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arXiv:1201.4884

Fine-grained uncertainty relation and entropic uncertainty relation in the presence of quantum memory

Tanumoy Pramanik, Priyanka Chowdhury

Published Jan 23, 2012• 0 citations
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Abstract

In presence of quantum memory [M. Berta, M. Christandl, R. Colbeck, J.M. Renes, and R. Renner, Nature Phys. 6, 659 (2010)] the lower bound of entropic uncertainty relation depends on amount of entanglement between the particle (on which two incompatible observable are measured) and the quantum memory. On other hand, fine grained uncertainty relation [J. Oppenheim and S. Wehner, Science 330, 1072 (2010)] gives the upper bound of winning non-local (CHSH) game. We construct similar kind of quantum game and relate two uncertainty relation for measurement of any two observable on the quantum system in presence of quantum memory and show there are certain region where the lower bound of entropic uncertainty relation in presents of quantum memory given by Berta et. al. should not be achieve by spin projective measurement.

DOI
10.48550/arxiv.1201.4884

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In presence of quantum memory [M. Berta, M. Christandl, R. Colbeck, J.M. Renes, and R. Renner, Nature Phys. 6, 659 (2010)] the lower bound of entropic uncertainty relation depends on amount of entanglement between the particle (on which two incompatible observable are measured) and the quantum memory. On other hand, fine grained uncertainty relation [J. Oppenheim and S. Wehner, Science 330, 1072 (2010)] gives the upper bound of winning non-local (CHSH) game. We construct similar kind of quantum game and relate two uncertainty relation for measurement of any two observable on the quantum system in presence of quantum memory and show there are certain region where the lower bound of entropic uncertainty relation in presents of quantum memory given by Berta et. al. should not be achieve by spin projective measurement.
The limitation on obtaining precise outcomes of measurements performed on two noncommuting observables of a particle as set by the uncertainty principle in its entropic form can be reduced in the presence of quantum memory. We derive a new entropic uncertainty relation based on fine graining, which leads to an ultimate limit on the precision achievable in measurements performed on two incompatible observables in the presence of quantum memory. We show that our derived uncertainty relation tightens the lower bound set by entropic uncertainty for members of the class of two-qubit states with maximally mixed marginals, while accounting for the recent experimental results using maximally entangled pure states and mixed Bell-diagonal states. An implication of our uncertainty relation on the security of quantum key generation protocols is pointed out.
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.
The uncertainty of measurement on a quantum system can be reduced in presence of quantum memory [M. Berta et. al. Nature Phys. {\bf 6}, 659 (2010)]. By measurement on quantum memory, some information (non-classical information) is inaccessible, hence, it limits the lower bound of entropic uncertainty relation. In the present work, we improve the lower bound of entropic uncertainty relation in presence of quantum memory by considering inaccessibility of non classical correlation at Bob side and also compare the lower bound given by M. Berta et. al. and our prescription for two different kind of shared bipartite state.
Abstract Berta et al ’s uncertainty principle in the presence of quantum memory (Berta et al 2010 Nat. Phys . 6 659 ) reveals uncertainties with quantum side information between the observables. In the recent important work of Coles and Piani (2014 Phys. Rev . A 89 022112 ), the entropic sum is controlled by the first and second maximum overlaps between the two projective measurements. We generalize the entropic uncertainty relation in the presence of quantum memory and find the exact dependence on all d largest overlaps between two measurements on any d -dimensional Hilbert space. Our bound is rigorously shown to be strictly tighter than previous entropic bounds in the presence of quantum memory, which have potential applications to quantum cryptography with entanglement witnesses and quantum key distributions.
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, which gives the constraints on obtaining precise outcomes for incompatible measurements, provides a new vision of the real world that we are not able to realize from the classical knowledge. In recent years, numerous theoretical and experimental developments about the new forms of the uncertainty principle have been achieved. Among these efforts, one attractive goal is to find tighter bounds of the uncertainty relation. Here, using an all optical setup, we experimentally investigate a most recently proposed form of uncertainty principle-the fine-grained uncertainty relation assisted by a quantum memory. The experimental results on the case of two-qubit state with maximally mixed marginal demonstrate that the fine-graining method can help to get a tighter bound of the uncertainty relation. Our results might contribute to further understanding and utilizing of the uncertainty principle.