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Continuous Variable Entropic Uncertainty Relations in the Presence of Quantum Memory

Mario Berta, Matthias Christandl, Fabian Furrer, Volkher B. Scholz, Marco Tomamichel

Published Aug 21, 2013• 4 citations
OpenAlex

Abstract

We generalize entropic uncertainty relations in the presence of quantum memory [Nature Physics 6 (659), 2010], and [Physical Review Letters 106 (110506), 2011] in two directions. First, we consider measurements with a continuum of outcomes, and, second, we allow for infinite-dimensional quantum memory. To achieve this, we introduce conditional differential entropies for classical-quantum states on von Neumann algebras, and show approximation properties for these entropies. As an example, we evaluate the uncertainty relations for position-momentum measurements, which has applications in continuous variable quantum cryptography and quantum information theory.

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We generalize entropic uncertainty relations in the presence of quantum memory [Nature Physics 6 (659), 2010], and [Physical Review Letters 106 (110506), 2011] in two directions. First, we consider measurements with a continuum of outcomes, and, second, we allow for infinite-dimensional quantum memory. To achieve this, we introduce conditional differential entropies for classical-quantum states on von Neumann algebras, and show approximation properties for these entropies. As an example, we evaluate the uncertainty relations for position-momentum measurements, which has applications in continuous variable quantum cryptography and quantum information theory.
A prominent formulation of the uncertainty principle identifies the fundamental quantum feature that no particle may be prepared with certain outcomes for both position and momentum measurements. Often the statistical uncertainties are thereby measured in terms of entropies providing a clear operational interpretation in information theory and cryptography. Recently, entropic uncertainty relations have been used to show that the uncertainty can be reduced in the presence of entanglement and to prove security of quantum cryptographic tasks. However, much of this recent progress has been focused on observables with only a finite number of outcomes not including Heisenberg’s original setting of position and momentum observables. Here, we show entropic uncertainty relations for general observables with discrete but infinite or continuous spectrum that take into account the power of an entangled observer. As an illustration, we evaluate the uncertainty relations for position and momentum measurements, which is operationally significant in that it implies security of a quantum key distribution scheme based on homodyne detection of squeezed Gaussian states.
Entropic uncertainty relations demonstrate the intrinsic uncertainty of nature from an information-theory perspective. Recently, a quantum-memory-assisted entropic uncertainty relation for multiple measurements was proposed by Wu et al. [Phys. Rev. A 106, 062219 (2022)]. Interestingly, the quantum-memory-assisted entropic uncertainty relation for multiple measurement settings can be further generalized. In this work, we propose two complementary multipartite quantum-memory-assisted entropic uncertainty relations and our lower bounds depend on values of complementarity of the observables, (conditional) von Neumann entropies, Holevo quantities, and mutual information. As an illustration, we provide several typical cases to exhibit that our bounds are tighter and outperform the previous bounds.
Entropic uncertainty relations provide an information-theoretic framework for quantifying the fundamental indeterminacy inherent in quantum mechanics. We propose more stringent quantum-memory-assisted entropic uncertainty relations for complete sets of mutually unbiased bases in multipartite scenarios. We present lower and upper bounds of the quantum uncertainties based on the complementarity of the observables, the purity of the measured state, the (conditional) von-Neumann entropies, the Holevo quantities and mutual information. The results are illustrated by several representative cases, showing that our bounds are tighter than and outperform previously existing bounds.
Abstract Uncertainty relations are central to quantum physics. While they were originally formulated in terms of variances, they have later been successfully expressed with entropies following the advent of Shannon information theory. Here, we review recent results on entropic uncertainty relations involving continuous variables, such as position x and momentum p . This includes the generalization to arbitrary (not necessarily canonically-conjugate) variables as well as entropic uncertainty relations that take x - p correlations into account and admit all Gaussian pure states as minimum uncertainty states. We emphasize that these continuous-variable uncertainty relations can be conveniently reformulated in terms of entropy power, a central quantity in the information-theoretic description of random signals, which makes a bridge with variance-based uncertainty relations. In this review, we take the quantum optics viewpoint and consider uncertainties on the amplitude and phase quadratures of the electromagnetic field, which are isomorphic to x and p , but the formalism applies to all such variables (and linear combinations thereof) regardless of their physical meaning. Then, in the second part of this paper, we move on to new results and introduce a tighter entropic uncertainty relation for two arbitrary vectors of intercommuting continuous variables that takes correlations into account. It is proven conditionally on reasonable assumptions. Finally, we present some conjectures for new entropic uncertainty relations involving more than two continuous variables.
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.