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AIP conference proceedings

A Quantification of Disturbance

Lorenzo Maccone

Published Jan 1, 2007• 0 citations
Publisher

Abstract

I introduce the “entropic disturbance” D that measures the quantity of disturbance induced by an arbitrary evolution of a quantum system. The quantity D satisfies the requirements that a disturbance should possess, as long as one focuses on the irreversibility of any disturbance. Other quantities customarily introduced to gauge disturbance do not weight irreversibility appropriately. An information‐disturbance tradeoff is introduced. It states that I ⩽ D, where I quantifies the information contained in the state of the system, i.e. the mutual information between the eigenvalues of the initial state and the measurement results.

DOI
10.1063/1.2713477

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I introduce the “entropic disturbance” D that measures the quantity of disturbance induced by an arbitrary evolution of a quantum system. The quantity D satisfies the requirements that a disturbance should possess, as long as one focuses on the irreversibility of any disturbance. Other quantities customarily introduced to gauge disturbance do not weight irreversibility appropriately. An information‐disturbance tradeoff is introduced. It states that I ⩽ D, where I quantifies the information contained in the state of the system, i.e. the mutual information between the eigenvalues of the initial state and the measurement results.
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 .
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Heisenberg's original intuition was that there should be a tradeoff between measuring a particle's position with greater precision and disturbing its momentum. Rigorous formulations of this idea have primarily focused on the question of how well two complementary observables can be jointly measured. Here, we provide an alternative approach based on how enhancing the predictability of one observable necessarily disturbs a complementary one. The tradeoff refers to a clear operational scenario capturing the effect of the measurement process on a single quantum system. Moreover, our relation is expressed by entropic quantities with clear statistical meaning evading recent criticism directed at some previous formulations. We discuss the performance of our tradeoff relation for existing experimental setups involving qubit measurements performed in Vienna and Toronto, and show that our relation is perfectly tight for all measurement strengths in the Toronto setup.
In the general theory of quantum measurement, one associates a positive-semidefinite operator on a ddd-dimensional Hilbert space to each of the nnn possible outcomes of an arbitrary measurement. In the special case of a projective measurement, these operators are pairwise Hilbert-Schmidt orthogonal, but when n>dn>dn>d, orthogonality is restricted by positivity. This restriction allows us to more precisely state the quantum adage: information gain of a system is always accompanied by unavoidable disturbance. Specifically, we investigate three properties of a measurement with L\"uders rule updating: its disturbance, a measure of how the expected postmeasurement state deviates from the input; its measurement strength, a measure of the intrinsic information producing capacity of the measurement; and its orthogonality, a measure of the degree to which the measurement operators differ from an orthonormal set. These quantities satisfy an information-disturbance tradeoff relation that highlights the additional role played by orthogonality. Finally, we assess several classes of measurements on these grounds and identify symmetric informationally complete quantum measurements as the unique quantum analogs of a perfectly informative and nondisturbing classical ideal measurement.
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.