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arXiv:1401.5887 · Physical Review Letters

Entanglement-Assisted Weak Value Amplification

Shengshi Pang, Justin Dressel, Todd A. Brun

Published Jul 15, 2014• 107 citations
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Abstract

Large weak values have been used to amplify the sensitivity of a linear response signal for detecting changes in a small parameter, which has also enabled a simple method for precise parameter estimation. However, producing a large weak value requires a low postselection probability for an ancilla degree of freedom, which limits the utility of the technique. We propose an improvement to this method that uses entanglement to increase the efficiency. We show that by entangling and postselecting n ancillas, the postselection probability can be increased by a factor of n while keeping the weak value fixed (compared to n uncorrelated attempts with one ancilla), which is the optimal scaling with n that is expected from quantum metrology. Furthermore, we show the surprising result that the quantum Fisher information about the detected parameter can be almost entirely preserved in the postselected state, which allows the sensitive estimation to approximately saturate the relevant quantum Cramér-Rao bound. To illustrate this protocol we provide simple quantum circuits that can be implemented using current experimental realizations of three entangled qubits.

DOI
10.1103/physrevlett.113.030401

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Large weak values have been used to amplify the sensitivity of a linear response signal for detecting changes in a small parameter, which has also enabled a simple method for precise parameter estimation. However, producing a large weak value requires a low postselection probability for an ancilla degree of freedom, which limits the utility of the technique. We propose an improvement to this method that uses entanglement to increase the efficiency. We show that by entangling and postselecting n ancillas, the postselection probability can be increased by a factor of n while keeping the weak value fixed (compared to n uncorrelated attempts with one ancilla), which is the optimal scaling with n that is expected from quantum metrology. Furthermore, we show the surprising result that the quantum Fisher information about the detected parameter can be almost entirely preserved in the postselected state, which allows the sensitive estimation to approximately saturate the relevant quantum Cramér-Rao bound. To illustrate this protocol we provide simple quantum circuits that can be implemented using current experimental realizations of three entangled qubits.
Weak-value amplification has been used to enhance the sensitivity of a linear detector response to small interaction parameters. However, the generation of a large weak value gives rise to reduction in postselection probability, which restricts the feasibility of the weak-value amplification technique. Recently, theoretical works have shown that the use of entanglement can increase the efficiency of the weak-value amplification method. Specifically, it is proven that by entangling nnn particles of a system, the maximum postselection probability scales quadratically with nnn [Pang, Dressel, and Brun, Phys. Rev. Lett. 113, 030401 (2014)]. In this work, we demonstrate an experimental realization of entanglement-assisted weak-value amplification in a photonic system, which shows that importing quantum entanglement can indeed improve the efficiency of weak-value amplification.
Weak value amplification is a popular method in quantum metrology for enhancing the sensitivity at the expense of the signal intensity. Recently, it was suggested that the trade-off between signal intensity and sensitivity can be improved by using an entangled auxiliary system. Here, we experimentally investigate such entanglement-assisted weak measurement of small conditional phase shifts induced by an interaction between ancilla and meter qubits. We utilize entangled photon pairs and implement the required three-qubit quantum logic circuit with linear optics. The circuit comprises a two-qubit controlled phase gate and a three-qubit controlled-controlled phase gate with fully tunable conditional phase shifts. We fully characterize the output states of our circuit by quantum state tomography and perform a comprehensive analysis of the trade-off between the measurement sensitivity and the success probability of the protocol. The observed experimental results are in good qualitative agreement with theoretical predictions, but the overall performance of our setup is limited by various experimental imperfections. We provide a detailed theoretical analysis of the influence of dephasing of the entangled ancilla state, which is one of the main sources of imperfections in the experiment. We also discuss the ultimate scaling with the dimension of the entangled ancilla system.
Postselected weak measurement has aroused broad interest for its distinctive ability to amplify small physical quantities. However, the low postselection efficiency to obtain a large weak value has been a big obstacle to its application in practice, since it can waste resources, and reduce the measurement precision. In this paper, we detail the entanglement-assisted postselected weak measurement protocol proposed in [Phys. Rev. Lett. 113, 030401 (2014)], and study its robustness against technical noises. We show that when the systems are properly entangled, postselection can be made dramatically more efficient, and as a consequence, the Fisher information can be increased to approximately saturate the well-known Heisenberg limit. We illustrate this protocol by a qubit example. Readout errors can greatly degrade the performance of postselected weak measurement schemes. We show that entanglement can significantly reduce two main detrimental effects of readout errors: inaccuracy in the measurement result, and loss of the Fisher information. We further compensate for readout errors by a majority vote scheme for postselection. With a proper threshold, almost no Fisher information will be lost. These results demonstrate the effectiveness of entanglement in protecting postselected weak measurement against readout errors.
Postselected weak measurement has aroused broad interest for its distinctive ability to amplify small physical quantities. However, the low postselection efficiency to obtain a large weak value has been a big obstacle to its application in practice since it may waste resources, and reduce the measurement precision. An improved protocol was proposed in Pang et al., Phys. Rev. Lett. 113, 030401 (2014) to make the postselected weak measurement dramatically more efficient by using entanglement. Such a protocol can increase the Fisher information of the measurement to approximately saturate the well-known Heisenberg limit. In this paper, we review the entanglement-assisted protocol of postselected weak measurement in detail, and study its robustness against technical noises. We focus on readout errors. Readout errors can greatly degrade the performance of postselected weak measurement, especially when the readout error probability is comparable to the postselection probability. We show that entanglement can significantly reduce the two main detrimental effects of readout errors: inaccuracy in the measurement result and the loss of Fisher information. We extend the protocol by introducing a majority vote scheme to postselection to further compensate for readout errors. With a proper threshold, almost no Fisher information will be lost. These results demonstrate the effectiveness of entanglement in protecting postselected weak measurement against readout errors.
Postselection strategies have been proposed with the aim of amplifying weak signals, which may help to overcome detection thresholds associated with technical noise in high-precision measurements. Here we use an optical setup to experimentally explore two different postselection protocols for the estimation of a small parameter: a weak-value amplification procedure and an alternative method that does not provide amplification but nonetheless is shown to be more robust for the sake of parameter estimation. Each technique leads approximately to the saturation of quantum limits for the estimation precision, expressed by the Cram\'er-Rao bound. For both situations, we show that parameter estimation is improved when the postselection statistics are considered together with the measurement device.
The weak-value amplification is extensively considered in precision metrology in order to achieve a higher sensitivity. Despite the practical benefits in amplifying small physical quantities, its metrological advantage still arouses a broad debate due to the low postselection probability of success. In this paper, by employing the quantum Fisher information metric, we show that the precision of estimating an unknown parameter can be improved by introducing a precoupling process with properly chosen interaction operators. We point out this result is credible for both real and imaginary weak values. By tracking the meter wave functions, we find this enhancement of estimation precision comes from a precoupling induced modulation of the meter wave function, thus the most sensitive regime with respect to the parameter is reached. In addition, the estimation error is investigated by considering the difference between the theoretical and estimated value. The analysis suggests that this kind of error can be effectively suppressed by averaging the estimations resulted from different initial meter states. These results are finally illustrated by an exact numerical simulation where the advantages of our proposal are displayed by the comparison to the standard weak-value amplification scheme.