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Entropy

A Survey of the Concept of Disturbance in Quantum Mechanics

Ernesto Benítez Rodríguez, L. M. Arévalo Aguilar

Published Feb 2, 2019• 12 citations
Publisher

Abstract

The concept of disturbance is of transcendental importance in Quantum Mechanics (QM). This key concept has been described in two different ways, the first one considering that the disturbance affects observables like x and p, as in the Heisenberg’s analysis of the measurement process and the other one takes into consideration that disturbance affects the state of the system instead. Entropic information measures have provided a path for studying disturbance in these both approaches; in fact, we found that initially it was studied by employing these entropic measures. In addition, in the last decade, there was an extensive amount of analyses and several new definitions of the disturbance concept emerged. Many crucial factors like this have inspired this concise paper which gathers the different concepts and definitions that have emerged through time for the better understanding of this topic.

DOI
10.3390/e21020142

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The concept of disturbance is of transcendental importance in Quantum Mechanics (QM). This key concept has been described in two different ways, the first one considering that the disturbance affects observables like x and p, as in the Heisenberg’s analysis of the measurement process and the other one takes into consideration that disturbance affects the state of the system instead. Entropic information measures have provided a path for studying disturbance in these both approaches; in fact, we found that initially it was studied by employing these entropic measures. In addition, in the last decade, there was an extensive amount of analyses and several new definitions of the disturbance concept emerged. Many crucial factors like this have inspired this concise paper which gathers the different concepts and definitions that have emerged through time for the better understanding of this topic.
The role of measurement-induced disturbance in weak measurements is of central importance for the interpretation of the weak value. Uncontrolled disturbance can interfere with the postselection process and make the weak value dependent on the details of the measurement process. Here we develop the concept of a generalized weak measurement for classical and quantum mechanics. The two cases appear remarkably similar, but we point out some important differences. A priori it is not clear what the correct notion of disturbance should be in the context of weak measurements. We consider three different notions and get three different results: (1) For a ``strong'' definition of disturbance, we find that weak measurements are disturbing. (2) For a weaker definition we find that a general class of weak measurements is nondisturbing, but that one gets weak values which depend on the measurement process. (3) Finally, with respect to an operational definition of the ``degree of disturbance,'' we find that the AAV weak measurements are the least disturbing, but that the disturbance is always nonzero.
The Heisenberg uncertainty principle, which underlies many quantum key features, is under close scrutiny regarding its applicability to new scenarios. Using both the Bell-Kochen-Specker theorem establishing that observables do not have predetermined values before measurements and the measurement postulate of quantum mechanics, we propose that in order to describe the disturbance produced by the measurement process, it is convenient to define disturbance by the changes produced on quantum states. Hence, we propose to quantify disturbance in terms of the square root of the Jensen-Shannon entropy distance between the probability distributions before and after the measurement process. Additionally, disturbance and statistical distinguishability of states are fundamental concepts of quantum mechanics that have thus far been unrelated; however, we show that they are intermingled thereupon we enquire into whether the statistical distinguishability of states, caused by statistical fluctuations in the measurement outcomes, is responsible for the disturbance's magnitude.
One of the most striking deviations of quantum mechanics from classical mechanics is that any measurement of a microscopic system involves the interaction with the measuring instruments. The interaction cannot be neglected and in fact leads to uncontrollable disturbance on the system. In 1927, Werner Karl Heisenberg in his celebrated paper established the quantitative expression for unavoidable momentum disturbance caused by any position measurement. The crucial word here is simultaneously. The uncertainty relation gives the lower bound on the product of the uncertainties of two pairs of variables. But it says nothing about the accuracy with which individual observables can be measured. The physical meaning of energy-time uncertainty is different from that of x-px uncertainty. Position and momentum are observables to which Hermitian operators are assigned and which are measured at a particular time. An important implication of the uncertainty principle is that quantum systems do follow classical trajectories which conflict with our everyday experience.
We show the flaws found in the customary fidelity-based definitions of disturbance in quantum measurements and evolutions. We introduce the "entropic disturbance" D and show that it adequately measures the degree of disturbance, intended essentially as an irreversible change in the state of the system. We also find that it complies with an information-disturbance tradeoff, namely the mutual information between the eigenvalues of the initial state and the measurement results is less than or equal to D .
Uncertainty relations play a central role in quantum mechanics. Entropic uncertainty relations in particular have gained significant importance within quantum information, providing the foundation for the security of many quantum cryptographic protocols. Yet, rather little is known about entropic uncertainty relations with more than two measurement settings. In this note we review known results and open questions. The uncertainty principle is one of the fundamental ideas of quantum mechanics. Since Heisenberg’s uncertainty relations for canonically conjugate variables, they have been one of the most prominent examples of how quantum mechanics differs from the classical world (Heisenberg, 1927). Uncertainty relations today are probably best known in the form given by (Robertson, 1929), who extended Heisenberg’s result to two arbitrary observables A and B. Robertson’s relation states that if we prepare many copies of the state |ψ〉, and measure each copy individually using either A or B, we have
From the early days of quantum theory, it is felt that the classical ideas about information may need reorientation under a quantum theoretical framework. In quantum theory, where Heisenberg uncertainty principle is the guiding philosophy, a non-commuting observable cannot simultaneously have precisely defined values. In fact, if two observations do not commute, performing a measurement of one observable will necessarily influence the outcome of a subsequent measurement of the other observable. Hence, the act of acquiring information about a physical system inevitably disturbs the state of the system. This is because of the fact that the outcome of a measurement has some random elements due to uncertainty of the initial state of the system. Moreover, acquiring information causes a disturbance, and due to this disturbance the information cannot be copied with perfect fidelity. The information under this framework is termed as quantum information [1].