The time-symmetric formalism endows the weak measurement and its outcome, the weak value,many unique features. In particular, it allows a direct tomography of quantum states without resort to complicated reconstruction algorithms and provides an operational meaning to wave functions and density matrices. Here, we propose and experimentally demonstrate the direct tomography of a measurement apparatus by taking the backward direction of weak measurement formalism. Our protocol works rigorously with the arbitrary measurement strength, which offers an improved accuracy and precision. The precision can be further improved by taking into account the completeness condition of the measurement operators, which also ensures the feasibility of our protocol for the characterization of the arbitrary quantum measurement. Our work provides new insight on the symmetry between quantum states and measurements, as well as an efficient method to characterize a measurement apparatus.
As a special experimental technique, weak measurement extracts very little information about the measured system and will not cause the measured state collapse. When coupling the orbital angular momentum (OAM) state with a well-defined pre-selected and post-selected system of a weak measurement process, there is an indirect coupling between position and topological charge (TC) of OAM state. Based on these ideas, we propose an experimental scheme that experimentally measure the TC of OAM beams from -14 to 14 through weak measurement.
The task of wavefront sensing is to measure the phase of the optical field. Here, we demonstrate that the widely used Shack-Hartmann wavefront sensor detects the weak value of transverse momentum, usually achieved by the method of quantum weak measurement. We extend its input states to partially coherent states and compare it with the weak measurement wavefront sensor, which has a higher spatial resolution but a smaller dynamic range. Since weak values are commonly used in investigating fundamental quantum physics and quantum metrology, our work would find essential applications in these field…
We propose the difference weak measurement scheme, and illustrate its advantages for measuring small longitude phase-shift in high precision. Compared to the standard interferometry and standard weak measurement schemes, the proposed scheme has much higher resolution in present of various practical imperfections, such as alignment error and light intensity variation error. Moreover, we highlight the advantage of utilizing complex weak value, where its imaginary part can reduce the harmful effect induced by channel decoherence. Finally, we propose closed-loop scenario to solve the narrow dynami…
Weak measurement is a novel technique for parameter estimation with higher precision. In this paper we develop a general theory for the parameter estimation based on weak measurement technique with arbitrary postselection. The previous weak value amplification model and the joint weak measurement model are two special cases in our theory. Applying the developed theory, the time-delay estimation is investigated in both theory and experiment. Experimental results shows that when the time-delay is ultra small, the joint weak measurement scheme outperforms the weak value amplification scheme, and …
Weak measurement is an important technique for detecting the tiny spin-dependent splitting in photonic spin Hall effect. The weak measurement is only valid when the probe wavefunction remains almost undisturbed during the procedure of measurements. However, it does not always satisfy such condition in some practical situations, such as in the strong-coupling regime or the preselected and postselected states are nearly orthogonal. In this paper, we develop a modified weak measurement for detecting photonic spin Hall effect when the probe wavefunction is distorted. We find that the measuring pro…
Photonic spin Hall effect (SHE) holds great potential applications in precision metrology. How to obtain a high measurement precision is an important issue to detect the photonic SHE. In this letter, we propose using optimal preselection and postselection in weak measurements to enhance the measurement precision. We find that the maximum weak value and pointer shift can be obtained with an optimal overlap of preselection and postselection states. These findings offer the possibility for improving the precision of weak measurements and thereby have possible applications for accurately character…
An optical analog of the quantum weak measurement scheme proved to be very useful for the observation of optical beam shifts. Here we adapt the weak value amplification method for the observation of the angular Goos-Hanchen shift. We observe this effect in the case of external air-dielectric reflection, the more fundamental case in which it occurs.We show that weak measurements allow for a faithful amplification of the effect at any angle of incidence, even at the Brewster angle of incidence.
A general formalism for joint weak measurements of a pair of complementary observables is given. The standard process of optical three-wave mixing in a nonlinear crystal (such as in parametric down-conversion) is suitable for such tasks. To obtain the weak value of a variable $A$ one performs weak measurements twice, with different initial states of the "meter" field. This seems to be a drawback, but as a compensation we get for free the weak value of a complementary variable $B$. The scheme is tunable and versatile: one has access to a continuous set of possible weak measurements of pair of o…
Heisenberg-limited and weak measurements are the two intriguing notions, used in recent times for enhancing the sensitivity of measurements in quantum metrology. Using a quantum cat state, endowed with sub-Planck structure, we connect these two novel concepts. It is demonstrated that these two phenomena manifest in complementary regimes, depending upon the degree of overlap between the mesoscopic states constituting the cat state under consideration. In particular, we find that when sub-Planck structure manifests, the imaginary weak value is obscured and vice-versa.
The role of measurement induced disturbance in weak measurements is of central importance for the interpretation of the weak value. Uncontrolled disturbance can interfere with the postselection process and make the weak value dependent on the details of the measurement process. Here we develop the concept of a generalized weak measurement for classical and quantum mechanics. The two cases appear remarkably similar, but we point out some important differences. A priori it is not clear what the correct notion of disturbance should be in the context of weak measurements. We consider three differe…
Methods for measuring the weak value of non local variables are investigated. We analyze local (indirect) measurement methods for obtaining the weak values. We also describe some new (direct) methods (Non local weak measurements) for measuring the weak values of some non local variables.
Traditional uncertainty relations dictate a minimal amount of noise in incompatible projective quantum measurements. However, not all measurements are projective. Weak measurements are minimally invasive methods for obtaining partial state information without projection. Recently, weak measurements were shown to obey an uncertainty relation cast in terms of entropies. We experimentally test this entropic uncertainty relation with strong and weak measurements of a superconducting transmon qubit. A weak measurement, we find, can reconcile two strong measurements' incompatibility, via backaction …
A protocol of quantum dense coding protection of two qubits is proposed in amplitude damping (AD) channel using weak measurement and reversal measurement. It is found that the capacity of quantum dense coding under the weak measurement and reversal measurement is always greater than that without weak measurement and reversal measurement. When the protocol is applied, for the AD channels with different damping coefficient, the result reflects that quantum entanglement can be protected and quantum dense coding becomes successful.
Weak measurements with mixed probe states were investigated in the presence of noise in the probe system. We show that the completely mixed state can be used as a probe state for weak measurements of imaginary weak values. The completely mixed probe state has some advantages in terms of noise tolerance. We also experimentally demonstrated weak measurements of polarization rotation via unpolarized light, which corresponds to weak measurements with completely mixed probe states in a two-state system.
Weak measurement is a novel quantum measurement scheme, which is usually characterized by the weak value formalism. To guarantee the validity of the weak value formalism, the fidelity between the pre-selection and the post-selection should not be too small generally. In this work, we study the weak measurement on a qubit system with exactly or asymptotically orthogonal pre- and post-selections. We shall establish a general rigorous framework for the weak measurement beyond the weak value formalism, and obtain the average output of a weak measurement when the pre- and post-selections are exactl…
Weak values are usually associated with weak measurements of an observable on a pre- and post-selected ensemble. We show that more generally, weak values are proportional to the correlation between two pointers in a successive measurement. We show that this generalized concept of weak measurements displays a symmetry under reversal of measurement order. We show that the conditions for order symmetry are the same as in classical mechanics. We also find that the imaginary part of the weak value has a counterpart in classical mechanics. This scheme suggests new experimental possibilities.
Three recent results on weak measurements are presented. They are: i) repeated measurements on a single copy can not provide any information on it and further, that in the limit of very large such measurements, weak measurements have exactly the same characterstics as strong measurements, ii) the apparent non-invasiveness of weak measurements is \emph{illusory} and they are no more advantageous than strong measurements even in the specific context of establishing Leggett-Garg inequalities, when errors are properly taken into account, and, finally, iii) weak value measurements are optimal, in t…
Weak values and measurements have been proposed as means to achieve dramatic enhancements in metrology based on the greatly increased range of possible measurement outcomes. Unfortunately, the very large values of measurement outcomes occur with highly suppressed probabilities. This raises three vital questions in weak-measurement-based metrology, namely, (Q1) Does post-selection enhance the measurement precision? (Q2) Does weak measurement offer better precision than strong measurement? (Q3) Is it possible to beat the standard quantum limit or to achieve the Heisenberg limit with weak measure…
Measurements are crucial in quantum mechanics, in fundamental research as well as in applicative fields like quantum metrology, quantum-enhanced measurements and other quantum technologies. In the recent years, weak-interaction-based protocols like Weak Measurements and Protective Measurements have been experimentally realized, showing peculiar features leading to surprising advantages in several different applications. In this work we analyze the validity range for such measurement protocols, that is, how the interaction strength affects the weak value extraction, by measuring different polar…
The Holevo bound is a keystone in many applications of quantum information theory. We propose "weak maximal Holevo quantity" with weak measurements as the generalization of the standard Holevo quantity which is defined as the optimal projective measurements. The scenarios that weak measurements is necessary are that only the weak measurements can be performed because for example the system is macroscopic or that one intentionally tries to do so such that the disturbance on the measured system can be controlled for example in quantum key distribution protocols. We evaluate systematically the we…
The main purpose of this short article is to give a brief overview of the development of the very interesting weak measurement protocol. I add some comments relating to the reality of weak values, and also comment on the allowed values of observables between measurements.
Weak measurements may result in extra quantity of quantumness of correlations compared with standard projective measurement on a bipartite quantum state. We show that the quantumness of correlations by weak measurements can be consumed for information encoding which is only accessible by coherent quantum interactions. Then it can be considered as a resource for quantum information processing and can quantify this quantum advantage. We conclude that weak measurements can create more valuable quantum correlation.
Since weak measurements are known to produce measurement values that can be much larger than the maximal eigenvalues of the measured observable, it is an interesting question how this enhancement of the measurement signal relates to the sensitivity of quantum measurements as investigated in the field of quantum metrology. In this presentation, it is pointed out that the estimation of a small interaction parameter using weak measurements actually corresponds to standard quantum metrology, where the logarithmic derivatives of the final measurement probabilities are proportional to the correspond…
We present a general framework of examining the validity of weak measurement -- the standard procedure to acquire Aharonov's weak value -- which has been used intensively in recent years for precision measurement, taking advantage of the amplification mechanism available for the weak value. Our framework allows one to deal systematically with various causes of uncertainties intrinsic to the actual measurement process as well as those found in the theoretical analysis employed to describe the system. Using our framework, we examine in detail the two seminal experiments, Hosten's detection measu…
We discuss two questions related to the concept of weak values as seen from the standard quantum-mechanics point of view. In the first part of the paper, we describe a scenario where unphysical results similar to those encountered in the study of weak values are obtained using a simple experimental setup that does not involve weak measurements. In the second part of the paper, we discuss the correct physical description, according to quantum mechanics, of what is being measured in a weak-value-type experiment.
Recently, weak measurements were used to measure small effects that are transverse to the propagation direction of a light beam. Here we address the question whether weak measurements are also useful for measuring small longitudinal phase shifts. We show that standard interferometry greatly outperforms weak measurements in a scenario involving a purely real weak value. However, we also present an interferometric scheme based on a purely imaginary weak value, combined with a frequency-domain analysis, which may have potential to outperform standard interferometry by several orders of magnitude.
We show that in wavepacket tunnelling localisation of the transmitted particle amounts to a quantum measurement of the delay it experiences in the barrier. With no external degree of freedom involved, the envelope of the wavepacket plays the role of the initial pointer state. Under tunnelling conditions such 'self measurement' is necessarily weak, and the Hartman effect just reflects the general tendency of weak values to diverge, as post-selection in the final state becomes improbable. We also demonstrate that it is a good precision, or 'not really weak' quantum measurement: no matter how wid…
Weak measurement with a coherent state pointer and in combination with an orthogonal postselection can lead to a surprising amplification effect, and we give a fire-new physical mechanism about the weak measurement in order to understand this effect. Moreover, this physical mechanism is a general result and based on it, we present a scheme of optomechanical system to implement weak measurement amplification on an orthogonal postselection.
In this paper, we propose a scheme to enhance trapping of entanglement of two qubits in the environment of a photonic band gap material. Our entanglement trapping promotion scheme makes use of combined weak measurements and quantum measurement reversals. The optimal promotion of entanglement trapping can be acquired with a reasonable finite success probability by adjusting measurement strengths.
The problem of measurement in quantum mechanics is studied within the Entropic Dynamics framework. We discuss von Neumann and Weak measurements, wavefunction collapse, and Weak Values as examples of bayesian and entropic inference.
We show how to optimally protect quantum states and quantum entanglement under non-zero temperature based on measurement reversal from weak measurement. In particular, we present explicit formulas of the protection.
For continuous weak measurement of qubits, we obtain exact expressions for weak values (WVs) from the post-selection restricted average of measurement outputs, by using both the quantumtrajectory- equation (QTE) and quantum Bayesian approach. The former is applicable to short-time weak measurement, while the latter can relax the measurement strength to finite. We find that even in the "very" weak limit the result can be essentially different from the one originally proposed by Aharonov, Albert and Vaidman (AAV), in a sense that our result incorporates non-perturbative correction which could be…
In this work, we consider the systematic error of quantum metrology by weak measurements under decoherence. We derive the systematic error of maximum likelihood estimation in general to the first-order approximation of a small deviation in the probability distribution, and study the robustness of standard weak measurement and postselected weak measurements against systematic errors. We show that, with a large weak value, the systematic error of a postselected weak measurement when the probe undergoes decoherence can be significantly lower than that of a standard weak measurement. This indicate…
Closed expressions are derived for the quantum measurement statistics of pre-and postselected gaussian particle beams. The weakness of the pre-selection step is shown to compete with the non-orthogonality of post-selection in a transparent way. The approach is shown to be useful in analyzing post-selection-based signal amplification, allowing measurements to be extended far beyond the range of validity of the well-known Aharonov-Albert-Vaidman limit.
Weak measurements can be seen as an attempt at answering the 'Which way?' question without destroying interference between the pathways involved. Unusual mean values obtained in such measurements represent the response of a quantum system to this 'forbidden' question, in which the 'true' composition of virtual pathways is hidden from the observer. Such values indicate a failure of a measurement where the uncertainty principle says it must fail, rather than provide an additional insight into physical reality.
We propose to use weak measurements away from the weak-value amplification regime to carry out precision measurements of time delays of light. Our scheme is robust to several sources of noise that are shown to only limit the relative precision of the measurement. Thus, they do not set a limit on the smallest measurable phase shift contrary to standard interferometry and weak-value based measurement techniques. Our idea is not restricted to phase-shift measurements and could be used to measure other small effects using a similar protocol.
Based on the quantum technique of weak measurement, we propose a scheme to protect the entanglement from correlated amplitude damping decoherence. In contrast to the results of memoryless amplitude damping channel, we show that the memory effects play a significant role in the suppression of entanglement sudden death and protection of entanglement under severe decoherence. Moreover, we find that the initial entanglement could be drastically amplified by the combination of weak measurement and quantum measurement reversal even under the correlated amplitude damping channel. The underlying mecha…
We present a scheme to estimate Gaussian states of one-dimensional continuous variable systems, based on weak (unsharp) quantum measurements. The estimation of a Gaussian state requires us to find position ($q$), momentum ($p$) and their second order moments. We measure $q$ weakly and follow it up with a projective measurement of $p$ on half of the ensemble, and on the other half we measure $p$ weakly followed by a projective measurement of $q$. In each case we use the state twice before discarding it. We compare our results with projective measurements and demonstrate that under certain condi…
Despite their important applications in metrology and in spite of numerous experimental demonstrations, weak measurements are still confusing for part of the community. This sometimes leads to unjustified criticism. Recent papers have experimentally clarified the meaning and practical significance of weak measurements, yet in [R.E. Kastner, Found. Phys. 47, 697-707 (2017)], Kastner seems to take us many years backwards in the debate, casting doubt on the very term "weak value" and the meaning of weak measurements. Kastner appears to ignore both the basics and frontiers of weak measurements and…
The use of weak measurements for performing quantum tomography is enjoying increased attention due to several recent proposals. The advertised merits of using weak measurements in this context are varied, but are generally represented by novelty, increased efficacy, and foundational significance. We critically evaluate two proposals that make such claims and find that weak measurements are not an essential ingredient for most of their advertised features.
Many quantum measurements, such as photodetection, can be destructive. In photodetection, when the detector clicks a photon has been absorbed and destroyed. Yet the lack of a click also gives information about the presence or absence of a photon. In monitoring the emission of photons from a source, one decomposes the strong measurement into a series of weak measurements, which describe the evolution of the state during the measurement process. Motivated by this example of destructive photon detection, a simple model of destructive weak measurements using qubits was studied in [1]. It has shown…
Weak values are typically obtained experimentally by performing weak measurements, which involve weak interactions between the measured system and a probe. However, the determination of weak values does not necessarily require weak measurements, and several methods without weak system-probe interactions have been developed previously. In this work, a framework for measuring weak values is proposed to describe the relationship between various weak measurement techniques in a unified manner. This framework, which uses a probe-controlled system transformation instead of the weak system-probe inte…
Recent empirical work in the field of 'weak measurements' has yielded novel ways of more directly accessing and exploring the quantum wavefunction. Measuring either position or momentum for a photon in a 'weak' manner yields a wide range of possible values for the measurement, and can be done in such a way as to only minimally effect the wavefunction rather than to collapse it to a specific precise value. Measuring the other complementary variable (position or momentum) precisely at a later time ('post-selection') and averaging the weak measurements can yield information about the wavefunction…
