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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).
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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.
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.
The non-relativistic quantum mechanics with a generalized uncertainty principle (GUP) is examined in $D$-dimensional free particle and harmonic oscillator systems. The Feynman propagators for these systems are exactly derived within the first order of the GUP parameter.
The quantum multiparameter estimation is very different from the classical multiparameter estimation due to Heisenbergs uncertainty principle in quantum mechanics. When the optimal measurements for different parameters are incompatible, they cannot be jointly performed. We find a correspondence relationship between the inaccuracy of a measurement for estimating the unknown parameter with the measurement error in the context of measurement uncertainty relations. Taking this correspondence relationship as a bridge, we incorporate Heisenbergs uncertainty principle into quantum multiparameter estimation by giving a tradeoff relation between the measurement inaccuracies for estimating different parameters. For pure quantum states, this tradeoff relation is tight, so it can reveal the true quantum limits on individual estimation errors in such cases. We apply our approach to derive the tradeoff between attainable errors of estimating the real and imaginary parts of a complex signal encoded in coherent states and obtain the joint measurements attaining the tradeoff relation. We also show that our approach can be readily used to derive the tradeoff between the errors of jointly estimating the phase shift and phase diffusion without explicitly parameterizing quantum measurements.
Unconditionally secure quantum bit commitment (QBC) was considered impossible. But the no-go proofs are based on the Hughston-Jozsa-Wootters (HJW) theorem (a.k.a. the Uhlmann theorem). Recently it was found that in high-dimensional systems, there exist some states which can display a chaos effect in quantum steering, so that the attack strategy based on the HJW theorem has to require the capability of discriminating quantum states with very subtle difference, to the extent that is not allowed by the uncertainty principle. With the help of this finding, here we propose a simple QBC protocol which manages to evade the no-go proofs.
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