The projection postulate has been used to predict a slow-down of the time evolution of the state of a system under rapidly repeated measurements, and ultimately a freezing of the state. To test this so-called quantum Zeno effect an experiment was performed by Itano et al. (Phys. Rev. A 41, 2295 (1990)) in which an atomic-level measurement was realized by means of a short laser pulse. The relevance of the results has given rise to controversies in the literature. In particular the projection postulate and its applicability in this experiment have been cast into doubt. In this paper we show analytically that for a wide range of parameters such a short laser pulse acts as an effective level measurement to which the usual projection postulate applies with high accuracy. The corrections to the ideal reductions and their accumulation over n pulses are calculated. Our conclusion is that the projection postulate is an excellent pragmatic tool for a quick and simple understanding of the slow-down of time evolution in experiments of this type. However, corrections have to be included, and an actual freezing does not seem possible because of the finite duration of measurements.
In this paper it is proposed the dynamical quantum Zeno Effect in quantum decision theory. The measurement postulate is not an essential ingredient for the explanation of the quantum Zeno effect, a dynamical account is given in quantum physics. In this account, the entanglement between the system of interest and the apparatus inhibit the quantum transition. The collapse postulate is not considered. It is show in this paper that the belief-action entanglement model provides a mathematical framework for the dynamical quantum Zeno effect in quantum decision theory. It is also shown that, in this …
The so-called quantum Zeno effect is essentially a consequence of the projection postulate for ideal measurements. To test the effect Itano et al. have performed an experiment on an ensemble of atoms where rapidly repeated level measurements were realized by means of short laser pulses. Using dynamical considerations we give an explanation why the projection postulate can be applied in good approximation to such measurements. Corrections to ideal measurements are determined explicitly. This is used to discuss in how far the experiment of Itano et al. can be considered as a test of the quantum …
Repeated observations inhibit the coherent evolution of quantum states through the quantum Zeno effect. In multi-qubit systems this effect provides new opportunities to control complex quantum states. Here, we experimentally demonstrate that repeatedly projecting joint observables of multiple spins creates coherent quantum Zeno subspaces and simultaneously suppresses dephasing caused by the environment. We encode up to two logical qubits in these subspaces and show that the enhancement of the dephasing time with increasing number of projections follows a scaling law that is independent of the …
Short pulses of a probe laser have been used in the past to measure whether a two-level atom is in its ground or excited state. The probe pulse couples the ground state to a third, auxiliary, level of the atom. Occurrence or absence of resonance fluorescence were taken to mean that the atom was found in its ground or excited state, respectively. In this paper we investigate to what extent this procedure results in an effective measurement to which the projection postulate can be applied, at least approximately. We discuss in detail the complications arising from an additional time development …
We perform stochastic simulations of the quantum Zeno and anti-Zeno effects for two level system and for the decaying one. Instead of simple projection postulate approach, a more realistic model of a detector interacting with the environment is used. The influence of the environment is taken into account using the quantum trajectory method. The simulation of the measurement for a single system exhibits the probabilistic behavior showing the collapse of the wave-packet. When a large ensemble is analysed using the quantum trajectory method, the results are the same as those produced using the de…
We exhibit a specific implementation of the creation of geometrical phase through the state-space evolution generated by the dynamic quantum Zeno effect. That is, a system is guided through a closed loop in Hilbert space by means a sequence of closely spaced projections leading to a phase difference with respect to the original state. Our goal is the proposal of a specific experimental setup in which this phase could be created and observed. To this end we study the case of neutron spin, examine the practical aspects of realizing the "projections," and estimate the difference between the ideal…
State-vectors resulting from collapse along the forward light cone from a measurement interaction can be used for the attribution of both local and non-local properties.
Modern development of quantum technologies based on quantum information theory stimulated analysis of proposed computational, cryptographic and teleportational schemes from the viewpoint of quantum foundations. It is evident that not all mathematical calculations performed in complex Hilbert space can be directly realized in physical space. Recently by analyzing the original EPR paper we found that they argument was based on the misuse of the von Neumann's projection postulate. Opposite to von Neumann, Einstein, Podolsky and Rosen (EPR) applied this postulate to observables represented by oper…
We present a decoherence-based interpretation for the quantum Zeno effect (QZE) where measurements are dynamically treated as dispersive couplings of the measured system to the apparatus, rather than the von Neumann's projections. It is found that the explicit dependence of the survival probability on the decoherence time quantitatively distinguishes this dynamic QZE from the usual one based on projection measurements. By revisiting the cavity-QED experiment of the QZE [J. Bernu, et al., Phys. Rev. Lett, 101, 180402 (2008)], we suggest an alternative scheme to verify our theoretical considerat…
We show that the projection postulate plays a crucial role in the discussion on the so called "quantum nonlocality", in particular in the EPR-argument. We stress that the original von Neumann projection postulate was crucially modified by extending it to observables with degenerate spectra (the Lüders postulate) and we show that this modification is highly questionable from a physical point of view, and it is the real source of "quantum nonlocality". The use of the original von Neumann postulate eliminates this problem: instead of "action at the distance"-nonlocality, we obtain a classical mea…
Quantum Zeno effect is a significant tool in quantum manipulating and computing. We propose its observation in superconducting phase qubit with two experimentally feasible measurement schemes. The conventional measurement method is used to achieve the proposed pulse and continuous readout of the qubit state, which are analyzed by projection assumption and Monte Carlo wave-function simulation, respectively. Our scheme gives a direct implementation of quantum Zeno effect in a superconducting phase qubit.
If frequent measurements ascertain whether a quantum system is still in a given subspace, it remains in that subspace and a quantum Zeno effect takes place. The limiting time evolution within the projected subspace is called quantum Zeno dynamics. This phenomenon is related to the limit of a product formula obtained by intertwining the time evolution group with an orthogonal projection. By introducing a novel product formula we will give a characterization of the quantum Zeno effect for finite-rank projections, in terms of a spectral decay property of the Hamiltonian in the range of the projec…
We study the measurement process by treating classical detectors entirely quantum mechanically. As a generic model we use a point-contact detector coupled to an electron in a quantum dot and tunneling into the continuum. Transition to the classical description and the mechanism of decoherence are investigated. We concentrate on the influence of the measurement on the electron decay rate to the continuum. We demonstrate that the Zeno (or the anti-Zeno) effect requires a nonuniform density of states in the continuum. In this case we show that the anti-Zeno effect relates only to the average deca…
Proponents of the Everett interpretation of Quantum Theory have made efforts to show that to an observer in a branch, everything happens as if the projection postulate were true without postulating it. In this paper, we will indicate that it is only possible to deduce this rule if one introduces another postulate that is logically equivalent to introducing the projection postulate as an extra assumption. We do this by examining the consequences of changing the projection postulate into an alternative one, while keeping the unitary part of quantum theory, and indicate that this is a consistent …
The paper discuss the structure of quantum mechanics and uniqueness of its postulates. The Born rule for quantum probabilities is fixed by requirement of nonexistence of quantum telepathy. Von Neumann projection postulate describes the transformation of quantum state under the condition of no-interaction measurement. Projection postulate could be considered as transition to conditional probability under the condition of a certain result of quantum measurement.
We consider a driven 2-level system with one level showing spontaneous decay to an otherwise uncoupled third level. Rabi transitions to the unstable level are strongly damped. This simple configuration can be used to demonstrate and to explore the quantum Zeno effect leading to a freezing of the system to the initial level. A comparison with repeated projection measurements is given. A treatment within a phenomenological theory of continuous measurements is sketched. The system visualizes the important role of null measurements (negative result measurements) and may serve as a good example for…
An apparatus model with discrete momentum space suitable for the exact solution of the problem is considered. The special Hamiltonian of its interaction with the object system under consideration is chosen. In this simple case it is easy to illustrate how difficulties in constructing the dynamical interpretation of selective collapse could be overcomed without any limiting procedure. For this purpose one can apply either averaging with respect to a non-quantum parameter or reducing the algebra of the joint system operators (i. e. passing to the quantum system subalgebra). The latter procedure …
Notwithstanding it is well known that von Neumann's projection postulate is inapplicable to most realistic measurement procedures, it keeps haunting the foundations of quantum mechanics. In particular its applicability to EPR experiments is often assumed. In the present contribution this problem is considered from the point of view of a quantum mechanical theory of measurement, allowing a treatment of projection in EPR experiments as a special case of conditional preparation. The conditions are spelled out under which the postulate may be applicable.
Quantum tunneling is a fundamental quantum mechanical effect involved in plenty of physical phenomena. Its control would impact these phenomena and the technologies based on them. We show that the quantum tunneling probability through a potential barrier can be increased to approach unity in a quantum Zeno dynamics undergone by the tunneling particle in which the direction of the momentum is frequently monitored. We first model the measurements of the momentum direction as selective von Neumann projections, and then as nonselective, direction-sensitive interactions of the particle with probe p…
We study transitions of a particle between two wells, separated by a reservoir, under the condition that the particle is not detected in the reservoir. Conventional quantum trajectory theory predicts that such no-result continuous measurement would not affect these transitions. We demonstrate that it holds only for Markovian reservoirs (infinite bandwidth $Λ$). In the case of finite $Λ$, the probability of the particle's interwell transition is a function of the ratio $Λ/ν$, where $ν$ is the frequency of measurements. This scaling tells us that in the limit $ν\to\infty$, the measurement freeze…
It is observed that the proofs of hidden-variable no-go theorems depend on the `projection postulate,' which is seen to be contradictory with respect to spin operators in directions orthogonal to the magnetic field direction. In this light it is argued that it is less costly to abandon the projection postulate than to abandon locality and this in turn renders Hidden-Variable No-Go Theorems evadible. To buttress this point, a local realist model of the EPR experiment which ignores the constraints of the projection postulate is presented.
An operational approach to quantum state reduction, the state change of the measured system caused by a measurement of an observable conditional upon the outcome of measurement, is founded without assuming the projection postulate in any stages of the measuring process. Whereas the conventional formula assumes that the probe measurement satisfies the projection postulate, a new formula for determining the state reduction shows that the state reduction does not depend on how the probe observable is measured, or in particular does not depend on whether the probe measurement satisfies the project…
We consider a point particle in one dimension initially confined to a finite spatial region whose state is frequently monitored by projection operators onto that region. In the limit of infinitely frequent monitoring, the state never escapes from the region -- this is the Zeno effect. The aim of this paper is to show how the Zeno effect disappears in the classical limit in this and similar examples. We give a general argument showing that the Zeno effect is suppressed in the presence of a decoherence mechanism which kills interference between histories. We show how this works explicitly by cou…
We prove a product formula which involves the unitary group generated by a semibounded self-adjoint operator and an orthogonal projection $P$ on a separable Hilbert space $\HH$, with the convergence in $L^2_\mathrm{loc}(\mathbb{R};\HH)$. It gives a partial answer to the question about existence of the limit which describes quantum Zeno dynamics in the subspace \hbox{$\mathrm{Ran} P$}. The convergence in $\HH$ is demonstrated in the case of a finite-dimensional $P$. The main result is illustrated in the example where the projection corresponds to a domain in $\mathbb{R}^d$ and the unitary group…
What is called "orthodox" quantum mechanics, as presented in standard foundational discussions, relies on two substantive assumptions --- the projection postulate and the eigenvalue-eigenvector link --- that do not in fact play any part in practical applications of quantum mechanics. I argue for this conclusion on a number of grounds, but primarily on the grounds that the projection postulate fails correctly to account for repeated, continuous and unsharp measurements (all of which are standard in contemporary physics) and that the eigenvalue-eigenvector link implies that virtually all interes…
We propose to use a quantum adiabatic and simulated-annealing framework to compute theground state of small molecules. The initial Hamiltonian of our algorithms is taken to be themaximum commuting Hamiltonian that consists of a maximal set of commuting terms in the fullHamiltonian of molecules in the Pauli basis. We consider two variants. In the first method, weperform the adiabatic evolution on the obtained time- or path-dependent Hamiltonian with theinitial state as the ground state of the maximum commuting Hamiltonian. However, this methoddoes suffer from the usual problems of adiabatic qua…
Given a nonnegative self-adjoint operator $H$ acting on a separable Hilbert space and an orthogonal projection $P$ such that $H_P := (H^{1/2}P)^*(H^{1/2}P)$ is densely defined, we prove that $\lim_{n\rightarrow \infty} (P\,\mathrm{e}^{-itH/n}P)^n = \mathrm{e}^{-itH_P}P$ holds in the strong operator topology. We also derive modifications of this product formula and its extension to the situation when $P$ is replaced by a strongly continuous projection-valued function satisfying $P(0)=P$.
The dynamics of a quantum system undergoing frequent measurements (quantum Zeno effect) is investigated. Using asymptotic analysis, the system is found to evolve unitarily in a proper subspace of the total Hilbert space. For spatial projections, the generator of the "Zeno dynamics" is the Hamiltonian with Dirichlet boundary conditions.
We apply the projection evolution approach to the particle detection process and calculation of the detection moment. Influence of the essential system properties on the evolution process is discussed. It is shown, that using only the projection postulate in the evolution scheme allows to understand the time as a kind of observable.
We demonstrated that final states of ultracold scattering between atom and molecule can be selectively produced using dynamic magnetic fields of multiple frequencies. The mechanism of the dynamic magnetic field control is based on a generalized quantum Zeno effect for the selected scattering channels. In particular, we use an atom-molecule spin flip scattering to show that the transition to the selected final spin projection of the molecule in the inelastic scattering can be suppressed by dynamic modulation of coupling between the Floquet engineered initial and final states.
From each of Feynman's and von Neumann's postulates to the restricted Feynman path integrals: a mathematical theory of temporally continuous quantum measurements
本文严格证明了在简单近似下，Mensky 的受限 Feynman 路径积分可由 Feynman 量子化公理和 von Neumann 瞬时测量公理导出，并应用于自旋系统及多种量子效应的表述。
This note is devoted to some foundational aspects of quantum mechanics (QM) related to quantum information (QI) theory, especially quantum teleportation and ``one way quantum computing.'' We emphasize the role of the projection postulate (determining post-measurement states) in QI and the difference between its Lüders and von Neumann versions. These projection postulates differ crucially in the case of observables with degenerate spectra. Such observables play the fundamental role in operations with entangled states: any measurement on one subsystem is represented by an observable with degener…
This is an attempt to apply Nagel's distinction between internal and external statements to the interpretation of quantum mechanics. I propose that this distinction resolves the contradiction between unitary evolution and the projection postulate. I also propose a more empirically realistic version of the projection postulate. The result is a version of Everett's relative-state interpretation, including a proposal for how probabilities are to be understood. Based on a talk given at the 9th UK Foundations of Physics meeting in Birmingham on 12 September 2000.
The measurement process in quantum mechanics is usually described by the von Neumann projection postulate, which forms a basic constituent of the laws of quantum mechanics. Since this postulate requires the outside observer of the system, it is hard to apply quantum mechanics to the whole Universe. Therefore we propose that the quantum measurement process is actually a physical process associated with the ubiquitous mechanism of spontaneous symmetry breaking. Based on this proposal, we construct a quantum measurement model in which the von Neumann projection is described as the dynamical pro-c…
We study an electrostatic qubit monitored by a point-contact detector. Projecting an entire qubit-detector wave function on the detector eigenstates we determine the precision limit for the qubit measurements, allowed by quantum mechanics. We found that this quantity is determined by qubit dynamics as well as decoherence, generated by the measurement. Our results show how the quantum precision limit can be improved by a proper design of a measurement procedure.
We study the measurement-induced enhancement of the spontaneous decay (called quantum anti-Zeno effect) for a two-level subsystem, where measurements are treated as couplings between the excited state and an auxiliary state rather than the von Neumann's wave function reduction. The photon radiated in a fast decay of the atom, from the auxiliary state to the excited state, triggers a quasi-measurement, as opposed to a projection measurement. Our use of the term "quasi-measurement" refers to a "coupling-based measurement". Such frequent quasi-measurements result in an exponential decay of the su…
In recent work, the so-called quasi-Zeno dynamics of a system has been investigated in the context of the quantum first passage problem. This dynamics considers the time evolution of a system subjected to a sequence of selective projective measurements made at small but finite intervals of time. This means that one has a sequence of steps, with each step consisting of a unitary transformation followed by a projection. The dynamics is non-unitary and, in recent work, it has been shown that it can be effectively described by two different non-Hermitian Hamiltonians. Here we explore this connecti…
If frequent measurements ascertain whether a quantum system is still in its initial state, transitions to other states are hindered and the quantum Zeno effect takes place. However, in its broader formulation, the quantum Zeno effect does not necessarily freeze everything. On the contrary, for frequent projections onto a multidimensional subspace, the system can evolve away from its initial state, although it remains in the subspace defined by the measurement. The continuing time evolution within the projected "quantum Zeno subspace" is called "quantum Zeno dynamics:" for instance, if the meas…
Recently foundational issues of applicability of the formalism of quantum mechanics (QM) to cognitive psychology, decision making, and psychophysics attracted a lot of interest. In particular, in \cite{DKBB} the possibility to use of the projection postulate and representation of "mental observables" by Hermitian operators was discussed in very detail. The main conclusion of the recent discussions on the foundations of "quantum(-like) cognitive psychology" is that one has to be careful in determination of conditions of applicability of the projection postulate as a mathematical tool for descri…
We modify the theory of the Quantum Zeno Effect to make it consistent with the postulates of quantum mechanics. This modification allows one, throughout a sequence of observations of an excited system, to address the nature of the observable and thereby to distinguish survival from non-decay, which is necessary whenever excited states are degenerate. As a consequence, one can determine which types of measurements can possibly inhibit the exponential decay of the system. We find that continuous monitoring taken as the limit of a sequence of ideal measurements will only inhibit decay in special …
We establish an exact differential equation for the operator describing time-dependent measurements continuous in time and obtain a series solution. Suppose the projection operator $E(t) = U(t) E U^\dagger(t)$ is measured continuously from t = 0 to T, where E is a projector leaving the initial state unchanged and U(t) a unitary operator obeying U(0) = 1 and some smoothness conditions in t. We prove that the probability of always finding E(t) = 1 from t = 0 to T is unity. If $U(t) \neq 1$, the watched kettle is sure to `boil'.
This paper is a comparison of two theories of the probability of a history in quantum mechanics. One is derived from Copenhagen quantum mechanics using the projection postulate and is the basis of the "consistent histories" interpretation; the other is based on a proposal by Bell, originally for the "pilot state" theory but here applied to pure unitary quantum mechanics. The first can be used for a wider class of histories but depends on the projection postulate, or "collapse", which is widely held to be an unsatisfactory feature of the theory; the second can be used in a theory of the univers…
It was predicted that frequently repeated measurements on an unstable quantum state may alter the decay rate of the state. This is called the quantum Zeno effect (QZE) or the anti-Zeno effect (AZE), depending on whether the decay is suppressed or enhanced. In conventional theories of the QZE and AZE, effects of measurements are simply described by the projection postulate, assuming that each measurement is an instantaneous and ideal one. However, real measurements are not instantaneous and ideal. For the QZE and AZE by such general measurements, interesting and surprising features have recentl…
Measurement is one of the most counter-intuitive aspects of quantum physics. Frequent measurements of a quantum system lead to quantum Zeno dynamics where time evolution becomes confined to a subspace defined by the projections. However, weak measurement performed at a finite rate is also capable of locking the system into such a Zeno subspace in an unconventional way: by Raman-like transitions via virtual intermediate states outside this subspace, which are not forbidden. Here, we extend this concept into the realm of non-Hermitian dynamics by showing that the stochastic competition between m…
This note is a part of my efforts for getting rid of nonlocality from quantum mechanics (QM). Quantum nonlocality is two faced Janus, one face is Lüders projection nonlocality, another face is Bell nonlocality. This paper is devoted to disillusion of the latter. The main casualty of Bell's model with hidden variables is that it straightforwardly contradicts to the Heinsenberg's uncertainty and generally Bohr's complementarity principles. Thus, we do not criticize the derivation or interpretation of the Bell inequality (as was done by numerous authors). Our critique is directed against the mode…
We derive differential equations for the modified Feynman propagator and for the density operator describing time-dependent measurements or histories continuous in time. We obtain an exact series solution and discuss its applications. Suppose the system is initially in a state with density operator $ρ(0)$ and the projection operator $E(t) = U(t) E U^\dagger(t)$ is measured continuously from $t = 0$ to $T$, where $E$ is a projector obeying $Eρ(0) E = ρ(0)$ and $U(t)$ a unitary operator obeying $U(0) = 1$ and some smoothness conditions in $t$. Then the probability of always finding $E(t) = 1$ fr…
A review of protective measurements. Protective measurement is a method for measuring an expectation value of an observable on a single quantum system. The quantum state of the system can be protected by a potential, when the state is a nondegenerate energy eigenstate with a known gap to neighboring states, or via Zeno effect by frequent projection measurements.
The evolution of a quantum system under observation becomes retarded or even impeded. We review this ``quantum Zeno effect'' in the light of the criticism that has been raised upon a previous attempt to demonstrate it, of later reexaminations of both the projection postulate and the significance of the observations, and of the results of a recent experiment on an individual cold atom. Here, the micro-state of the quantum system gets unveiled with the observation, and the effect of measurement is no longer mixed up with dephasing the object's wave function by the reactive effect of the detectio…
The well known interpretational difficulties with nonlinear Schrödinger and von Neumann equations can be reduced to the problem of computing multiple-time correlation functions in the absence of Heisenberg picture. Having no Heisenberg picture one often resorts to Zeno-type reasoning which explicitly involves the projection postulate as a means of computing conditional and joint probabilities. Although the method works well in linear quantum mechanics, it completely fails for nonlinear evolutions. We propose an alternative way of performing the same task in linear quantum mechanics and show th…
We consider a spin chain extending from Alice to Bob with next neighbors interactions, initially in its ground state. Assuming that Bob measures the last spin of the chain, the energy of the spin chain has to increase, at least on average, due to the measurement disturbance. Presumably, the energy is provided by Bob's measurement apparatus. Assuming now that, simultaneously to Bob's measurement, Alice measures the first spin, we show that either energy is not conserved, - implausible - or the projection postulate doesn't apply, and that there is signalling. An explicit measurement model shows …
In this paper, without any priori assumption about the post-measurement state of system, we will examine how this state is restricted by assuming each of these following assumptions. First, by using this reasonable assumption that two successive measurements should be describable as one measurement. Second, by assuming the impossibility of faster than light signaling, "No-signaling condition". However, only by using these assumptions it is not possible to obtain the usual projection postulate. Instead, by means of a simple lemma, we will show that the density operator of system after a measure…
Everettian Quantum Mechanics, or the Many Worlds Interpretation, lacks an explanation for quantum probabilities. We show that the values given by the Born rule equal projection factors, describing the contraction of Lebesgue measures in orthogonal projections from the complex line of a quantum state to eigenspaces of an observable. Unit total probability corresponds to a complex Pythagorean theorem: the measure of a subset of the complex line is the sum of the measures of its projections on all eigenspaces. To show that projection factors can work as probabilities, we postulate the existence o…
The Projection Postulate from Standard Quantum Mechanics relies fundamentally on measurements. But measurements implicitly suggest the existence of anthropocentric notions like measuring devices, which should rather emerge from the theory. This article proposes an alternative formulation of the Standard Quantum Mechanics, in which the Projection Postulate is replaced with a version in which measurements and observations are not assumed as fundamental. More precisely, the Wigner functions representing the quantum states on the phase space are required to be tightly constrained to regions of the…
