling of the sensitivity as 1/√N, known in quantum measurement theory as the “standard quantum limit” (SQL). It is known that if one puts the N systems into an entangled state, a scaling as 1/N can be achieved, the socalled “Heisenberg limit” (HL), but decoherence problems have so far prevented implementation of such protocols for large N. Here we show that a method of coherent averaging inspired by a recent entanglement-free quantum enhanced measurement protocol is capable of achieving a sensitivity that scales as 1/N in a purely classical setup. This may substantially improve the measurement of very weak interactions in the classical realm, and, in particular, open a novel route to measuring the gravitational constant with enhanced precision.
The Heisenberg scaling is an ultimate precision limit of parameter estimation allowed by the principles of quantum mechanics, with no counterpart in the classical realm, and has been a long-pursued goal in quantum metrology. It has been known that interactions between the probes can help reach the Heisenberg scaling without entanglement. In this paper, we show that interactions between the probes and the additional dimensions of an ancillary system may also increase the precision of parameter estimation to surpass the standard quantum limit and attain the Heisenberg scaling without entanglement, if the measurement scheme is properly designed. The quantum Fisher information exhibits periodic patterns over the evolution time, implying the existence of optimal time points for measurements that can maximize the quantum Fisher information. By implementing optimizations over the Hamiltonian, the initial states of the probes and the ancillary system, the interaction strength, and the time points for measurements, our protocol achieves the Heisenberg scaling for the parameter of the probe Hamiltonian, in terms of both evolution time and probe number. Our protocol features two aspects: (i) the Heisenberg scaling can be achieved by a product state of the probes and (ii) mere local measurement on the ancilla is sufficient, both of which reduce the quantum resources and the implementation complexity to achieve the Heisenberg scaling. The paper is concluded by the investigation of the effects of noise on this protocol.
We show that the recently discovered quantum-enhanced measurement protocol of coherent averaging that is capable of achieving Heisenberg-limited sensitivity without using entanglement, has a classical analogue. The classical protocol uses N harmonic oscillators coupled to a central oscillator and one measures the signal from the latter. We propose an application to the measurement of very weak interactions, and, in particular, a novel route to measuring the gravitational constant with enhanced precision.
Questions about quantum limits on measurement precision were once viewed from the perspective of how to reduce or avoid the effects of quantum noise. With the advent of quantum information science came a paradigm shift to proving rigorous bounds on measurement precision. These bounds have been interpreted as saying, first, that the best achievable sensitivity scales as 1/n, where n is the number of particles one has available for a measurement and, second, that the only way to achieve this Heisenberg-limited sensitivity is to use quantum entanglement. We review these results and show that using quadratic couplings of n particles to a parameter to be estimated, one can achieve sensitivities that scale as 1/n^2 if one uses entanglement, but even in the absence of any entanglement at any time during the measurement protocol, one can achieve a super-Heisenberg scaling of 1/n^(3/2)
We propose an adiabatic Ramsey interferometry technique for detecting weak nonlinearities with trapped ions. The method relies on using the quantum Rabi model as a probe, which is sensitive to nonlinear symmetry-breaking perturbations. We show that the couplings which arise either from anharmonic terms of the trapping potential or due to higher order terms in the Coulomb interaction expansion can be efficiently estimated by measuring the spin state probabilities alone. We show that the spin signal is amplified by the mean-phonon excitations, which results in the estimation precision reaching the super-Heisenberg limit. Notably, achieving such high-precision estimation does not require specific entangled state preparation and can be reached even for initial thermal motion state. Furthermore, we show that the super-Heisenberg scaling can be observed even in the presence of weak spin-dephasing.