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Register a new userThe Uncertainty Principle Revisited
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We study the quantum-mechanical uncertainty relation originating from the successive measurement of two observables $hat{A}$ and $hat{B}$, with eigenvalues $a_n$ and $b_m$, respectively, performed on the same system. We use an extension of the von Neumann model of measurement, in which two probes interact with the same system proper at two successive times, so we can exhibit how the disturbing effect of the first interaction affects the second measurement. Detecting the statistical properties of the second {em probe} variable $Q_2$ conditioned on the first {em probe} measurement yielding $Q_1$ we obtain information on the statistical distribution of the {em system} variable $b_m$ conditioned on having found the system variable $a_n$ in the interval $delta a$ around $a^{(n)}$. The width of this statistical distribution as function of $delta a$ constitutes an {em uncertainty relation}. We find a general connection of this uncertainty relation with the commutator of the two observables that have been measured successively. We illustrate this relation for the successive measurement of position and momentum in the discrete and in the continuous cases and, within a model, for the successive measurement of a more general class of observables.
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We propose a test for certifying the dimension of a quantum system: store in it a random $n$-bit string, in either the computational or the Hadamard basis, and later check that the string can be mostly recovered. The protocol tolerates noise, and the verifier only needs to prepare one-qubit states. The analysis is based on uncertainty relations in the presence of quantum memory, due to Berta et al. (2010).
The fundamental sources of noise in a vacuum-tunneling probe used as an electromechanical transducer to monitor the location of a test mass are examined using a first-quantization formalism. We show that a tunneling transducer enforces the Heisenberg uncertainty principle for the position and momentum of a test mass monitored by the transducer through the presence of two sources of noise: the shot noise of the tunneling current and the momentum fluctuations transferred by the tunneling electrons to the test mass. We analyze a number of cases including symmetric and asymmetric rectangular potential barriers and a barrier in which there is a constant electric field. Practical configurations for reaching the quantum limit in measurements of the position of macroscopic bodies with such a class of transducers are studied.
Toy models for quantum evolution in the presence of closed timelike curves (CTCs) have gained attention in the recent literature due to the strange effects they predict. The circuits that give rise to these effects appear quite abstract and contrived, as they require non-trivial interactions between the future and past which lead to infinitely recursive equations. We consider the special case in which there is no interaction inside the CTC, referred to as an open timelike curve (OTC), for which the only local effect is to increase the time elapsed by a clock carried by the system. Remarkably, circuits with access to OTCs are shown to violate Heisenbergs uncertainty principle, allowing perfect state discrimination and perfect cloning of coherent states. The model is extended to wave-packets and smoothly recovers standard quantum mechanics in an appropriate physical limit. The analogy with general relativistic time-dilation suggests that OTCs provide a novel alternative to existing proposals for the behaviour of quantum systems under gravity.
Donoho and Stark have shown that a precise deterministic recovery of missing information contained in a time interval shorter than the time-frequency uncertainty limit is possible. We analyze this signal recovery mechanism from a physics point of view and show that the well-known Shannon-Nyquist sampling theorem, which is fundamental in signal processing, also uses essentially the same mechanism. The uncertainty relation in the context of information theory, which is based on Fourier analysis, provides a criterion to distinguish Shannon-Nyquist sampling from compressed sensing. A new signal recovery formula, which is analogous to Donoho-Stark formula, is given using the idea of Shannon-Nyquist sampling; in this formulation, the smearing of information below the uncertainty limit as well as the recovery of information with specified bandwidth take place. We also discuss the recovery of states from the domain below the uncertainty limit of coordinate and momentum in quantum mechanics and show that in principle the state recovery works by assuming ideal measurement procedures. The recovery of the lost information in the sub-uncertainty domain means that the loss of information in such a small domain is not fatal, which is in accord with our common understanding of the uncertainty principle, although its precise recovery is something we are not used to in quantum mechanics. The uncertainty principle provides a universal sampling criterion covering both the classical Shannon-Nyquist sampling theorem and the quantum mechanical measurement.
This paper deduces universal uncertainty principle in different quantum theories after about one century of proposing uncertainty principle by Heisenberg, i.e., new universal uncertainty principle of any orders of physical quantities in quantum physics, overcomes the difficulty that current quantum computer, quantum communication, quantum control, quantum mechanics and so on theories cannot give exact values of general uncertainty of any orders of physical quantities, further gives all relevant different expressions of the universal uncertainty principle and their applications. In fact, our studies are consistent with current theories and physical factual experiments, e.g., relevant to hydrogen atom physics experiments. Using the new universal uncertainty principle, people can give all applications to atomic physics, quantum mechanics, quantum communication, quantum calculations, quantum computer and so on.
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