The exact conditions on valid pointer states for weak measurements are derived. It is demonstrated that weak measurements can be performed with any pointer state with vanishing probability current density. This condition is found both for weak measurements of noncommuting observables and for $c$-number observables. In addition, the interaction between pointer and object must be sufficiently weak. There is no restriction on the purity of the pointer state. For example, a thermal pointer state is fully valid.
Comment on "Weak Measurements with Orbital-Angular-Momentum Pointer states"
在最近的一封信（物理学。109，040401（2012）），G. Puentes，N. Hermosa和J. P. Torres报道了一个利用轨道角动量态作为指针态来提取高阶弱值的方案。他们声称这样的弱值只能用高斯指针状态来访问。在这篇评论中，我们表明，高斯指针状态本身可以提供访问高阶弱值，如果适当的指针位移观察。
The transition from von Neumann's projective strong measurement to Aharonov's weak measurement has recently received large attention, theoretical and experimental. In this work, we present a general approach to describe the weak-to-strong measurement transition for Fock-state-based pointer pointer states, and analyze in some details the case of coherent pointer states. A possible realization of our measurement scheme using trapped ions is also discussed.
In the weak measurement (WM) scenario involving weak interaction and postselection by projective measurement, the empirical significance of weak values is manifested in terms of shifts in the measurement pointer's mean position and mean momentum. In this context, a general quantitative treatment is presented in this paper by taking into account the hitherto unexplored effect of correlations among the pointer degrees of freedom which pertain to an arbitrary multidimensional preselected pointer state. This leads to an extension of the earlier results, showing that, for complex weak values, the c…
While the novel applications of weak values have recently attracted wide attention, weak measurement, the usual way to extract weak values, suffers from risky approximations and severe quantum noises. In this paper, we show that the weak-value information can be obtained exactly in strong measurement with postselections, via measuring the coupling-deformed pointer observables, i.e., the observables selected according to the coupling strength. With this approach, we keep all the advantages claimed by weak-measurement schemes and at the same time solve some widely criticized problems thereof, su…
The purpose of the present paper is to derive the pointer states of a macro-object using a simple perturbation method. We study the model Hamiltonian involving the weak interaction between the center of mass and its environment. The main conclusion is that the pointer states emerge as the small part of the total state vector when the interaction Hamiltonian is not dominant: the degrees of freedom being at the center of interaction can be localized. In the weak interaction case, if the disappearance of the energy contribution from the off-diagonal terms is caused by the decoherence, it threaten…
Quantum metrology is being gradually studied for weak measurement systems. For weak measurement systems with thermal state pointer, we find that in the displacement space corresponding to imaginary weak values, the maximal QFI after successful postselection can attain the level of thermal fluctuations, without surpassing total QFI, and that QFI which increases with increasing temperature can constantly improve the measurement precision. These results are much better than that of weak measurement with pure state (i.e., Gaussian state) pointer. On the other hand, in Kerr nonlinear interaction sy…
It is argued that a weak value of an observable is a robust property of a single pre- and post-selected quantum system rather than a statistical property. During an infinitesimal time a system with a given weak value affects other systems as if it were in an eigenstate with eigenvalue equal to the weak value. This differs significantly from the action of a system pre-selected only and possessing a numerically equal expectation value. The weak value has a physical meaning beyond a conditional average of a pointer in the weak measurement procedure. The difference between the weak value and the e…
Modular values are quantities that described by pre- and postselected states of quantum systems like weak values but are different from them: The associated interaction is not necessary to be weak. We discuss an optimal modular-value-based measurement with a spin coherent pointer: A quantum system is exposed to a field in which strength is to be estimated through its modular value. We consider two cases, with a two-dimensional and a higher-dimensional pointer, and evaluate the quantum Fisher information. The modular-value-based measurement has no merit in the former case, while its sensitivity…
Weak values are traditionally obtained using a weak interaction between the measured system and a pointer state. It has, however, been pointed out that weak coupling can be replaced by a carefully tailored strong interaction. This paper provides a direct comparison of two strong interaction-based approaches (strong interaction accompanied by either a suitably prepared pointer state or quantum erasure) and the traditional weak interaction-based method. Presented theoretical derivations explicitly prove analytical equivalence of these approaches which was subsequently certified by an experiment …
Direct state tomography (DST) using weak measurements has received wide attention. Based on the concept of coupling-deformed pointer observables presented by Zhang \emph{et al}.[Phys. Rev. A \textbf{93}, 032128 (2016)], a modified direct state tomography (MDST) is proposed, examined, and compared with other typical state tomography schemes. MDST has exact validity for measurements of any strength. We identify the strength needed to attain the highest efficiency level of MDST by using statistical theory. MDST is much more efficient than DST in the sense that far fewer samples are needed to reac…
Most studies for postselected weak measurement focus on using pure Gaussian state as a pointer, which can only give an amplification limit reaching the level of the ground state fluctuation. When the pointer is initialised in a thermal state, we find that the amplification limit after the postselection can reach the level of thermal fluctuation, indicating that the amplification effect achieving the level of thermal fluctuation is also increased with the temperature grow, and also give the amplification mechanism different from the one with pure Gaussian state pointer. To illustrate these resu…
In this work, the measurement transition as well as precision measurement advantages of single-photon-added coherent state after postselected von Neumann measurement are investigated. We noticed that the weak-to-strong measurement transition characterized by the shifts of pointer's position and momentum variables occurred in continuously by controlling a dimensionless parameter associated with system-pointer coupling. We calculate the ratio between the signal-to-noise ratio of nonpostselected and postselected measurements, and the latter is used to find the quantum Fisher information. We found…
In a weak measurement the real and imaginary parts of a weak value participate in the shifts of the complementary variables of a pointer. While the real part represents the value of an observable in the limit of zero measurement strength, the imaginary one is regarded as the back-action due to the measurement with a post-selection, which has an influence on the post-selection probability. In this paper I give a case in which a real part could also appear as such a back-action in a post-selection probability on an equal footing with an imaginary one. It is also shown that both of the real and i…
We study protective quantum measurements in the presence of an environment and decoherence. We consider the model of a protectively measured qubit that also interacts with a spin environment during the measurement. We investigate how the coupling to the environment affects the two characteristic properties of a protective measurement, namely, (i) the ability to leave the state of the system approximately unchanged and (ii) the transfer of information about expectation values to the apparatus pointer. We find that even when the interaction with the environment is weak enough not to lead to appr…
A large literature has grown up around the proposed use of 'weak measurements' (i.e., unsharp measurements followed by post-selection) to allegedly provide information about hidden ontological features of quantum systems. This paper attempts to clarify the fact that 'weak measurements' involve strong (projective) measurements on one (pointer) member of an entangled system. The only thing 'weak' about such measurements is that the correlation established via the entanglement does not correspond to eigenstates of the 'weakly measured observable' for the remaining component system(s) subject to t…
The notion of weak measurement in quantum mechanics has gained a significant and wide interest in realizing apparently counterintuitive quantum effects. In recent times, several theoretical and experimental works have been reported for demonstrating the joint weak value of two observables where the coupling strength is restricted to the second order. In this paper, we extend such a formulation by providing a complete treatment of joint weak measurement scenario for all-order-coupling for the observable satisfying $A^2=\mathbb{I}$ and $A^2=A$, which allows us to reveal several hitherto unexplor…
Recently, weak value $\rm A_w$ derived in the pre- and post-selected weak measurement has been shown to be powerful in measuring minute physical effects. In principle, the decrease in the post-selection probability will increase the sensitivity. Besides, the sensitivity which characterizes the pointer position shift is proportional to the real part of $\rm A_w$, and the sensitivity which characterizes the pointer shift in momentum space is proportional to the imaginary part of $\rm A_w$. However, we find that the relationship between the post-selection probability and the sensitivity is true w…
