We show through a series of arguments that, given a total average photon number 〈N〉, the fundamental limit in the precision phase measurement is set by quantum mechanics to be the so-called Heisenberg limit, i.e., 1/〈N〉. Some specific types of phase measurement are considered in the discussion. However, the proof based on the general principle of complementarity of quantum mechanics applies to any scheme of phase measurement. From the general argument by the complementarity principle, we are able to find a necessary condition for those states that can achieve the Heisenberg limit if they are employed for precision phase measurement. A general guideline is given for the search of the measurement schemes in which the Heisenberg limit is achieved. We demonstrate the procedure by applying it to a few specific examples.
We show through a series of arguments that, given a total average photon number 〈N〉, the fundamental limit in the precision phase measurement is set by quantum mechanics to be the so-called Heisenberg limit, i.e., 1/〈N〉. Some specific types of phase measurement are considered in the discussion. However, the proof based on the general principle of complementarity of quantum mechanics applies to any scheme of phase measurement. From the general argument by the complementarity principle, we are able to find a necessary condition for those states that can achieve the Heisenberg limit if they are employed for precision phase measurement. A general guideline is given for the search of the measurement schemes in which the Heisenberg limit is achieved. We demonstrate the procedure by applying it to a few specific examples.
By analyzing a single-photon interference experiment together with a quantum nondemolition measurement scheme for the which-path information, we prove that given the total mean number 〈n〉 of available photons, the fundamental limit in precision measurement of a phase shift is the Heisenberg limit, i.e., 1/〈n〉. The analysis is based on the complementarity principle and is independent of the scheme for the measurement of the phase shift. We also show that to achieve the Heisenberg limit states with photon number fluctuations of the order of or larger than the mean photon number have to be exploited.
Optical phase measurement through its application in quantum metrology has pushed the precision limit with which some physical quantities can be measured accurately. At the very fundamental level, the laws of quantum mechanics dictate that the uncertainty in phase estimations scales as 1/N, where N is the number of quantum resources employed in the protocol [1]. This is the well known Heisenberg limit (HL) which is quadratically better than the traditional precision limit known as the standard quantum limit (SQL) with uncertainty asymptotically scaling as 1/sqrt{N} [1]. Several experiments have demonstrated that the SQL can be beaten by using an entangled state as the probe and a specific measurement scheme for ab initio estimation of unknown phases [2,3]. It has also been shown experimentally that even in the absence of the entanglement one can measure an unknown phase with imprecision scaling at the HL [4]. In this work we first present a new protocol able to estimate an optical phase at the Heisenberg limit, and then experimentally explore fundamental and practical issues in generating high-quality novel entangled states, for use in this protocol and beyond. Our aim in this study is to measure an unknown phase in the interval [0,2pi) with uncertainty attaining the exact HL. There is a condition that should be met to address this objective: preparation of an optimal state [5]. This would cover part of the presentation through which we explain how to experimentally realise such an optimal state with the current technological limitations and the feasibility of the scheme. In particular, we generate an entangled 3-photon (2-photon) state of specific superposition of GHZ (Bell) states. Our numerical simulation of the phase measurement gate together with the experimental outcomes show that the created state should have a high fidelity and purity to be able to have the phase uncertainty achieving the exact HL. Therefore, we briefly explain the modelling for experimental imperfections and finally present the results of experimental phase measurements.
The use of quantum resources can provide measurement precision beyond the shot noise limit (SNL). The task of ab initio optical phase measurement-the estimation of a completely unknown phase-has been experimentally demonstrated with precision beyond the SNL, and even scaling like the ultimate bound, the Heisenberg limit (HL), but with an overhead factor. However, existing approaches have not been able-even in principle-to achieve the best possible precision, saturating the HL exactly. Here we demonstrate a scheme to achieve true HL phase measurement, using a combination of three techniques: entanglement, multiple samplings of the phase shift, and adaptive measurement. Our experimental demonstration of the scheme, with photonic qubits and N = 3 photon-passes, achieves a precision that is within 4% of the HL, surpassing the best precision theoretically achievable with simpler techniques. This work represents a fundamental achievement of the ultimate limits of metrology, and the scheme can be extended to higher N and other physical systems.
Abstract The use of quantum resources can provide measurement precision beyond the shot-noise limit (SNL). The task of ab initio optical phase measurement—the estimation of a completely unknown phase—has been experimentally demonstrated with precision beyond the SNL, and even scaling like the ultimate bound, the Heisenberg limit (HL), but with an overhead factor. However, existing approaches have not been able—even in principle—to achieve the best possible precision, saturating the HL exactly. Here we demonstrate a scheme to achieve true HL phase measurement, using a combination of three techniques: entanglement, multiple samplings of the phase shift, and adaptive measurement. Our experimental demonstration of the scheme uses two photonic qubits, one double passed, so that, for a successful coincidence detection, the number of photon-passes is N = 3. We achieve a precision that is within 4% of the HL. This scheme can be extended to higher N and other physical systems.
The use of quantum resources can provide measurement precision beyond the shot-noise limit (SNL). The task of ab initio optical phase measurement-the estimation of a completely unknown phase-has been experimentally demonstrated with precision beyond the SNL, and even scaling like the ultimate bound, the Heisenberg limit (HL), but with an overhead factor. However, existing approaches have not been able-even in principle-to achieve the best possible precision, saturating the HL exactly. Here we demonstrate a scheme to achieve true HL phase measurement, using a combination of three techniques: entanglement, multiple samplings of the phase shift, and adaptive measurement. Our experimental demonstration of the scheme uses two photonic qubits, one double passed, so that, for a successful coincidence detection, the number of photon-passes is N = 3. We achieve a precision that is within 4% of the HL. This scheme can be extended to higher N and other physical systems.
We propose and demonstrate experimentally a projection scheme to measure the quantum phase with a precision beating the standard quantum limit. The initial input state is a twin Fock state | N , N ⟩ proposed by Holland and Burnett ( Phys. Rev. Lett. , 71 (1993) 1355) but the phase information is extracted by a quantum state projection measurement. The phase precision is about 1.4/ N for large photon number N , which approaches the Heisenberg limit of 1/ N . Experimentally, we employ a four-photon state from type-II parametric down-conversion and achieve a phase uncertainty of 0.291±0.001 beating the standard quantum limit of for four photons.