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Chapman & Hall/CRC eBooks

Observability and State Determination

Domenico D'Alessandro

Published Aug 3, 2007• 0 citations
Publisher

Abstract

As is known from measurement theory (cf. section 1.2), the measurement of an observable gives a certain result with a probability depending on the state. This suggests that measurements of appropriate observables on several copies of the same system can be used to determine the state of the system with some level of confidence. In systems theory, the study of the observability of a system describes to what extent it is possible to determine the initial state by a combined action of dynamics and output measurement.

DOI
10.1201/9781584888833-8

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As is known from measurement theory (cf. section 1.2), the measurement of an observable gives a certain result with a probability depending on the state. This suggests that measurements of appropriate observables on several copies of the same system can be used to determine the state of the system with some level of confidence. In systems theory, the study of the observability of a system describes to what extent it is possible to determine the initial state by a combined action of dynamics and output measurement.
In systems theory, the study of the observability of a system describes to what extent it is possible to determine the initial state by a combined action of dynamics and output measurement. This chapter briefly summarizes the main ideas of quantum state tomography which is the technique used to determine a quantum state via a series of measurements. It discusses a system theoretic treatment of the problem of state determination namely to the study of the observability of quantum systems. The chapter partitions the space of density matrices into equivalence classes of states that cannot be distinguished by appropriate control and measurement.
The term measurement theory refers to that part of a physical theory in which the empirical and operational content of the concepts of the theory is determined. Measurements are analyzed both as operational procedures defining the → observables of the theory and as physical processes which are themselves subject to the laws of physics. In classical physics, measurements are performed in order to deter-mine the values of one or several observables of the physical system un-der consideration. Classical physics allowed the idealized notion that every physical quantity has a definite value at any time, and that this value can be determined with certainty by measurement without influ-encing the object system in a significant way. By contrast, in quantum mechanics both features fail to hold without strong qualifications. Ac-cordingly, in their seminal paper of 1935 [1], Einstein, Podolsky and Rosen used elements of this description as a sufficient criterion of phys-ical reality, applicable both in classical and quantum mechanics:
The extent to which a temperature can be appropriately assigned to a small quantum system, as an internal property but not as a property of any large environment, is still an open problem. In this paper, a method is proposed for solving this problem, by which a studied small system is coupled to a two-level system as a probe, the latter of which can be measured by measurement devices. A main difficulty in the determination of possible temperature of the studied system comes from the back-action of the probe-system coupling to the system. For small quantum chaotic systems, we show that a temperature can be determined, the value of which is sensitive to neither the form, location, and strength of the probe-system coupling, nor the Hamiltonian and initial state of the probe. The temperature thus obtained turns out to have the form of Boltzmann temperature.