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arXiv:1509.05973

Entropic Uncertainty Principle and Information Exclusion Principle for multiple measurements in the presence of quantum memory

Jun Zhang, Yang Zhang, Chang‐shui Yu

Published Sep 20, 2015• 0 citations
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Abstract

The Heisenberg uncertainty principle shows that no one can specify the values of the non-commuting canonically conjugated variables simultaneously. However, the uncertainty relation is usually applied to two incompatible measurements. We present tighter bounds on both entropic uncertainty relation and information exclusion principle for multiple measurements in the presence of quantum memory. As applications, three incompatible measurements on Werner state and Horodecki's bound entangled state are investigated in details.

DOI
10.48550/arxiv.1509.05973

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The Heisenberg uncertainty principle shows that no one can specify the values of the non-commuting canonically conjugated variables simultaneously. However, the uncertainty relation is usually applied to two incompatible measurements. We present tighter bounds on both entropic uncertainty relation and information exclusion principle for multiple measurements in the presence of quantum memory. As applications, three incompatible measurements on Werner state and Horodecki's bound entangled state are investigated in details.
The Heisenberg uncertainty principle shows that no one can specify the values of the non-commuting canonically conjugated variables simultaneously. However, the uncertainty relation is usually applied to two incompatible measurements. We present tighter bounds on both entropic uncertainty relation and information exclusion relation for multiple measurements in the presence of quantum memory. As applications, three incompatible measurements on Werner state and Horodecki's bound entangled state are investigated in details.
One unique feature of quantum mechanics is the Heisenberg uncertainty principle, which states that the outcomes of two incompatible measurements cannot simultaneously achieve arbitrary precision. In an information-theoretic context of quantum information, the uncertainty principle can be formulated as entropic uncertainty relations with two measurements for a quantum bit (qubit) in two-dimensional system. New entropic uncertainty relations are studied for a higher-dimensional quantum state with multiple measurements, the uncertainty bounds can be tighter than that expected from two measurements settings and cannot result from qubits system with or without a quantum memory. Here we report the first room-temperature experimental testing of the entropic uncertainty relations with three measurements in a natural three-dimensional solid-state system: the nitrogen-vacancy center in pure diamond. The experimental results confirm the entropic uncertainty relations for multiple measurements. Our result represents a more precise demonstrating of the fundamental uncertainty principle of quantum mechanics.
One unique feature of quantum mechanics is the Heisenberg uncertainty principle, which states that the outcomes of two incompatible measurements cannot simultaneously achieve arbitrary precision. In an information-theoretic context of quantum information, the uncertainty principle can be formulated as entropic uncertainty relations with two measurements for a quantum bit (qubit) in two-dimensional system. New entropic uncertainty relations are studied for a higher-dimensional quantum state with multiple measurements, and the uncertainty bounds can be tighter than that expected from two measurements settings and cannot result from qubits system with or without a quantum memory. Here we report the first room-temperature experimental testing of the entropic uncertainty relations with three measurements in a natural three-dimensional solid-state system: the nitrogen-vacancy center in pure diamond. The experimental results confirm the entropic uncertainty relations for multiple measurements. Our result represents a more precise demonstrating of the fundamental uncertainty principle of quantum mechanics.
We present the entropic uncertainty relations for multiple measurement settings which demonstrate the uncertainty principle of quantum mechanics. Those uncertainty relations are obtained for both cases with and without the presence of quantum memory, and can be proven by a unified method. Our results recover some well known entropic uncertainty relations for two observables, which show the uncertainties about the outcomes of two incompatible measurements. The bounds of those relations which quantify the extent of the uncertainty take concise forms and are easy to calculate. Those uncertainty relations might play important roles in the foundations of quantum theory. Potential experimental demonstration of those entropic uncertainty relations is discussed.
The uncertainty principle sets a bound on our ability to predict the measurement outcomes of two incompatible observables which are measured on a quantum particle simultaneously. In quantum information theory, the uncertainty principle can be formulated in terms of the Shannon entropy. Entropic uncertainty bound can be improved by adding a particle which correlates with the measured particle. The added particle acts as a quantum memory. In this work, a method is provided for obtaining the entropic uncertainty relations in the presence of a quantum memory by using quantum coherence. In the method, firstly, one can use the quantum relative entropy of quantum coherence to obtain the uncertainty relations. Secondly, these relations are applied to obtain the entropic uncertainty relations in the presence of a quantum memory. In comparison with other methods this approach is much simpler. Also, for a given state, the upper bounds on the sum of the relative entropies of unilateral coherences are provided, and it is shown which one is tighter. In addition, using the upper bound obtained for unilateral coherence, the nontrivial upper bound on the sum of the entropies for different observables is derived in the presence of a quantum memory.
The uncertainty principle is a fundamental principle in quantum physics. It implies that the measurement outcomes of two incompatible observables cannot be predicted simultaneously. In quantum information theory, this principle can be expressed in terms of entropic measures. M. Berta et al. [Nat. Phys. 6, 659 (2010)] have indicated that uncertainty bound can be altered by considering a particle as a quantum memory correlating with the primary particle. In this article, we obtain a lower bound for entropic uncertainty in the presence of a quantum memory by adding an additional term depending on the Holevo quantity and mutual information. We conclude that our lower bound will be tightened with respect to that of Berta et al. when the accessible information about measurements outcomes is less than the mutual information about the joint state. Some examples have been investigated for which our lower bound is tighter than Berta et al.'s lower bound. Using our lower bound, a lower bound for the entanglement of formation of bipartite quantum states has been obtained, as well as an upper bound for the regularized distillable common randomness.