Even though measurement results obtained in the real world are generally both noisy and continuous, quantum measurement theory tends to emphasize the ideal limit of perfect precision and quantized measurement results. In this article, a more general concept of noisy measurements is applied to investigate the role of quantum noise in the measurement process. In particular, it is shown that the effects of quantum noise can be separated from the effects of information obtained in the measurement. However, quantum noise is required to ``cover up'' negative probabilities arising as the quantum limit is approached. These negative probabilities represent fundamental quantum-mechanical correlations between the measured variable and the variables affected by quantum noise.
Even though measurement results obtained in the real world are generally both noisy and continuous, quantum measurement theory tends to emphasize the ideal limit of perfect precision and quantized measurement results. In this article, a more general concept of noisy measurements is applied to investigate the role of quantum noise in the measurement process. In particular, it is shown that the effects of quantum noise can be separated from the effects of information obtained in the measurement. However, quantum noise is required to ``cover up'' negative probabilities arising as the quantum limit is approached. These negative probabilities represent fundamental quantum-mechanical correlations between the measured variable and the variables affected by quantum noise.
Heisenberg's uncertainty principle has been understood to set a limitation on measurements, whereas the long standing mathematical formulation does not allow such an interpretation. Recently, a new relation was found by the present author to give a universally valid relation between noise and disturbance in general quantum measurements, and it has become clear that the new relation plays a role of the first principle to derive various quantum limits on measurement and information processing. Here, we discuss the state-dependent notion of precise measurements of a given observable, and consider a perfect distance such that zero distance implies the perfect correlation. Then, we shall show that even in the perfect error notions both the position measuring noise and the momentum disturbance can be arbitrarily small. We also show that it is possible to generalize the new noise-disturbance uncertainty relation to perfect error distances.
(i) is therefore a good reflection of the crux of the measurement process. Here H(X) is the which characterizes the a priori of the quantity being measured, and H(X/Y) is the total conditional which characterizes the of the quantity being measured which remains after the measurement result Y has been obtained. Let us examine the application of the concept in the case of continuous distributions. The value of the as a numerical characteristic of a continuous random variable is rejected in [12] on the basis that the equals ~ for continuous random variables. The introduction of the probability density p(x) could be rejected on approximately the same basis by considering that this quantity does not reflect the probabilistic characteristic by not being a probability; and only the product p(x)dx possesses the probabilistic interpretation, but equals 0 for continuous quantities and hence loses meaning. Wherever required and possible, the probability is approximated by the expression p(x)Ax, where &x, albeit small, is a finite quantity. The concepts of reduced (differential) entropy and quantization step have an analogous (but not identical) meaning [16]. If is understood to be the mathematical expectation of the logarithm of the probability of random variable values, then it is indeed impossible to relate this concept to continuous random variables. On the other hand, the initial physical meaning, uncertainty, vanishes in the reduced entropy concept. The initial concept uncertainty not only remains real for continuous random variables in the examination of the quantization step concept, but stands out more, exposing most completely the content of the Heisenberg principle and the negative principle of information [5, 7], according to which no quantity being observed can be measured with arbitrarily high accuracy. The limit is not set on this accuracy by the imperfection of the instruments but by objective laws of nature.
One of the broadest concepts of measurement in quantum theory is the generalized measurement. Another paradigm of measurement---arising naturally in quantum optics, among other fields---is that of continuous-time measurements, which can be seen as the limit of a consecutive sequence of weak measurements. They are naturally described in terms of stochastic processes, or time-dependent random variables. We show that any generalized measurement can be decomposed as a sequence of weak measurements with a mathematical limit as a continuous stochastic process. We give an explicit construction for any generalized measurement, and prove that the resulting continuous evolution, in the long-time limit, collapses the state of the quantum system to one of the final states generated by the generalized measurement, being decomposed, with the correct probabilities. A prominent feature of the construction is the presence of a feedback mechanism---the instantaneous choice weak measurement at a given time depends on the outcomes of earlier measurements. For a generalized measurement with n outcomes, this information is captured by a real n-vector on an n-simplex, which obeys a simple classical stochastic evolution.
We introduce information-theoretic definitions for noise and disturbance in quantum measurements and prove a state-independent noise-disturbance tradeoff relation that these quantities have to satisfy in any conceivable setup. Contrary to previous approaches, the information-theoretic quantities we define are invariant under the relabelling of outcomes and allow for the possibility of using quantum or classical operations to "correct" for the disturbance. We also show how our bound implies strong tradeoff relations for mean square deviations.
An analysis of quantum measurement is presented that relies on an information-theoretic description of quantum entanglement. In a consistent quantum information theory of entanglement, entropies (uncertainties) conditional on measurement outcomes can be negative, implying that measurement can be described via unitary, entropy-conserving, interactions, while still producing randomness in a measurement device. In such a framework, quantum measurement is not accompanied by a wave-function collapse, or a quantum jump. The theory is applied to the measurement of incompatible variables, giving rise to a stronger entropic uncertainty relation than heretofore known. It is also applied to standard quantum measurement situations such as the Stern-Gerlach and double-slit experiments to illustrate how randomness, inherent in the conventional quantum probabilities, arises in a unitary framework. Finally, the present view clarifies the relationship between classical and quantum concepts.
We present a detailed motivation for and definition of the contextual values of an observable, which were introduced by Dressel et al. [Phys. Rev. Lett. 104, 240401 (2010)]. The theory of contextual values is a principled approach to the generalized measurement of observables. It extends the well-established theory of generalized state measurements by bridging the gap between partial state collapse and the observables that represent physically relevant information about the system. To emphasize the general utility of the concept, we first construct the full theory of contextual values within an operational formulation of classical probability theory, paying special attention to observable construction, detector coupling, generalized measurement, and measurement disturbance. We then extend the results to quantum probability theory built as a superstructure on the classical theory, pointing out both the classical correspondences to and the full quantum generalizations of both L\"uder's rule and the Aharonov-Bergmann-Lebowitz rule in the process. As such, our treatment doubles as a self-contained pedagogical introduction to the essential components of the operational formulations for both classical and quantum probability theory. We find in both cases that the contextual values of a system observable form a generalized spectrum that is associated with the independent outcomes of a partially correlated and generally ambiguous detector; the eigenvalues are a special case when the detector is perfectly correlated and unambiguous. To illustrate the approach, we apply the technique to both a classical example of marble color detection and a quantum example of polarization detection. For the quantum example we detail two devices: Fresnel reflection from a glass coverslip, and continuous beam displacement from a calcite crystal. We also analyze the three-box paradox to demonstrate that no negative probabilities are necessary in its analysis. Finally, we provide a derivation of the quantum weak value as a limit point of a pre- and postselected conditioned average and provide sufficient conditions for the derivation to hold.