What Weak Measurements and Weak Values Really Mean: Reply to Kastner

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 Kastner (Found Phys 47:697–707, 2017), Kastner seems to take us many years backwards in the 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 misinterprets the weak measurement process and its outcomes. In addition, she accuses the authors of Aharonov et al. (Ann Phys 355:258–268, 2015) in statements completely opposite to the ones they have actually made. There are many points of disagreement between Kastner and us, but in this short reply I will leave aside the ontology (which is indeed interpretational and far more complex than that described by Kastner) and focus mainly on the injustice in her criticism. I shall add some general comments regarding the broader theory of weak measurements and the two-state-vector formalism, as well as supporting experimental results. Finally, I will point out some recent promising results, which can be proven by (strong) projective measurements, without the need of employing weak measurements.     

On Fuzzy Probability Theory

Our main aim from this work is to see which theorems in classical probability theory are still valid in fuzzy probability theory. Following Gudder's approach [Demonestratio Mathematica 31(3), 1998, 235–254; Foundations of Physics, 30, 1663–1678] to fuzzy probability theory, the basic concepts of the theory, that is of fuzzy probability measures and fuzzy random variables (observables), are presented. We show that fuzzy random variables extend the usual ones. Moreover, we prove that for any separable metrizable space, the crisp observables coincide with random variables. Then we prove the existence of a joint observable for any collection of observables, and we prove the weak law of large numbers and the central limit theorem in the fuzzy context. We construct a new definition of almost everywhere convergence. After proving that Gudder's definition implies ours and presenting an example that indicates that the converse is not true, we prove the strong law of large numbers according to this definition.     

Weak Solutions of the Boltzmann Equation Without Angle Cutoff

The definition of the concept of weak solution of the nonlinear Boltzmann equation, recently introduced by the author, is used to prove that, without any cutoff in the collision kernel, the Boltzmann equation for Maxwell molecules in the one-dimensional case has a global weak solution in this sense. Global conservation of energy follows.     

Quantum Theory Without Hilbert Spaces

Quantum theory does not only predict probabilities, but also relative phases for any experiment, that involves measurements of an ensemble of systems at different moments of time. We argue, that any operational formulation of quantum theory needs an algebra of observables and an object that incorporates the information about relative phases and probabilities. The latter is the (de)coherence functional, introduced by the consistent histories approach to quantum theory. The acceptance of relative phases as a primitive ingredient of any quantum theory, liberates us from the need to use a Hilbert space and non-commutative observables. It is shown, that quantum phenomena are adequately described by a theory of relative phases and non-additive probabilities on the classical phase space. The only difference lies on the type of observables that correspond to sharp measurements. This class of theories does not suffer from the consequences of Bell's theorem (it is not a theory of Kolmogorov probabilities) and Kochen–Specker's theorem (it has distributive logic). We discuss its predictability properties, the meaning of the classical limit and attempt to see if it can be experimentally distinguished from standard quantum theory. Our construction is operational and statistical, in the spirit of Copenhagen, but makes plausible the existence of a realist, geometric theory for individual quantum systems.     

Quantum Weak Values and Logic: An Uneasy Couple

Quantum mechanical weak values of projection operators have been used to answer which-way questions, e.g. to trace which arms in a multiple Mach–Zehnder setup a particle may have traversed from a given initial to a prescribed final state. I show that this procedure might lead to logical inconsistencies in the sense that different methods used to answer composite questions, like “Has the particle traversed the way X or the way Y?”, may result in different answers depending on which methods are used to find the answer. I illustrate the problem by considering some examples: the “quantum pigeonhole” framework of Aharonov et al., the three-box problem, and Hardy’s paradox. To prepare the ground for my main conclusion on the incompatibility in certain cases of weak values and logic, I study the corresponding situation for strong/projective measurements. In this case, no logical inconsistencies occur provided one is always careful in specifying exactly to which ensemble or sample space one refers. My results cast doubts on the utility of quantum weak values in treating cases like the examples mentioned.     

Correlations and Entanglement in Probability Theory

We generalize the classical probability frame by adopting a wider family of random variables that includes non-deterministic ones. The frame that emerges is known to host a “classical” extension of quantum mechanics. We discuss the notion of probabilistic correlation and show that it includes two kinds of correlation: a classical one, which occurs for both deterministic and indeterministic observables, and a non-classical one, which occurs only for indeterministic observables. The latter will be called probabilistic entanglement and represents a property of intrinsically random systems, not necessarily quantum. It appears possible to separate the two kinds of correlation and characterize them by numerical functions which satisfy a simple product rule. This paper was written a few months before the death of S. Bugajski: the first author recalls him as a creative scientist, a great human personality, and a dear friend.     

Controlling Chaos Probability of a Bose-Einstein Condensate in a Weak Optical Superlattice

The spatial chaos probability of a Bose-Einstein condensate perturbed by a weak optical superlattice is studied. It is demonstrated that the spatial. chaotic solution appears with a certain probability in a given parameter region under a random boundary condition. The effects of the lattice depths and wave vectors on the chaos probability are illustrated, and different regions associated with different chaos probabilities are found. This suggests a feasible scheme for suppressing and strengthening chaos by adjusting the optical superlattice experimentaJly.     

From Weak to Strong Coupling in ABJM Theory

The partition function of N=6{\mathcal{N}=6} supersymmetric Chern–Simons-matter theory (known as ABJM theory) on \mathbbS³{\mathbb{S}^3} , as well as certain Wilson loop observables, are captured by a zero dimensional super-matrix model. This super–matrix model is closely related to a matrix model describing topological Chern–Simons theory on a lens space. We explore further these recent observations and extract more exact results in ABJM theory from the matrix model. In particular we calculate the planar free energy, which matches at strong coupling the classical IIA supergravity action on AdS₄×\mathbbC\mathbbP³{{\rm AdS}_4\times\mathbb{C}\mathbb{P}^3} and gives the correct N ^3/2 scaling for the number of degrees of freedom of the M2 brane theory. Furthermore we find contributions coming from world-sheet instanton corrections in \mathbbC\mathbbP³{\mathbb{C}\mathbb{P}^3} . We also calculate non-planar corrections, both to the free energy and to the Wilson loop expectation values. This matrix model appears also in the study of topological strings on a toric Calabi–Yau manifold, and an intriguing connection arises between the space of couplings of the planar ABJM theory and the moduli space of this Calabi–Yau. In particular it suggests that, in addition to the usual perturbative and strong coupling (AdS) expansions, a third natural expansion locus is the line where one of the two ’t Hooft couplings vanishes and the other is finite. This is the conifold locus of the Calabi–Yau, and leads to an expansion around topological Chern–Simons theory. We present some explicit results for the partition function and Wilson loop observables around this locus.     

Towards a Quantum Analog of Weak KAM Theory

We discuss a quantum analogue of Mathers minimization principle for Lagrangian dynamics, and provide some formal calculations suggesting the corresponding Euler–Lagrange equation. We then rigorously construct from the dual eigenfunctions of a certain non-selfadjoint operator a candidate for a minimizer, and recover aspects of weak KAM theory in the limit as h0. Regarding our state as a quasimode, we furthermore derive some error estimates, although it remains an open problem to improve these bounds.Supported in part by NSF Grant DMS-0070480 and by the Miller Institute for Basic Research in Science, UC Berkeley     

Operational Phase-Space Probability Distribution in Quantum Communication Theory

Operational phase-space probability distributions are useful tools for describing quantum mechanical systems, including quantum communication and quantum information processing systems. It is shown that quantum communication channels with Gaussian noise and quantum teleportation of continuous variables are described by operational phase-space probability distributions. The relation of operational phase-space probability distribution to the extended phase-space formalism proposed by Chountasis and Vourdas is discussed.     

正确理解算数平均值巧解物理题

先看一例．【例题】一质量为／／7,的金属杆。6以一定的初速度从一光滑平行金属导轨底端向上滑行,导轨平面与水平面成30°,两导轨上端用一电阻R相连,如图1所示．磁场垂直斜面向上,导轨与杆的电阻不计,金属杆上行到某一高度之后又返回到底端,则在此过程中,     

On Expected Values and “Negative Probability” in 4-Space QED

A proposed 4-space Dirac theory requires modified definitions of expected value and Hermitian operator, because the charge density is not positive definite. However, this does not imply negative probability.     

Electromagnetic Field Theory Without Divergence Problems 2. A Least Invasively Quantized Theory

Classical electrodynamics based on the Maxwell–Born–Infeld field equations coupled with a Hamilton–Jacobi law of point charge motion is partially quantized. The Hamilton–Jacobi phase function is supplemented by a dynamical amplitude field on configuration space. Both together combine into a single complex wave function satisfying a relativistic Klein–Gordon equation that is self-consistently coupled to the evolution equations for the point charges and the electromagnetic fields. Radiation-free stationary states exist. The hydrogen spectrum is discussed in some detail. Upper bounds for Born's “aether constant” are obtained. In the limit of small velocities of and negligible radiation from the point charges, the model reduces to Schrödinger's equation with Coulomb Hamiltonian, coupled with the de Broglie–Bohm guiding equation.     

Cosmological Background in Higgs Scalar-Tensor Theory Without Higgs Particles

The scalar background field and its consequencesare discussed for the Friedmann-type cosmologicalsolutions of the scalar-tensor theory of gravity withthe Higgs field of the Standard Model as the scalar gravitational field.     

New Electroweak Model Without a Higgs Particle

A new unified electroweak model is proposed in this paper. In this unified electroweak model, Higgsmechanism is not used, so no Higgs particle exists in the model. In order to keep the masses of intermediate gaugebosons non-zero, two sets of gauge fields will be introduced. In order to introduce symmetry breaking and to help tointroduce the masses of all fields, a vacuum potential is needed. Except for those terms concerning Higgs particle, thefundamental dynamical properties of this model are similar to those of the standard model. And in a proper limit, thismodel will approximately return to the standard model. The purpose of this paper is not to say that the Higgs particledoes not exist in Nature, it is only to prove that, without a Higgs particle, we can also set up a unified electroweak modelwhich is consistent with present experiments.     

New Challenges for Classical and Quantum Probability

The discovery that any classical random variable with all moments gives rise to a full quantum theory (that in the Gaussian and Poisson cases coincides with the usual one) implies that a quantum–type formalism will enter into practically all applications of classical probability and statistics. The new challenge consists in finding the classical interpretation, for different types of classical contexts, of typical quantum notions such as entanglement, normal order, equilibrium states, etc. As an example, every classical symmetric random variable has a canonically associated conjugate momentum. In usual quantum mechanics (associated with Gaussian or Poisson classical random variables), the interpretation of the momentum operator was already clear to Heisenberg. How should we interpret the conjugate momentum operator associated with classical random variables outside the Gauss–Poisson class? The Introduction is intended to place in historical perspective the recent developments that are the main object of the present exposition.     

A New Family of Continuous Probability Distributions

In this paper, a new parametric compound G family of continuous probability distributions called the Poisson generalized exponential G (PGEG) family is derived and studied. Relevant mathematical properties are derived. Some new bivariate G families using the theorems of “Farlie-Gumbel-Morgenstern copula”, “the modified Farlie-Gumbel-Morgenstern copula”, “the Clayton copula”, and “the Renyi’s entropy copula” are presented. Many special members are derived, and a special attention is devoted to the exponential and the one parameter Pareto type II model. The maximum likelihood method is used to estimate the model parameters. A graphical simulation is performed to assess the finite sample behavior of the estimators of the maximum likelihood method. Two real-life data applications are proposed to illustrate the importance of the new family.     

Experimental Evidence for a Dynamical Non-locality Induced Effect in Quantum Interference Using Weak Values

The quantum theoretical concepts of modular momentum and dynamical non-locality, which were introduced four decades ago, have recently been used to explain single particle quantum interference phenomena. Although the non-local exchange of modular momentum associated with such phenomena cannot be directly observed, it has been suggested that effects induced by this exchange can be measured experimentally using weak measurements of pre- and post-selected ensembles of particles. This paper reports on such an optical experiment that yielded measured weak values that were consistent with the theoretical prediction of an effect induced by a non-local exchange of modular momentum.