Bounds for entanglement via an extension of strong. Optimal transport and renyi informational divergence. What distinguishes the swiveled renyi entropies from the prior proposal of berta et al. Shannon entropy this chapter is a digression in information theory. Renyi2 entropy provides a natural measure of information for arbitrary gaussian states it is operationally linked to the phase space sampling entropy of the wigner distribution it satisfies the strong subadditivity inequality it leads us to define bona fide quantifiers of all types. In this work, we focus on the role of subadditivity of entropy in probability. Dimensionally sharp inequalities for the linear entropy. Entanglement entropy is an interesting quantity with many applications. By using the proof technique that establishes the left side of the triangle inequality above, one can show that the strong subadditivity. Shannon entropy the entropy power inequality can also be expressed for the renyi class of entropies, based on the extended youngs inequality.
Subadditivity of matrix entropy and concentration of. Also, the renyi entropy is a monotonic function of the information. Entropy is a central concept in both classical and quantum information theory, measuring the uncertainty and the information content in the. This condition is the inequality available for tsallis entropy of the bipartite system state and tsallis entropies of two subsystem states. Let be a bipartite state on tensor product hilbert space. This paper introduces swiveled renyi entropies as an alternative to the renyi entropic quantities put forward in berta et al. Entropy in quantum information theory communication and. Measuring gaussian quantum information and correlations. Therefore, they can generally only be used in comparative or di. Shannon entropy, renyi entropy, and information tina. Instead, we will show the socalled property of subadditivity. There are numerous examples of subadditive functions in various areas of mathematics, particularly norms and square roots.
In the present paper, we focus on the hypoentropy introduced by carlo ferreri and the tsallis entropy introduced by constantino tsallis. Renyi entropy and entanglement spectrum renyi entropy is defined by. The renyi entropy function and the large deviation of short. Many extensions of shannon entropy have been studied. Pdf note on multiple additivity of minimal renyi entropy. We also discuss the question of the existence of the entropy for sums of independent random variables. We also prove convexityconcavity properties of some other, related functionals. Conditional renyi entropy mathematical institute universiteit leiden. In this sense, it is the strongest way to measure the information content of a discrete random variable. We investigate the universal inequalities relating the. In the present paper we shed light on this subject by providing a detailed survey of renyi entropies for bosonic and fermionic gaussian states. The arithmetic mean is not the only interesting average and the shannon entropy is not the only interesting entropy.
A strengthened monotonicity inequality of quantum relative. There is extensive literature on the applications of the renyi entropy in many. For simplicity you may wish to focus on the case n 1 and. Entropy in classical and quantum information theory william fedus physics department, university of california, san diego. Bounds for entanglement via an extension of strong subadditivity of entropy article in letters in mathematical physics 1011 march 2012 with 15 reads how we measure reads. Covariance matrix description of the hbt effect in order to. Then the maximized value is the transfer entropy that presents the strength of causal relations from y to x and the. Note on multiple additivity of minimal renyi entropy output of the wernerholevo channels. For a random variable x with probability distribution function pdf f x. Shannon entropy versus renyi entropy from a cryptographic. Kim we prove an operator inequality that extends strong subadditivity of entropy. Pdf counterexamples to additivity of minimum output p.
Revisiting conditional renyi entropies and generalizing shannons. This is a fascinating subject, which arose once the notion of information got precise and quantifyable. For some sequences of random quantum channels, we compute almost surely the limit of their. Entropy 2008, 10 262 no attempt at completeness is made, the references cover only part of the historically important contributions, but are believed to be representative for the development of the main ideas in the. We want also to stress that the argument of the log, which is the 2. Adscft and strong subadditivity of entanglement entropy tomoyoshi hirata1 and tadashi takayanagi2 department of physics, kyoto university, kyoto 6068502, japan abstract recently, a holographic computation of the entanglement entropy in conformal. A renyi entropy convolution inequality with application. Neumann entropy, renyi entropy, tsallis entropy, min entropy, max. However, so far there is no generally accepted conditional version of the renyi entropy corresponding to the one of the shannon entropy. The image shows our two new inequalities, the inhomogeneous subadditivity and the dimensionally sharp subadditivity, which give an additional constraint. Unveiling the hanbury brown and twiss effect through renyi. The structure of renyi entropic inequalities proceedings of. Deformed entropic and information inequalities for states. Axiomatic characterizations of information measures.
It follows from strong subadditivity that the entanglement entropy s. Schumacher 1996 or quantum communication ohya and volovich 2003, ozawa and yuen 1993, holevo 1998. The structure of renyi entropic inequalities proceedings. A portfolio selection application illustrates that minimizing renyi entropy yields portfolios that outperform stateoftheart minimum variance portfolios in terms of riskreturnturnover trade. Shannon entropy versus renyi entropy from a cryptographic viewpoint. For the linear entropy the set of all entropy vectors entropy body is a subset of the box x,y,z 0. Application of renyi entropy for ultrasonic molecular imaging. Entropy in classical and quantum information theory. The probability density function pdf is estimated by a kernel estimator, which can be found in bauers work in 2007. Chistyakov abstractthe classical entropy power inequality is extended to the r. The tsallis and renyi entropies are related to the bgs entropy. The shannon entropy for positive probabilities is the weighted arithmetic mean with the probabilities as weights of the quantities log 2 p k k 1,n which can be considered see note 1 entropies of single events. If you relied on shannon entropy for security youd be in trouble in that case.
Ganesh ajjanagadde, yury polyanskiy adder mac and estimates for r enyi entropy september 30, 2015 11 28. This is a more difficult theorem and was proved in 1973 by elliott h. Strong subadditivity ssa liebruskai 73 when for any, actually, these two inequalities are equivalent. Somewhat surprisingly, we find for 0 entropy is monotone under completely positive, trace preserving maps. The complexity of estimating r enyi entropy jayadev acharya 1, alon orlitskyy 2, ananda theertha sureshz, and himanshu tyagix2 1massachusetts institute of technology 2university of california, san diego abstract it was recently shown that estimating the shannon entropy hp of a discrete ksymbol distribution p re. For the proof, we use an interpolating family of l pnorms with respect to a state. Renyi entropy power inequalities via normal transport and. We investigate the universal inequalities relating the alpha renyi entropies of the marginals of a multipartite quantum state. Strong subadditivity of entropy ssa was long known and appreciated in classical probability theory and information theory. Other applications include quantum algorithms, quantum cryptography, or statistical physics as discussed in ohya and watanabe 2010. We also give some improved entropy inequalities based on renyi relative entropy. The most widely used quantum renyi entropies are the conditional min.
Somewhat surprisingly, we find for 0 entropy and information alfrped rrnyi mathematical institute hungarian academy of sciences 1. Notes on entanglement and computational com plexity. We develop a unifying treatment toward the improvement of some quantum entropy inequalities. The survey lie75 contains a discussion of subadditivity in statistical mechanics, and the monograph rs describes examples in information theory. However, the concept of shanon entropy shares some intuition with boltzmanns. This was done using both continuous entropy, h f, and renyi entropy, i f 1. Lieb and mary beth ruskai, using a matrix inequality of elliott h. However, it does not have the meaning of a distance, and the inequality itself should be. Renyi entropies, mutual information, relative entropy also powerful when combined with semiclassical gravity generalize entanglement entropy to deal with nonminimal bulk surfaces. The tsallis entropy of a bipartite qudit system was shown to satisfy the generalized subadditivity condition 9, 10.
This is no longer true when the entanglement is measured by some renyi entropy. We demonstrate that the renyi 2 entropy provides a natural measure of information for any multimode gaussian state of quantum harmonic systems, operationally linked to the phasespace shannon sampling entropy of the wigner distribution of the state. In information theory, entropy is a measure of randomness or uncertainty in the system. A minkowski type trace inequality and strong subadditivity. However, as kendall states 10 these measures are scaledependent when applied to continuous distributions, and so their absolute values are meaningless.
On the strong subadditivity of the renyi entropies for bosonic and fermionic gaussian states. Ee in qfts tadashi takayanagi ipmu, the university of tokyo. For arbitrary gaussian states, the renyi entropy of order. Mutual information renyi entropy relative entropy modular hamiltonian quantum renyi relative entropy finite replica compare thermal. Among other things this leads to a definition of an lqlp norm for operators when 1. Optimal transport and renyi informational divergence here, the righthand side has a strong relationship with the renyi divergence power. From a physical point of view, information theory has nothing to do with physics. We provide a new inequality that links two important entropy notions. Adscft and strong subadditivity of entanglement entropy. Let us begin with a precise statement of this fact. In this work, we focus on the role of subadditivity of entropy. In particular, an emphasis is put on a lower bound of quantum conditional mutual information qcmi as it gives a pinskerlike lower bound for the qcmi. Subadditivity of matrix entropy and concentration of random.
Dvoretzkys theorem concerns the existence of almost spherical crosssections of highdimensional convex bodies, and the additivity problem for renyi entropy can be restated in precisely this form. This doesnt really alter anything, as both sides can be converted to the appropriate base by multiplying with a constant, and the above goes through with no change. The tsallis entropy is often used in finitesize or especially correlated systems. As we change h, the value of equation 1 will change and we can find an h to maximize the value. The renyi entropy as a measure of uncertainty the renyi entropy is a oneparameter generalization of the shannon entropy. We prove that, in the gaussian scenario, such an entropy satisfies the strong subadditivity inequality, a key requirement for quantum information. On the strong subadditivity of the r\enyi entropies for bosonic and. Aspects of renormalized entanglement entropy tatsuma nishioka u. Second, it is also known that renyi entropies do not satisfy the subadditivity since only the additivity for independent random variables is required for renyi. Abstract it was recently shown that estimating the shannon entropy hp of a discrete ksymbol. In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the functions values at each element. In general, the limit 1 does not exist anymore, and the values of the lim inf and lim sup depend upon the arithmetic properties of the map. Roos entanglement is a key feature of manybody quantum systems.
Physica scripta paper unveiling the hanbury brown and twiss effect through renyi entropy correlations to cite this article. Characterization of shannonsmeasureof entropy let d pi, p2, pn, be a finite discrete probability distribution, that. Janos aczel, in modern information processing, 2006. For instance, by relaxing the third of shannons requirements, that of additivity, renyi 11 was able to extend shannon entropy to a continuous. The mathematical results until the 1970s are well written in the book 8.
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