Searcharxiv⌕ Search

arXiv subjects

N. D. Hari Dass

Publications and source records attributed to N. D. Hari Dass.

At least 19 recordsLinked to original sources
arXiv2025

POVM generated quantum trajectories without stochastic differential equations

In this paper we examine the issue of quantum trajectories generated by QND-POVM's on {\it single} copies of unknown states. After an introduction to various aspects of quantum measurements, we discuss an earlier approach by one of us(NDH) based on Gaussian QND measurement operators that addressed the asymptotic behaviour of such trajectories showing the impossibility of determining the unknown state of a single copy from the statistics of such repeated measurements.The essence of our present work is the so called martingale and super-martingale properties of certain observables, and the consequent martingale convergence theorem. The main result is that asymptotically all trajectories approach either the non-degenerate eigenstates of the system observable, or,density matrices spanned by the degenerate eigenstates of the observable. The proofs given by us are very transparent..A unified treatment of both the degenerate and non-degenerate cases is given with the help of projectors of arbitrary dimensionalities.In the degenerate case we reproduce the Lüders prescription. The distribution of the trajectories is shown to be given exactly by the Born rule.Similar conclusions were reached, earlier to us, by Bauer et al on the one hand, and, by Amini et al on the other. A detailed comparison of the three approaches is given. A distinctive feature of all three approaches is that no use is made of stochastic differential equations and the conclusions follow directly from quantum mechanics. Alter and Yamomoto were the first to investigate repeated QND measurements on single copies in unknown states. We make detailed comparisons with their works too. We end with a brief discussion of i) the robustness of the results against free evolutions of both the system as well as the probe and ii) the anti-Zeno aspects of the results.

quant-ph↗
arXiv2024

Effective String Theories (EST) of Yang-Mills Flux Tubes

This chapter explains the concept of \emph{Effective String Theories}(EST), and their success in explaining the results that Yang-Mills flux tubes behave, to a high degree of accuracy, like Bosonic Strings(BST). It describes EST's of Lüscher and Weisz, and their principal conclusions. It then discusses the Polchinski-Strominger EST's. which are valid in all dimensions. It then describes the works by Drummond, and, the author and Peter Matlock, which extended the analysis to $R^{-3}$ order and showed, that even at that order the spectrum is that of BST. The chapter analyses the issues of string momentum in higher orders. It discusses at length the powerful covariant calculus, to systematically construct EST's to arbitrary orders. The most general actions to $R^{-7}$ order are shown to be governed by just two parameters. The works of Aharony and collaborators on the spectrum of EST's, both in static and conformal gauges,to $R^{-5}$ order, and their results that even at that order the ground state energy remains the same as that of BST, but the excited spectrum gets corrected for $D\,>\,3$, are explained. It discusses the simulation results for excited states. It also discusses the AdS-CFT approaches and thickness of flux tubes.Recent works by Ambjorn, Makeenko, and Sedrakyan on the path-integral approaches to this issue are also discussed and compared with the other approaches. It concludes with remarks on the significance of the results for QCD.

hep-th↗
arXiv2020

Dirac and the Path Integral

Through a very careful analysis of Dirac's 1932 paper on the Lagrangian in Quantum Mechanics as well as the second and third editions of his classic book {\it The Principles of Quantum Mechanics}, I show that Dirac's contributions to the birth of the path-integral approach to quantum mechanics is not restricted to just his seminal demonstration of how Lagrangians appear naturally in quantum mechanics, but that Dirac should be credited for creating a path-integral which I call {\it Dirac path-integral} which is far more general than Feynman's while possessing all its desirable features. On top of it, the Dirac path-integral is fully compatible with the inevitable quantisation ambiguities, while the Feynman path-integral can never have that full consistency. In particular, I show that the claim by Feynman that for infinitesimal time intervals, what Dirac thought were analogues were actually proportional can not be correct always. I have also shown the conection between Dirac path-integrals and the Schrödinger equation. In particular, it is shown that each choice of Dirac path-integral yields a {\it quantum Hamiltonian} that is generically different from what the Feynman path-integral gives, and that all of them have the same {\it classical analogue}. Dirac's method of demonstrating the least action principle for classical mechanics generalizes in a most straightforward way to all the generalized path-integrals.

physics.hist-ph↗
arXiv2018

Optimal weak measurements: Mixed States

In an earlier publication we had given an exhaustive analysis of the criteria for weak value measurements of pure states to be optimal in the sense considered by Wootters and Fields. We had proved, for arbitrary spin cases, that the measurements are optimal when the post-selected state is mutually unbiased wrt the eigenstates of the observable being measured.Here we extend the discussion to mixed states. For these, weak value measurements have several problems which we illustrate with the protocol proposed by Shengjun Wu. We discuss tomography of mixed states based on weak measurements and show that while the principal results of Wootters and Fields hold, namely, the set of observables needed for complete tomography are such that their eigenstates form a mutually unbiased bases, weak tomography removes a serious lacuna from the Wootters and Fields analysis i.e the need to consider only state averaged error volumes or information. We also consider another proposal for weak tomography of mixed states by Lundeen and Bamber, and reach similar conclusions about MUB.

quant-ph↗
arXiv2017

On repeated (continuous) weak measurements of a single copy of an unknown quantum state

In this paper we investigate repeated weak measurements,without post-selection, on a \emph{single copy} of an \emph{unknown} quantum state. The resulting random walk in state space is precisely characterised in terms of joint probabilities for outcomes. We conclusively answer, in the negative, the very important question whether the statistics of such repeated measurements can determine the unknown state. We quantify the notion of error in this context as the departure of a suitably averaged density matrix from the initial state. When the number of weak measurements is small the original state is preserved to a great degree, but only an ensemble of such measurements, of a complete set of observables, can determine the unknown state. By a careful analysis of errors, it is shown that there is a precise tradeoff between errors and \emph{invasiveness}. Lower the errors, greater the invasiveness. Though the outcomes are not independently distributed, an analytical expression is obtained for how averages are distributed, which is shown to be the way outcomes are distributed in a \emph{strong measurement}. An \emph{error-disturbance} relation, though not of the Ozawa-type, is also derived. In the limit of vanishing errors, the invasiveness approaches what would obtain from strong measurements.

quant-ph↗
arXiv2017

Optimal weak value measurements: Pure states

We apply the notion of \emph{optimality} of measurements for state determination(tomography) as originally given by Wootters and Fields to \emph{weak value tomography} of \emph{pure states}. They defined measurements to be optimal if they 'minimised' the effects of statistical errors. For technical reasons they actually maximised the state averaged information, precisely quantified as the negative logarithm of 'error volume'. In this paper we optimise both the state averaged information as well as error volumes. We prove, for Hilbert spaces of arbitrary (finite) dimensionality, that varieties of weak value measurements are optimal when the post-selected bases are \emph{mutually unbiased} with respect to the eigenvectors of the observable being measured. We prove a number of important results about the geometry of state spaces when expressed through the weak values as coordinates. We derive an expression for the Kaëhler potential for the N-dimensional case with the help of which we give an exact treatment of the arbitrary-spin case.

quant-ph↗
arXiv2016

Bell-like examples for spin-1 hidden variable theories

Though John Bell had claimed that his spin-1/2 example of a hidden-variable theory(HV) is an \emph{explicit} counterexample to von Neumann's proof of the non-existence of hidden variable theories empirically equivalent to quantum mechanics, such examples can be so construed only if they met all of von Neumann's requirements. In particular, that they reproduced all the observed predictions of quantum theory. To shed light on these aspects, we have, on the one hand, simplified and critically examined Bell's original example and on the other hand, constructed explicit such examples for spin-1 systems. We have clarified the relation of our example to the Kochen-Specker and Bell's powerful earlier results. Our spin-1 examples are manifestly non-contextual, yet violating the K-S constraints configuration by configuration. Nevertheless, they reproduce the correct quantum expectation values and variances for arbitrary linear combinations of the beables in one case, and for close approximants to the beables representing K-S constraints. In conformity with the K-S theorem, the variance of the K-S constraint is nonzero. The implications of this non-vanishing variance are analysed in detail. In the other example, we show how this variance can be made arbitrarily small but not zero.

quant-ph↗
arXiv2016

Killing vectors of FLRW metric (in comoving coordinates) and zero modes of the scalar Laplacian

Based on an examination of the solutions to the Killing Vector equations for the FLRW-metric in co moving coordinates , it is conjectured and proved that the components(in these coordinates) of Killing Vectors, when suitably scaled by functions, are \emph{zero modes} of the corresponding \emph{scalar} Laplacian. The complete such set of zero modes(infinitely many) are explicitly constructed for the two-sphere. They are parametrised by an integer n. For $n\,\ge\,2$, all the solutions are \emph{irregular} (in the sense that they are neither well defined everywhere nor are \emph{square-integrable}). The associated 2-d vectors are also \emph{not normalisable}. The $n=0$ solutions being constants (these correspond to the zero angular momentum solutions) are regular and normalizable. Not all of the $n=1$ solutions are regular but the associated vectors are normalizable. Of course, the action of scalar Laplacian coordinate independent significance only when acting on scalars. However, our conclusions have an unambiguous meaning as long as one works in this coordinate system. As an intermediate step, the covariant Laplacians(vector Laplacians) of Killing vectors are worked out for four-manifolds in two different ways, both of which have the novelty of not explicitly needing the connections. It is further shown that for certain maximally symmetric sub-manifolds(hypersurfaces of one or more constant comoving coordinates) of the FLRW-spaces also, the scaled Killing vector components are zero modes of their corresponding scalar Laplacians. The Killing vectors for the maximally symmetric four-manifolds are worked out using the elegant embedding formalism originally due to Schrödinger . Some consequences of our results are worked out. Relevance to some very recent works on zero modes in AdS/CFT correspondences , as well as on braneworld scenarios is briefly commented upon.

gr-qc↗
arXiv2015

Are weak measurements really necessary for Leggett-Garg type measurements?

Leggett-Garg inequalities are an important milestone in our quest to bridge the classical-quantum divide. An experimental investigation of these inequalities requires the so called \emph{non-invasive measurements}(NIM). It has become popular to invoke weak measurements as the means of realising NIMs to very good approximation, because of their allegedly low disturbance of systems under measurement. In this note, this is shown to be a myth; it is shown, by simple estimates of errors, that for comparable levels of statistical errors, even the strong or projective measurements can be used. In fact, it is shown that resource-wise, strong measurements are even preferable.

quant-ph↗
arXiv2015

Three results on weak measurements

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 the precise sense of Wootters and Fields, when the post-selected states are mutually unbiased with respect to the eigenstates of the observable whose weak values are being measured. Notion of weak value coordinates for state spaces are introduced and elaborated.

quant-ph↗
arXiv2015

Protective Measurements: Probing Single Quantum Systems

Making measurements on single quantum systems is considered difficult, almost impossible if the state is a-priori unknown. Protective measurements suggest a possibility to measure single quantum systems and gain some new information in the process. Protective measurement is described, both in the original and generalized form. The degree to which the system and the apparatus remain entangled in a protective measurement, is assessed. A possible experimental test of protective measurements is discussed.

quant-ph↗
arXiv2014

Ontology and Quantum Mechanics

The issue of ontology in quantum mechanics, or equivalently the issue of the reality of the wave function is critically examined within standard quantum theory. It is argued that though no strict ontology is possible within quantum theory, ingenious measurement schemes may still make the notion of a \emph{FAPP Ontology} i.e ontology for all practical purposes (a phrase coined by John Bell), meaningful and useful.

quant-ph↗
arXiv2013

Freeman Dyson and Gravitational Spin Precession

In 1974 Hulse and Taylor discovered the binary pulsar. At that time Prof. Dyson was visiting the Max Planck Institute for Physics at Munich, where I was also working. He initiated a number of discussions on this object. During them it occurred to me that this system could be used to test Geodetic Precession in Einsteins theory, which, even after years of work by the Stanford gyroscope expt, had remained a challenge. I showed some preliminary calculations to Prof Dyson and he encouraged me to do a more refined job. To be applicable to the binary pulsar, one needed to generalise the general relativistic calculations to beyond the so called test particle assumption. Barker and OConnell had obtained such a result from analysing the gravitational interactions of spin half Dirac fermions in linearized spin 2 theories of gravitation. With C.F. Cho I produced a purely classical calculation, using Schwingers Source theory. Boerner, Ehlers and Rudolf confirmed this result with their general relativistic calculations shortly after. With V. Radhakrishnan, I gave a detailed model for the pulse width and polarization sweep as a means of observing this effect. All throughout Prof. Dyson was supportive with reading the manuscripts and his critical comments. In 2005, coincidentally the centennial of the Annus Mirabilis(1905), Hotan, Bailes and Ord observed this in the binary pulsar J1141 6545.

gr-qc↗
arXiv2013

The Superposition Principle in Quantum Mechanics - did the rock enter the foundation surreptitiously?

The superposition principle forms the very backbone of quantum theory. The resulting linear structure of quantum theory is structurally so rigid that tampering with it may have serious, seemingly unphysical, consequences. This principle has been succesful at even the highest available accelerator energies. Is this aspect of quantum theory forever then? The present work is an attempt to understand the attitude of the founding fathers, particularly of Bohr and Dirac, towards this principle. The Heisenberg matrix mechanics on the one hand, and the Schrodinger wave mechanics on the other, are critically examined to shed light as to how this principle entered the very foundations of quantum theory.

physics.hist-ph↗
arXiv2013

Does the first part of the second law also imply its second part?

Sommerfeld called the first part of the second law to be the entropy axiom, which is about the existence of the state function entropy. It was usually thought that the second part of the second law, which is about the non-decreasing nature of entropy of thermally isolated systems, did not follow from the first part. In this note, we point out the surprise that the first part in fact implies the second part.

physics.hist-ph↗
arXiv2013

Minimum Uncertainty and Entanglement

We address the question, does a system A being entangled with another system B, put any constraints on the Heisenberg uncertainty relation (or the Schrodinger-Robertson inequality)? We find that the equality of the uncertainty relation cannot be reached for any two noncommuting observables, for finite dimensional Hilbert spaces if the Schmidt rank of the entangled state is maximal. One consequence is that the lower bound of the uncertainty relation can never be attained for any two observables for qubits, if the state is entangled. For infinite-dimensional Hilbert space too, we show that there is a class of physically interesting entangled states for which no two noncommuting observables can attain the minimum uncertainty equality.

quant-ph↗
arXiv2012

The Turbulent 'Mixing' Layer as a Problem in the Non-equilibrium Statistical Mechanics of a Vortex Gas

The objective of this paper is to unravel any relations that may exist between turbulent shear flows and statistical mechanics, through a detailed numerical investigation in the simplest case where both can be well defined. The shear flow considered for the purpose is the 2D temporal mixing layer, which is a time-dependent flow that is statistically homogeneous in the streamwise direction (x) and evolves from a plane vortex sheet in the direction normal to it (y) in a periodic-in-x domain with period L. The connections to statistical mechanics are explored by revisiting, via extensive computer simulations, an appropriate initial value problem for a finite but large collection of (N) point vortices of same strength (γ) and sign constituting a 'vortex gas'. Such connections may be expected to be meaningful as hydrodynamics, since the flow associated with the vortex gas is known to provide weak solutions of the Euler equation. Over ten different initial conditions classes are investigated using simulations involving up to 10^4 vortices, with ensemble averages evaluated over up to 10^3 realizations and integration over 10^4 L/(ΔU) (where ΔU is the velocity differential across the layer, given by Nγ/L). (see PDF for complete abstract)

physics.flu-dyn↗