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Quantum Measurements, Instruments, and Uncertainty Principle ( The Proceedings of the Second Sendai Workshop on Quantum Probability and Quantum Information)

Masanao Ozawa

Published Mar 1, 2004• 0 citations
OpenAlex

Abstract

The Heisenberg uncertainty relation for measurement noise and disturbance states that any position measurement with noise ε brings the momentum disturbance not less than h/2ε. However, this relation holds only for restricted class of measurements. Here, we discuss universally valid uncertainty relations for measurement noise and disturbance, which hold for all the possible quantum measurements.

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Heisenberg's uncertainty principle has been understood to set a limitation on measurements; however, the long standing mathematical formulation established by Heisenberg, Kennard, and Robertson does not allow such an interpretation. Recently, a new relation was found 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 in a unified treatment. This paper examines the above development on the noise-disturbance uncertainty principle in the model independent approach based on measurement operator formalism, which is widely accepted to describe a class of generalized measurements in the field of quantum information. We obtain explicit formulas for noise and disturbance of measurements given by measurement operators, and show that projective measurements do not satisfy the Heisenberg-type noise-disturbance relation that is typical in the gamma-ray microscope thought experiments. We also show that the disturbance on a Pauli operator of a projective measurement of another Pauli operator constantly equals to the square root of 2, and examine how this measurement violates the Heisenberg-type relation but satisfies the new noise-disturbance relation.

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The Heisenberg uncertainty relation for measurement noise and disturbance states that any position measurement with noise ε brings the momentum disturbance not less than h/2ε. However, this relation holds only for restricted class of measurements. Here, we discuss universally valid uncertainty relations for measurement noise and disturbance, which hold for all the possible quantum measurements.
The Heisenberg uncertainty relation for measurement noise and disturbance states that any position measurement with noise /spl epsiv/ brings the momentum disturbance not less than h/2/spl epsiv/. However, this relation holds only for restricted class of measurements. Here, a generalized uncertainty relation for measurement noise and disturbance is formalized and proven, which holds for all the possible quantum measurements. For this purpose, all the possible quantum measurements are characterized by naturally acceptable axioms. The measurement noise and disturbance are defined rigorously for any such general quantum measurements.
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.
A prevailing formulation of the Heisenberg uncertainty principle is re-examined that measuring position with noise epsilon\\epsilonepsilon brings the momentum disturbance no less than hbar/2epsilon\\hbar/2\\epsilonhbar/2epsilon. With rigorous formulations of noise and disturbance, a model of a measuring interaction demonstrates that this formulation of the Heisenberg uncertainty principle is by no means universally valid. An experimental realization of the model is also suggested.
The Heisenberg uncertainty principle states that the product of the noise in a position measurement and the momentum disturbance caused by that measurement should be no less than the limit set by Planck's constant $\ensuremath{\Elzxh}/2$ as demonstrated by Heisenberg's thought experiment using a \ensuremath{\gamma}-ray microscope. Here it is shown that this common assumption is not universally true: a universally valid trade-off relation between the noise and the disturbance has an additional correlation term, which is redundant when the intervention brought by the measurement is independent of the measured object, but which allows the noise-disturbance product much below Planck's constant when the intervention is dependent. A model of measuring interaction with dependent intervention shows that Heisenberg's lower bound for the noise-disturbance product is violated even by a nearly nondisturbing precise position measurement. An experimental implementation is also proposed to realize the above model in the context of optical quadrature measurement with currently available linear optical devices.
Heisenberg's uncertainty principle has been understood to set a limitation on measurements; however, the long standing mathematical formulation established by Heisenberg, Kennard, and Robertson does not allow such an interpretation. Recently, a new relation was found 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 in a unified treatment. This paper examines the above development on the noise-disturbance uncertainty principle in the model independent approach based on measurement operator formalism, which is widely accepted to describe a class of generalized measurements in the field of quantum information. We obtain explicit formulas for noise and disturbance of measurements given by measurement operators, and show that projective measurements do not satisfy the Heisenberg-type noise-disturbance relation that is typical in the gamma-ray microscope thought experiments. We also show that the disturbance on a Pauli operator of a projective measurement of another Pauli operator constantly equals to the square root of 2, and examine how this measurement violates the Heisenberg-type relation but satisfies the new noise-disturbance relation.