A theoretical result for correlated quantum systems is presented which leads to a noise commutation relation and a generalized Heisenberg uncertainty relation. These relations imply an inherent and unavoidable extra noise in quantum measurements beyond that already included in the Heisenberg lower bound. These relations lead directly to model-independent lower bounds on inherent noise, useful in a variety of applications, including balanced homodyne detection and quantum optical linear amplifiers.
A theoretical result for correlated quantum systems is presented which leads to a noise commutation relation and a generalized Heisenberg uncertainty relation. These relations imply an inherent and unavoidable extra noise in quantum measurements beyond that already included in the Heisenberg lower bound. These relations lead directly to model-independent lower bounds on inherent noise, useful in a variety of applications, including balanced homodyne detection and quantum optical linear amplifiers.
Recent advances, particularly in the area of quantum optics, have caused heightened interest in the fundamental limitations on the achievable accuracy that may be obtained in measuring quantum mechanical systems. We present a quantum correlation result and a noise commutation relation for correlated quantum systems. The measurement of quantum system observables requires the correlation of these micro-observables and the measuring apparatus (usually a macroscopic system). The noise commutation relation is applied to quantum measurements leading to a generalized Heisenberg uncertainty relation. The generalized Heisenberg uncertainty relation yields a lower bound on the inherent unavoidable extra noise in quantum measurements, which is due to the measuring process itself. To indicate the utility of the noise commutation relation, two different applications in optics are discussed. The first, an application of the generalized Heisenberg uncertainty relation, indicates how a model-independent noise limit for balanced homodyne detection can be obtained. Another application is discussed in which the noise commutation relation is applied to develop a model-independent lower bound for the inherent noise of a quantum optical linear amplifier.
Uncertainty relations involving incompatible observables are one of the cornerstones of quantum mechanics. Aside from their fundamental significance, they play an important role in practical applications, such as detection of quantum correlations and security requirements in quantum cryptography. In continuous variable systems, the spectra of the relevant observables form a continuum and this necessitates the coarse graining of measurements. However, these coarse-grained observables do not necessarily obey the same uncertainty relations as the original ones, a fact that can lead to false results when considering applications. That is, one cannot naively replace the original observables in the uncertainty relation for the coarse-grained observables and expect consistent results. As such, several uncertainty relations that are specifically designed for coarse-grained observables have been developed. In recognition of the 90th anniversary of the seminal Heisenberg uncertainty relation, celebrated last year, and all the subsequent work since then, here we give a review of the state of the art of coarse-grained uncertainty relations in continuous variable quantum systems, as well as their applications to fundamental quantum physics and quantum information tasks. Our review is meant to be balanced in its content, since both theoretical considerations and experimental perspectives are put on an equal footing.
Uncertainty relations involving incompatible observables are one of the cornerstones of quantum mechanics. Aside from their fundamental significance, they play an important role in practical applications, such as detection of quantum correlations and security requirements in quantum cryptography. In continuous variable systems, the spectra of the relevant observables form a continuum and this necessitates the coarse graining of measurements. However, these coarse-grained observables do not necessarily obey the same uncertainty relations as the original ones, a fact that can lead to false results when considering applications. That is, one cannot naively replace the original observables in the uncertainty relation for the coarse-grained observables and expect consistent results. As such, several uncertainty relations that are specifically designed for coarse-grained observables have been developed. In recognition of the 90th anniversary of the seminal Heisenberg uncertainty relation, celebrated last year, and all the subsequent work since then, here we give a review of the state of the art of coarse-grained uncertainty relations in continuous variable quantum systems, as well as their applications to fundamental quantum physics and quantum information tasks. Our review is meant to be balanced in its content, since both theoretical considerations and experimental perspectives are put on an equal footing.
In quantum sensing and metrology, an important class of measurement is the continuous linear measurement, in which the detector is coupled to the system of interest linearly and continuously in time. One key aspect involved is the quantum noise of the detector, arising from quantum fluctuations in the detector input and output. It determines how fast we acquire information about the system and also influences the system evolution in terms of measurement backaction. We therefore often categorize it as the so-called imprecision noise and quantum backaction noise. There is a general Heisenberg-like uncertainty relation that constrains the magnitude of and the correlation between these two types of quantum noise. The main result of this paper is to show that, when the detector becomes ideal, i.e., at the quantum limit with minimum uncertainty, not only does the uncertainty relation takes the equal sign as expected, but also there are two new equalities. This general result is illustrated by using the typical cavity QED setup with the system being either a qubit or a mechanical oscillator. Particularly, the dispersive readout of a qubit state, and the measurement of mechanical motional sideband asymmetry are considered.
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.
We analyze the Schwarz inequality and its generalizations, as well as inequalities resulting from the Jensen inequality. They are used in quantum theory to derive the Heisenberg-Robertson (HR) and Schroedinger-Robertson (SR) uncertainty relation for two non-commuting observables and their generalizations to three or more non-commuting observables. Jensen's inequality, in turn, is helpful in deriving various the "sum uncertainty relations" for two or more observables. Using these inequalities, we derive various types of generalized uncertainty relations for more than two non--commuting observables and analyze their properties and critical points. We also study the connections between the generalizations of the HR and SR uncertainty relations for two and more observables and the correlations of these observables in the state of the quantum system under study. In this analysis, we pay special attention to the consequences of the generalized SR uncertainty relation for three non--commuting observables on their correlations in a given state of a quantum system and to the connections of this relation with the appropriate correlation matrix, whose matrix elements are the quantum versions of the Pearson coefficient. We show also that the SR uncertainty relation (including the generalized ones) can be written in an equivalent way using these Pearson coefficients.