Postselection Probability Research Articles

Optimization of parameter measurement precision with precoupling-assisted weak-value amplification

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.

Optimizing the trade off between precision and probability in post-selection estimation of the coupling of trapped ion system

In post-selection quantum measurement, it is a dilemma to obtain both high precision and large post-selection probability. Here,we deeply examine quantum Fisher information and probability of post-selection quantum measurement based on the trapped ion system, where the internal electronic state of the trapped ion acts as an interested system and the external motional state as a pointer. Based on the bichromatic excitation of the trapped ion, in the Lamb-Dicke regime we observe the dynamical behaviors of quantum Fisher information of estimating coupling parameter and post-selection probability under the various phase difference of the bichromatic laser. We find that the oscillations of quantum Fisher information and post-selection probability can be simultaneously enhanced by selecting appropriately the phase difference of the bichromatic laser, the amplitude and the phase of the pointer state. Although the competition between quantum Fisher information and the post-selection probability still exists in the whole coupling range, a trade-off point in the moderate-coupling regime where the precision and the probability surpass their convergence values in the strong coupling regime can be obtained. It means that there is a possibility that the desired precision and post-selection probability can be simultaneously achieved.

Experimental investigation of high-efficiency weak-value amplification of nonunitary evolution

High-sensitivity metrology plays an important role in parameter estimation, and weak-value amplification is a significant tool used to enhance sensitivity. However, the efficiency of weak-value amplification is always affected by the postselection probability and meter loss. In this article we report on the realization of a device for weak interaction that realizes the nonunitary evolution based on a spatial light modulator and applies the weak-value amplification of nonunitary evolution. The weak-value amplification of nonunitary evolution can achieve efficiency nearly ninefold that of unitary evolution experimentally under specific settings, because of the quadratic relation between the meter shift and the norm of the weak value $|{A}_{w}|$. This work demonstrates the superiority of nonunitary evolution in weak-value amplification, provides an experimental scheme of weak interaction for other forms of nonunitary evolution, and possesses the potential to be combined with other techniques for higher efficiency.

Enhancing signal and signal-to-noise ratio with post-selection and nonclassical states

We propose a weak measurement scheme with post-selection based on the cross-Kerr interaction between two optical fields. We demonstrate that our scheme can measure smaller physical effects and obtain a larger signal-to-noise ratio compared to the weak measurement scheme without post-selection. We also show that the initial pointer states with nonclassical properties can increase the successful post-selection probability, offer a bigger signal-to-noise ratio compared to the coherent states.

Implementation of weak measurement amplification with the weak coherent light and optomechanical system

Most studies for postselected weak measurement in optomechanical system focus on using a single photon as a measured system. However, we find that using weak coherent light instead of a single photon can also amplify the mirror’s position displacement of one photon. In the weak value amplification (WVA) regime, the weak value of one photon can lie outside the eigenvalue spectrum, proportional to the difference of the mirror’s position displacement between the successful and failed postselection and its successful postselection probability is dependent of the mean photon number of the coherent light and improved by adjusting it accordingly. Outside the WVA regime, the amplification limit can reach the level of the vacuum fluctuations, and when the mean photon number and the optomechanical coupling parameter are selected appropriately, its successful postselection probability becomes higher, which is beneficial to observe the maximum amplification value under the current experimental conditions. This result breaks the constraint that it is difficult to detect outside the WVA regime. This opens up a new regime for the study of a single photon nonlinearity in optomechanical system.

Enhanced Weak-Value Amplification via Photon Recycling.

In a quantum-noise limited system, weak-value amplification using postselection normally does not produce more sensitive measurements than standard methods for ideal detectors: the increased weak value is compensated by the reduced power due to the small postselection probability. Here, we experimentally demonstrate recycled weak-value measurements using a pulsed light source and optical switch to enable nearly deterministic weak-value amplification of a mirror tilt. Using photon counting detectors, we demonstrate a signal improvement by a factor of 4.4±0.2 and a signal-to-noise ratio improvement of 2.10±0.06, compared to a single-pass weak-value experiment, and also compared to a conventional direct measurement of the tilt. The signal-to-noise ratio improvement could reach around six for the parameters of this experiment, assuming lower loss elements.

Weak value amplification for angular velocity measurements.

The weak-value-amplification technique has shown great importance in the measurement of tiny physical effects. Here we introduce a polarization-dependent angular velocity measurement system consisting of two Glan prisms and a true zero-order half-wave plate, where a non-Fourier-limited Gaussian pulse acts as the meter. The angular velocities measurements results agree well with theoretical predictions, and its uncertainties are bounded by the Cramér-Rao bound. We also investigate uncertainties of angular velocities for different numbers of detected photons and the smallest reliable postselection probability, which can reach ${3.42*10^{- 6}}$.

Influence of entanglement on precision of parameter estimation in quantum weak measurement

An appropriate weak value can be used to amplify weak physical parameters and improve the precision of parameter estimation in the process of quantum weak measurement. The increase of the precision of such a parameter estimation may originate from the entanglement in the system. This paper uses Fisher information to study the influence of the entanglement in the system and the entanglement between the system and the pointer on the estimation precision of the coupling parameters between the system and the pointer in the process of weak measurement. The entanglement of the entangled pure state of the GHZ-like state and the entangled mixed state affected by the depolarization noise and the influence of the entanglement between the system and the pointer on the parameter estimation are analyzed. The results show that the Fisher information quantity increases with the increase of the initial state entanglement degree of the system when the initial state of the system is an entangled pure state or an entangled mixed state affected by depolarization noise, and both the Fisher information quantity and the post-selection probability reach their maximum values when the initial and final state of the system are both the maximum entangled states; but the weaker the entanglement between the system and the pointer, the more the Fisher information obtained in the measurement will be and the higher the accuracy of parameter estimation. These research results show that the entanglement in the system will improve the precision of parameter estimation, while the entanglement between the system and the pointer will reduce the precision of parameter estimation.

State distinguishability under weak measurement and postselection: A unified system and device perspective

We quantify the disturbance of a quantum state undergoing a sequence of observations, and particularly focus on a weak measurement followed by post-selection and compare these results to the projective counterpart. Taking into account the distinguishability of both, the system and the device, we obtain the exact trade-off between the system state disturbance and the change of the device pointer state. We show that for particular post-selection procedures the coupling strength between the system and the device can be significantly reduced without loosing measurement sensitivity, which is directly transferred to a reduced state disturbance of the system. We observe that a weak measurement alone does not provide this advantage but only in combination with post-selection a significant improvement in terms of increased measurement sensitivity and reduced state disturbance is found. We further show that under realistic experimental conditions this state disturbance is small, whereas the exact post-selection probability is considerably larger than the approximate value given by the overlap of the initial and final state when neglecting the system state disturbance.

Weak-value-amplification analysis beyond the Aharonov-Albert-Vaidman limit

The weak-value (WV) measurement proposed by Aharonov, Albert and Vaidman (AAV) has attracted a great deal of interest in connection with quantum metrology. In this work, we extend the analysis beyond the AAV limit and obtain a few main results. (i) We obtain non-perturbative result for the signal-to-noise ratio (SNR). In contrast to the AAV's prediction, we find that the SNR asymptotically gets worse when the AAV's WV $A_w$ becomes large, i.e., in the case $g|A_w|^2>>1$, where $g$ is the measurement strength. (ii) With the increase of $g$ (but also small), we find that the SNR is comparable to the result under the AAV limit, while both can reach -- actually the former can slightly exceed -- the SNR of the standard measurement. However, along a further increase of $g$, the WV technique will become less efficient than the standard measurement, despite that the postselection probability is increased. (iii) We find that the Fisher information can characterize the estimate precision qualitatively well as the SNR, yet their difference will become more prominent with the increase of $g$. (iv) We carry out analytic expressions of the SNR in the presence of technical noises and illustrate the particular advantage of the imaginary WV measurement. The non-perturbative result of the SNR manifests a favorable range of the noise strength and allows an optimal determination.

Operational formulation of weak values without probe systems

Weak values are the fundamental values for observables in a pre- and post-selected system. Weak values are typically measured by weak measurement, in which weak values appear in the change of not the pre- and post-selected system but the probe system. This indirect characteristic of weak measurement obscures the meaning of weak values for the pre- and post-selected system, in contrast to conventional physical quantities, which have a clear operational meaning. In this study, we operationally formulate weak values as the sensitivity of post-selection probability amplitude to small transformation in a pre- and post-selected system. This formulation of weak values, which is free from the concept of probe shift assumed in weak measurement, gives a direct interpretation of strange weak values for the pre- and post-selected system. We further explain that this formulation can simplify weak-value measurement experiments because no probe system is required.

Experimental Demonstration of Quantum Fully Homomorphic Encryption with Application in a Two-Party Secure Protocol

A fully homomorphic encryption system hides data from unauthorized parties, while still allowing them to perform computations on the encrypted data. Aside from the straightforward benefit of allowing users to delegate computations to a more powerful server without revealing their inputs, a fully homomorphic cryptosystem can be used as a building block in the construction of a number of cryptographic functionalities. Designing such a scheme remained an open problem until 2009, decades after the idea was first conceived, and the past few years have seen the generalization of this functionality to the world of quantum machines. Quantum schemes prior to the one implemented here were able to replicate some features in particular use-cases often associated with homomorphic encryption but lacked other crucial properties, for example, relying on continual interaction to perform a computation or leaking information about the encrypted data. We present the first experimental realisation of a quantum fully homomorphic encryption scheme. We further present a toy two-party secure computation task enabled by our scheme. Finally, as part of our implementation, we also demonstrate a post-selective two-qubit linear optical controlled-phase gate with a much higher post-selection success probability (1/2) when compared to alternate implementations, e.g. with post-selective controlled-$Z$ or controlled-$X$ gates (1/9).

Amplifying single-photon nonlinearity using thermal light

Single-photon nonlinearity effect can be greatly amplified using a thermal light meter. The change of the meter light is just induced by postselection. Large postselection probability can be obtained when the mean photon number of the thermal light is large. A similar reduction phenomenon is revealed, and the large-probability enhancement phenomenon is also found. This scheme can be realized via electromagnetically induced transparency.

Realization of entanglement-assisted weak-value amplification in a photonic system

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 $n$ particles of a system, the maximum postselection probability scales quadratically with $n$ [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.

Optical qubit generation via atomic postselection in a Ramsey interferometer

We propose a realizable experimental scheme to prepare a superposition of the vacuum and one-photon states using a typical cavity QED-setup. This is different from previous schemes, where the superposition state of the field is generated by resonant atom-field interaction and the cavity is initially empty. Here, we consider only dispersive atom-field interaction and the initial state of the cavity field is coherent. Then, we determine the parameters to prepare the desired state via atomic postselection. We also include the effect of cavity losses and detection imperfections in our analysis, against which this preparation of the optical qubit in a real Fabry–Pérot superconducting cavity is robust. Additionally, we show that this scheme can be used for the preparation of other photon number Fock state superpositions. In summary, our task is achieved with a high fidelity and a postselection probability within experimental reach.

Various pointer states approaches to polar modular values

We theoretically analyze the polar decomposition for quantum modular values under various pointer states approaches. We consider both the finite-dimensional discrete pointer state and continuous pointer state cases. In contrast to that, a weak value of an observable is usually divided into its real and imaginary parts; here, we show that separation from the modulus and phase is necessary to a modular value. We show that the modulus of the modular value is related to the pointer post-selection conditional probability, and the phase of the modular value is connected to the summation of a geometric phase and an intrinsic phase. We also discuss a relationship between the modulus and phase, and therein, the derivative of the phase is related to the derivative of the logarithm of the modulus via a Berry-Simon-like connection which is in the form of a weak value. As a consequence, the modulus-phase relation allows us to obtain these polar components whenever the connection is specified. One of the possible applications of our results is to evaluate the weak value (the Berry-Simon-like connection) when the polar modular value is experimentally obtained.

Merlinization of complexity classes above BQP

We study how complexity classes above BQP, such as postBQP, postBQPFP, and SBQP, change if we “Merlinize” them, i.e., if we allow an extra input quantum state (or classical bit string) given by Merlin as witness. Our main results are the following: First, the Merlinized version of postBQP is equal to PSPACE. Second, if the Merlinized postBQP is restricted in such a way that the postselection probability is equal for all witness states, then the class is equal to PP. Finally, the Merlinization does not change the class SBQP.

Finite-key analysis of a practical decoy-state high-dimensional quantum key distribution

Compared with two-level quantum key distribution (QKD), high-dimensional QKD enables two distant parties to share a secret key at a higher rate. We provide a finite-key security analysis for the recently proposed practical high-dimensional decoy-state QKD protocol based on time-energy entanglement. We employ two methods to estimate the statistical fluctuation of the postselection probability and give a tighter bound on the secure-key capacity. By numerical evaluation, we show the finite-key effect on the secure-key capacity in different conditions. Moreover, our approach could be used to optimize parameters in practical implementations of high-dimensional QKD.

Suppressing technical noise in weak measurements by entanglement

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 [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.

Entanglement-assisted weak value amplification.

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.