Research in Theoretical Physics —
Professor M. D. Srinivas
After completing M. Sc. Physics from Bangalore University, I obtained a fellowship from the University of Rochester and worked on Quantum Optics and the Foundations of Quantum Mechanics under the supervision of Professor Emil Wolf and completed the Ph.D. Thesis on Quantum Mechanics as a Generalized Stochastic Process in Phase Space in 1976. Since then I worked as a faculty member, in the Department of Theoretical Physics, University of Madras, till 1996.
My research work in Theoretical Physics has been mainly in the area of Conceptual and Mathematical Foundations of Quantum Theory. While basically of foundational interest, this work has also found significant applications in the areas of Quantum Optics and Quantum Information Processing. The following brief overview summarizes the research work according to the major themes which have been pursued.
Non-Classical Features of Quantum Mechanics
One of the early concerns of our work has been to investigate how quantum theory and classical stochastic theory differ at the fundamental level. An interesting way of comparing quantum theory with classical stochastic theories is by formulating quantum mechanics as a generalized statistical theory in phase space. This reveals several non-classical features inherent in quantum mechanics. One crucial feature is the theorem of Wigner that the joint probability densities characterizing quantum states are not necessarily non-negative. Our work TP5 provides a new proof of Wigner's theorem and reveals its deep links with the uncertainty principle. TP1, TP3, TP7 and TP8, present a detailed analysis of several non-classical features, which show up when quantum mechanics is formulated as a generalized stochastic theory in phase space. TP2 and TP4 present some important applications of this formalism in quantum statistical mechanics.
Quantum Probability Theory and Quantum Information Theory
The basic joint probabilities predicted by quantum theory are the joint probabilities associated with realizing different outcomes in a sequence of observations. The fundamental non-classical feature of these joint probabilities is that the probability distribution associated with any observable depends not only on the state of the system as prepared initially, but also on the entire sequence of observations carried out on the system - this has been termed the quantum interference of probabilities by de Broglie. This is reflected in the fact that the structure of the event space associated with Quantum Probability Theory is very different from the Boolean algebraic structure characteristic of the event space in Classical Probability Theory as axiomatized by Kolmogorov. Our analysis of the foundations of quantum probability theory in TP6 has been considered a landmark contribution. The paper has been reprinted in a collection of basic papers on the logico-algebraic foundations of quantum theory over the past seventy years. The fact that, in the quantum event space, conjunction is non-commutative and that there exist several maximal elements has many interesting consequences regarding the conditional probabilities and so on, as shown in TP10.
Investigations into the fundamentals of quantum probability theory led us to the formulation of the basic principles of quantum information theory in TP11. One crucial application of these studies is in the understanding of the uncertainty relations for successive measurements. The usual formulation of the uncertainty relation (say for position and momentum) consider measurements which are performed on two distinct but identically prepared ensembles of systems. The conventional wisdom associated with the uncertainty relation that it expresses the unavoidable interference of one measurement on the other can be formalized only when one considers sequential measurements performed on the same ensemble, as has been done in TP14, TP24, TP25 and TP30. The later papers consider the Entropic Formulation of the Uncertainty Relation for Successive Measurements and reveal that the optimal bound on the sum of sum of entropic uncertainties of two or more observables when they are sequentially measured on the same ensemble of systems is greater than or equal to the sum of entropic uncertainties of these observables when they are measured on distinct but identically prepared ensembles of systems.
Relativity and Quantum Mechanics: Local Causality and Hidden Variable Theories
The notion of successive observations in relativistic quantum theory is analyzed in TP13. In TP16 we discuss the issue of time as an observable in quantum theory, and show that the notion of the "time of occurrence" of an event can be formulated consistently only in the context of an observation carried out over an extended interval of time.
It is a celebrated result of Bell that any hidden variable formulation of quantum mechanics also necessarily violates the requirement of local causality. In TP21, we show that any hidden variable formulation of quantum mechanics necessarily distinguishes two ensembles of systems which, though represented by the same density operator, have been prepared by mixing systems in different set of pure states. Using this result we present in TP22 a remarkably simple and revealing proof of Bell's theorem without taking recourse to any inequalities.
While Bell's theorem demonstrates the violation of local causality in hidden variable formulations of quantum theory, the predictions of quantum theory are known to satisfy local causality at the statistical level. In TP28 we investigate the conditions under which quantum theory satisfies local causality at the statistical level, when one considers a more general class of measurements.
Measurement Theory for Observables with Continuous Spectra
The conventional formulation of the collapse postulate due to von Neumann and Lüders is applicable only for observables with a purely discrete spectrum. Unless this postulate is extended to observables with continuous spectra, it is not possible to discuss the post measurement state of a system consequent to the measurement of even standard observables such as position and momentum. The only prescription of standard quantum theory, which refers to the successive measurement of such observables, is the prescription for the joint probabilities associated with two compatible observables, which is very general and does not impose any restriction on the spectrum of the observables. In TP15 it is shown that one can use this prescription to work out possible extensions of the Collapse Postulate for Observables with Continuous Spectrum. We obtain the remarkable "negative" result that any such extension is not possible within the conventional formulation of quantum mechanics, which restricts the allowed states to so-calld normal states characterized by density operators. In this paper, as well as in TP26, we show how a generalization of the collapse postulate is possible if one allows non-normal states also. The resultant statistics of successive observations is also investigated and it is shown that one will have to accept joint probabilities which are only finitely additive in the case of continuous spectrum observables. Whether such a generalization of the conventional framework is acceptable or not depends on whether one takes continuous spectrum observables seriously or otherwise.
We also use the generalized collapse postulate to discuss the successive measurement of position immediately followed by momentum, something which was not possible in the conventional framework of quantum mechanics. We are able to exactly formalize the intuitive content of the Heisenberg microscope experiment and show that a precise measurement of position leads to the situation that, immediately afterwards, the momentum is concentrated entirely at plus-minus infinity.
Quantum Theory of Continuous Measurements and Applications in Quantum Optics
Another limitation of the conventional formulations of quantum theory is that it is restricted to the consideration of a single instantaneous measurement or, at best, to a sequence of such instantaneous measurements. There are many situations frequently met with where one performs an experiment over an extended interval of time and records when a particular event occurs. This is for instance the case in counting experiments which record when different particles are detected by a counter which is continuously active over an extended interval of time. We discuss the quantum theory of counting experiments in TP9 and TP12, based on a formulation developed by Davies for discussing such "continuous measurements".
In TP17 we show that the Quantum Theory of Continuous Measurements is the appropriate framework for discussing photon-counting experiments in quantum optics. This paper, which has been widely cited, presents what has now become the standard framework (sometimes referred to as the Srinivas - Davies model) for discussing counting experiments in quantum optics. It has led to the clarification of the fact (see TP20, TP29) that the conventional discussion of photon-counting (as given for instance by Kelley and Kleiner, which leads to the so called quantum Mandel formula which gives rise to negative probabilities), is inadequate because it does not take into account the change in the state of the electromagnetic field in those intervals where no counts are recorded. Our analysis has led to the development of the correct quantum counting formula by Chmara.
In TP18 and TP19 we show how the above framework can be used to discuss dead time effects in quantum counting experiments. Dead times are precisely those intervals when the detector is inactive and does not perform any measurement. In TP18 we present a complete solution to the corresponding classical problem of obtaining the dead time corrections to the counting statistics of a doubly stochastic Poisson process. In the case of quantum theory, it is possible to discuss such dead time effects only in the framework of the quantum theory of continuous measurements as is done in TP19.
In the last two decades there have been several applications of the above framework in diverse contexts in quantum optics. Apart from photon counting experiments another important class of phenomena which are best discussed in the framework of the quantum theory of continuous measurements is the so called "quantum jumps", which deal with the random emission times of photons in the resonance fluorescence of atomic systems. In the review article TP29, we review these and other applications of the quantum theory of continuous measurements in quantum optics.
Some of the above studies on the conceptual and mathematical foundations of quantum mechanics Have been compiled in the book M D Srinivas, Measurements and Quantum Probabilities, Universities Press, Hyderabad, 2001