Weak measurements offer insights into the behavior of quantum systems. Combined with postselection, quantum mechanics predicts a range of experimentally testable phenomena. In this paper I consider weak measurements performed on time-dependent preselected and postselected ensembles, with emphasis on the decay of excited states. The results show that the standard exponential decay law is a limiting case of a more general law that depends on both the time of postselection and the choice of final state. The generalized law is illustrated for two interesting choices of postselection.
Weak measurements offer insights into the behavior of quantum systems. Combined with postselection, quantum mechanics predicts a range of experimentally testable phenomena. In this paper I consider weak measurements performed on time-dependent preselected and postselected ensembles, with emphasis on the decay of excited states. The results show that the standard exponential decay law is a limiting case of a more general law that depends on both the time of postselection and the choice of final state. The generalized law is illustrated for two interesting choices of postselection.
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 preselection and the postselection 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 postselections. 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 postselections are exactly orthogonal. We shall also study the asymptotic behavior of a weak measurement in the limiting process that the pre- and postselections tend to be orthogonal.
We investigate the sequential composition of weak values in the framework of time-symmetric quantum mechanics. Specifically, we consider a forward'' weak measurement from a preselected state ∣ψ⟩ to a post-selected state ∣ϕ⟩, followed by a reverse'' weak measurement. We show that the product of these two weak values corresponds to the normalized expectation value of a strong, state-conditioned observable B=APψA, where Pψ=∣ψ⟩⟨ψ∣ is the projector onto the preselected state. Analyzing the structure of B, we demonstrate how it encodes interference information, particularly when ∣ψ⟩ is a superposition rather than an eigenstate of A. This formulation extends naturally to mixed states by replacing Pψ with a generic density matrix ρ, linking the construction to the formalism of generalized quantum measurements. We illustrate practical applications in quantum information, including state-specific error witnessing in quantum computing, and show how the phase of a weak value can be inferred via strong measurements in the pure-state case.
A description of quantum systems at the time interval between two successive measurements is presented. Two wave functions, the first preselected by the initial measurement and the second postselected by the final measurement, describe quantum systems at a single time. It is shown how this approach leads to a new concept: a weak value of an observable. Weak values represent novel characteristics of quantum systems between two measurements. They are outcomes of a standard measurement procedure that fulfills certain requirements of ``weakness.'' We call it weak measurement. Physical meaning, underlying mathematical structure, and prospects of practical usage of weak measurements are explored.
Entanglement and nonlocality are studied in the framework of pre-/postselected ensembles with the aid of weak measurements and the two-state-vector formalism. In addition to the EPR–Bohm experiment, we revisit the Hardy and Cheshire Cat experiments, whose entangled pre- or postselected states give rise to curious phenomena. We then turn to even more peculiar phenomenon suggesting “emerging correlations” between independent preand postselected ensembles of particles. This can be viewed as a quantum violation of the classical “pigeonhole principle.” Introduction It is seldom acknowledged that seven years before the celebrated Bell paper [1], Bohm and Aharonov [2] published an analysis of the EPR paradox [3]. They suggested an experimental setup, based on Compton scattering, for testing nonlocal correlations between the polarizations of two annihilation photons. In 1964, Bell proposed his general inequality, thereby excluding local realism. During the same time, Aharonov et al. constructed the foundations of a time-symmetric formalism of quantum mechanics [4]. While Bell's proof utilizes entanglement to demonstrate nonlocal correlations,wewill describe inwhat follows the emergence of nonlocal correlations between product states. For this purpose, however, we shall invoke weak measurements of pre- and postselected ensembles. In classical mechanics, initial conditions of position and velocity for every particle fully determine the time evolution of the system. Therefore, trying to impose an additional final condition would lead either to redundancy or to inconsistency with the initial conditions. This is radically different in quantum mechanics. Because of the uncertainty principle, an initial state vector does not fully determine, in general, the outcome of a future measurement. However, adding a final (backward-evolving) state vector results in a more complete description of the quantum system in between these two boundary conditions, which has a bearing on the determination of measurement outcomes. The basis for this time-symmetric formulation of quantum mechanics was laid down by Aharonov, Bergman, and Lebowitz (ABL), who derived a symmetric probability rule concerning measurements performed on systems, while taking into account the final state of the system, in addition to the usual initial state [4].
We use homodyne detection to monitor the radiative decay of a superconducting qubit. According to the classical theory of conditional probabilities, the excited-state population differs from an exponential decay law if it is conditioned upon a later projective qubit measurement. Quantum trajectory theory accounts for the expectation values of general observables, and we use experimental data to show how a homodyne detection signal is conditioned upon both the initial state and the finally projected state of a decaying qubit. We observe, in particular, how anomalous weak values occur in continuous weak measurement for certain pre- and postselected states. Subject to homodyne detection, the density matrix evolves in a stochastic manner, but it is restricted to a specific surface in the Bloch sphere. We show that a similar restriction applies to the information associated with the postselection, and thus bounds the predictions of the theory.
We study quantum measurements with preselection and postselection, and derive the precise expressions of the measurement results without any restriction on the coupling strength. For a qubit system, we derive the maximum measurement results by choosing appropriate initial and finial states. Significant amplification effect is obtained when the interaction between the system and the measuring device is very weak; and typical ideal quantum measurement results are obtained when the interaction is strong. The improvement of the signal-to-noise ratio (SNR) and the enhancement of the measure sensitivity (MS) by weak measurements are studied. Without considering the probability decrease due to postselection, the SNR and the MS can both be significantly improved by weak measurements; surprisingly, neither SNR nor MS can be effectively improved when the probability decrease is considered.