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The Wehrl entropy is an entropy associated to the Husimi quasi-probability distribution. We discuss how it can be used to formulate entropic uncertainty relations and for a quantification of entanglement in continuous variables. We show that the Wehrl-Lieb inequality is closer to equality than the usual Bia{l}ynicki-Birula and Mycielski entropic uncertainty relation almost everywhere. Furthermore, we show how a Wehrl mutual information can be used to obtain a measurable perfect witness for pure state bipartite entanglement, which additionally provides a lower bound on the entanglement entropy.
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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 take 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.
We derive entropic uncertainty relations for successive generalized measurements by using general descriptions of quantum measurement within two {distinctive operational} scenarios. In the first scenario, by merging {two successive measurements} into one we consider successive measurement scheme as a method to perform an overall {composite} measurement. In the second scenario, on the other hand, we consider it as a method to measure a pair of jointly measurable observables by marginalizing over the distribution obtained in this scheme. In the course of this work, we identify that limits on ones ability to measure with low uncertainty via this scheme come from intrinsic unsharpness of observables obtained in each scenario. In particular, for the L{u}ders instrument, disturbance caused by the first measurement to the second one gives rise to the unsharpness at least as much as incompatibility of the observables composing successive measurement.
How violently do two quantum operators disagree? Different fields of physics feature different measures of incompatibility: (i) In quantum information theory, entropic uncertainty relations constrain measurement outcomes. (ii) In condensed matter and high-energy physics, the out-of-time-ordered correlator (OTOC) signals scrambling, the spread of information through many-body entanglement. We unite these measures, proving entropic uncertainty relations for scrambling. The entropies are of distributions over weak and strong measurements possible outcomes. The weak measurements ensure that the OTOC quasiprobability (a nonclassical generalization of a probability, which coarse-grains to the OTOC) governs terms in the uncertainty bound. The quasiprobability causes scrambling to strengthen the bound in numerical simulations of a spin chain. This strengthening shows that entropic uncertainty relations can reflect the type of operator disagreement behind scrambling. Generalizing beyond scrambling, we prove entropic uncertainty relations satisfied by commonly performed weak-measurement experiments. We unveil a physical significance of weak values (conditioned expectation values): as governing terms in entropic uncertainty bounds.
Our knowledge of quantum mechanics can satisfactorily describe simple, microscopic systems, but is yet to explain the macroscopic everyday phenomena we observe. Here we aim to shed some light on the quantum-to-classical transition as seen through the analysis of uncertainty relations. We employ entropic uncertainty relations to show that it is only by the inclusion of imprecision in our model of macroscopic measurements that we can prepare a system with two simultaneously well-defined quantities, even if their associated observables do not commute. We also establish how the precision of measurements must increase in order to keep quantum properties, a desirable feature for large quantum computers.
We formulate an entanglement criterion using Peres-Horodecki positive partial transpose operations combined with the Schrodinger-Robertson uncertainty relation. We show that any pure entangled bipartite and tripartite state can be detected by experimentally measuring mean values and variances of specific observables. Those observables must satisfy a specific condition in order to be used, and we show their general form in the $2times 2$ (two qubits) dimension case. The criterion is applied on a variety of physical systems including bipartite and multipartite mixed states and reveals itself to be stronger than the Bell inequalities and other criteria. The criterion also work on continuous variable cat states and angular momentum states of the radiation field.
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