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An Isaac Newton Institute Workshop

Quantum Statistics - quantum measurements, estimation and related topics

Universal Uncertainty Principle

Author: Masanao Ozawa (Tohoku University)


In 1927 Heisenberg [1] proposed a reciprocal relation for measurement noise and disturbance, or equivalently for joint measurement noises, by the famous gamma ray microscope thought experiment and claimed that the relation is a straightforward mathematical consequence of the commutation relation. However, his proposed proof did not really consider the measurement noise or disturbance, and was immediately reformulated by Kennard [2] as the famous inequality for the standard deviations of position and momentum. Kennard's relation was soon generalized by Robertson [3] to arbitrary pairs of observables. In spite of the above development, text books of quantum mechanics have treated Robertson's relation as a mathematical expression of Heisenberg's original relation for noise and disturbance.

In this talk, we discuss a new relation (the universal uncertainty principle) found and rigorously proven recently by the present speaker [8] for the well-defined root-mean-square noise and disturbance in all the physically possible quantum measurements. In the previous work [4], it has been shown that jointly unbiased measurements satisfy the reciprocal joint measurement noise relation analogous to Robsertson's. However, it should be pointed out that there is no jointly unbiased spin measurements for two different components, and that no projective measurements of finite level systems satisfy the reciprocal noise disturbance relation. Counter examples of the original Heisenberg relation also include the contractive state position measurement and the indirect position measurement for the EPR pair [6,8]. The universal uncertainty relation reveals the correct bound for the noise in non-disturbing measurements. Using this noise bound, we construct a proof of a quantitative generalization of the Wigner-Araki-Yanase theorem, setting a lower bound for the accuracy of measurement under conservation laws [5,9]. Then, this relation is used to give a lower bound for the accuracy of quantum gate operations realized by interactions between qubits and the controller (or the ancilla) obeying given conservation laws [7,9]. The relations to the no-cloning theorem and to security arguments for quantum cryptography will be also discussed briefly.

References. 1. W. Heisenberg, Z. Phys. 43, 172 (1927). 2. E. H. Kennard, Z. Phys. 44, 326 (1927). 3. H. P. Robertson, Phys. Rev. 34, 163 (1929). 4. M. Ozawa, in Lecture Notes in Physics 378 (Springer, Berlin), 3 (1991); S. Ishikawa, Rep. Math. Phys. 29, 257 (1991). 5. M. Ozawa, Phys. Rev. Lett. 88, 050402 (2002). 6. M. Ozawa, Phys. Lett. A 299, 1 (2002). 7. M. Ozawa, Phys. Rev. Lett. 89, 057902 (2002); 91, 089802 (2003). 8. M. Ozawa, Phys. Rev. A 67, 042105 (2003); Phys. Lett. A, 318, 21 (2003); Phys. Lett. A 320, 367 (2004); Ann. Phys. 31, 350 (2004). 9. M. Ozawa, Intern. J. Quant. Inf. (IJQI) 1, 569 (2003).