be found using our quantum relative entropy with a suitable uniform prior density matrix.. 27.[r]

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Given two pairs of **quantum** states, we want to decide if there exists a quan- tum channel that transforms one pair into the other. The theory of **quantum** statistical comparison and **quantum** **relative** majorization provides necessary and sufficient conditions for such a transformation to exist, but such conditions are typically difficult to check in practice. Here, by building upon work by Keiji Matsumoto, we relax the problem by allowing for small errors in one of the transformations. In this way, a simple sufficient condition can be formulated in terms of one-shot **relative** entropies of the two pairs. In the asymptotic set- ting where we consider sequences of state pairs, under some mild convergence conditions, this implies that the **quantum** **relative** **entropy** is the only relevant quantity deciding when a pairwise state transformation is possible. More pre- cisely, if the **relative** **entropy** of the initial state pair is strictly larger compared to the **relative** **entropy** of the target state pair, then a transformation with exponentially vanishing error is possible. On the other hand, if the **relative** **entropy** of the target state is strictly larger, then any such transformation will have an error converging exponentially to one. As an immediate consequence, we show that the rate at which pairs of states can be transformed into each other is given by the ratio of their **relative** entropies. We discuss applications to the resource theories of athermality and coherence, where our results imply an exponential strong converse for general state interconversion.

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Let us discuss related work in this area. In [9, 17] it was noted that the von Neumann index may be interpreted as a measure of network regularity. In [10], it was then shown that for scale free networks the von Neumann index of a graph is linearly related to the Shannon **entropy** of the graph’s ensemble. Correlations between these entropies were observed when the graph’s degree distribution displayed heterogeneity in [2]. In [6], it was shown that the von Neumann index of an n vertex graph is almost always minimized by the star graph. In that work, the star graph was also conjectured to minimize the **entropy** over connected graphs. Other studies of the von Neuamnn **entropy** of Laplacians have been performed in [11, 12, 13, 14, 15, 18]. Also of interest is [16], in which it is shown that the **quantum** **relative** **entropy** of a graph’s Laplacian with another particular **quantum** state can be related to the number of spanning trees of the graph.

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encounter a state of high energy density commonly associated with the big bang. The origin of this state is an open question of cosmogony (Khoury et al. 2001) and may motivate a big bounce (Brandenberger and Peter 2017). Another puzzle arises when we assume that this state initially contained equal amounts of matter and antimatter, while unequal amounts are observed in the universe today. To account for this matter-antimatter asymmetry, a mechanism for baryogenesis is necessary (Sakharov 1991). Additionally, the initially hot and dense matter cooled sufficiently to become transparent to radiation ∼ 10 5 years after the big bang. In the ΛCDM model, this matter emits the cosmic microwave background (CMB) radiation (Penzias and Wilson 1965). However, the CMB is more isotropic than expected, which is known as the horizon problem. The theory of cosmological inflation addresses this by introducing a period of exponentially accelerating expansion up to 10 −32 s after the big bang (Guth 1981; Linde 1982; Albrecht and Steinhardt 1982). This could allow any two regions of the CMB to become thermalized in the early universe. This also addresses the question of why our expanding metric appears to be spatially flat, known as the flatness problem. However, inflation suffers from problems such as the **entropy** problem (Penrose 1989) or the multiverse problem (Ijjas et al. 2014). Moreover, the ΛCDM model interpretation of CMB data gives an expansion rate of the universe (H 0 ) that is in tension with cosmology-independent measurements

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The above definitions of **entropy** practically cover all known types, although with each of the existing definitions of **entropy** has its field of application. But they all share a common limitation. This limitation is that with their help, you cannot answer the question of: how the **entropy** changes over time in the evolution of systems in accordance with changes in external constraints. They are difficult to use and to analyze the evolution of the system. Moreover, the use of statistical regularities and averaging in (2-3) makes it difficult to study the possibility of actually observed in nature irreversible dynamics in the framework of the laws of mechanics.

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Although the definitions of the classical and the **quantum** channel capacities are similar, these two notions differ in several ways. To show this consider the dephasing channel, which for a given basis removes all off–diagonal elements of the density matrix. This channel transforms any coherent superposition of pure orthogonal states into their statistical mixture, however, any classical state remains unchanged. Therefore, the classical capacity of this channel can be positive, while its **quantum** capacity is equal to zero, as there does not exist even a two-dimensional Hilbert subspace which survives the action of the channel [8].

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controlled by an attacker, therefore unique random numbers can be yielded by measuring the quadrature amplitude of the vacuum state [16, 17]. All the components in this scheme, including laser source, beam splitter and photo detectors have been integrated on a single chip recently [18]. Meanwhile, bit conversion and post-processing are easy to be implemented in virtual “hardware” inside the field-programmable gate array (FPGA). Chip-size integration of the QRNG is expectable. Several dedicated researches have been developed to enhance the generation rate of random bits in this proposal, such as schemes based on optimization of the digitization algorithm [19], implementation of fast randomness extraction in the post- processing [20], application of squeezing vacuum state to increase **entropy** in raw data [21], and optimization of ADC parameters to improve the **quantum** **entropy** in the raw data [22]. In this paper, considering the effects of the classical noise, we discuss the role of homodyne gain in enhancing **quantum** **entropy** in the vacuum-based **quantum** RNG working in the optimum dynamical ADC range scenario. Conditional min-**entropy** is applied to critically assess the **quantum** **entropy** in the **quantum** RNG. It is the key input parameter of randomness extractor and determines the extraction ratio of true randomness from the raw random sequence, thereby affects the generation speed of **quantum** RNG significantly.

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1. Introduction. It is known that the equilibrium states of Hamiltonian models of interacting particle systems correspond to states with extremal entropic properties. The transition to equilibrium occurs through the metastable states, related to prob- abilistic limit results, such as the law of large numbers, the central limit theorem, large deviations. The principle of maximum **entropy** is well known and widely used in the construction of mathematical models. According to it, under the uncertainty of the choice of a speciﬁc type of distribution to be used as a model of a physical phe- nomenon, maximum **entropy** distributions (and/or their mixtures) should be tested ﬁrst. Conversely, if a distribution ﬁts a real phenomenon well, its relationship with maximum **entropy** distributions should be investigated.

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= − ∑ (7) which shows the degree of randomness of the system. For example, when p 1 = p 2 = = p n = 1 n , every state happens in the equal probability, which is a random system. In this situation, the Shannon **entropy** takes its maximum value ln n . If p 1 = 1 , the system is completely predictable, i.e., the first state always happens and the **entropy** takes its minimum value 0. We can also regard the **entropy** as the superposition of the uncertainties

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Pruning is one approach to address this problem, where models are made more compact by discarding entries from the model, based on additional selection criteria. The challenge in this task is to choose the entries that will least degenerate the quality of the task for which the model is used. For language models, an effective algorithm based on **relative** **entropy** is described in (Seymore and Rosenfeld, 1996; Stolcke, 1998; Moore and Quirk, 2009). In these approaches, a criteria based on the KL divergence is applied, so that higher order n-grams are only included in the model when they provide enough additional information to the model, given the lower order n-grams. Recently, this concept was applied for translation model pruning (Ling et al., 2012; Zens et al., 2012), and results indicate that this method yields a better phrase table size and translation quality ratio than previous methods, such as the well known method in (Johnson et al., 2007), which uses the Fisher’s exact test to calculate how well a phrase pair is supported by data.

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Werner-Popescu state is found to be the reason behind the matching of the 1: N − 1 separability ranges due to commutative AR-criterion and non-commutative CSTRE criterion. The relatively smoother convergence of the parameter x with respect to increasing q is observed in the case of implicit plots of CSTRE in comparison with the convergence in the case of AR q -conditional **entropy** thus establishing the supremacy of CSTRE criterion over the AR-criterion. The 1: N − 1 separability range obtained for N -qudit Werner Popescu states using entropic criteria is seen to match with that obtained using an algebraic necessary and sufficient condition for separability.

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law it follows that the dimensionless quantity M 2 /K is the adiabatic invariant, which in principle can be quantized if to follow the Bekenstein conjecture. From the Euclidean action for the black hole it follows that K and A serve as dynamically conjugate variables. This allows us to calculate the **quantum** tunneling from the black hole to the white hole, and determine the temperature and **entropy** of the white hole.

Functional version for the so-called Furuta parametric **relative** operator **entropy** is here investigated. Some related functional inequalities are also discussed. The theoretical results obtained by our functional approach immediately imply those of operator versions in a simple, fast, and nice way.

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The information contained within a **quantum** system is of great importance for various practical implementations of **quantum** mechanics, most importantly for the development of **quantum** computers (e.g., Refs. [1–4]). In order to un- derstand the **quantum** information, one must find a way of measuring the entanglement of a state. Entanglement is a defining feature of **quantum** mechanics that distinguishes it from classical mechanics, and there has been much work in recent years into the evolution of entanglement with time, particularly the observation of the abrupt decay of entangled states, coined as “sudden death” [5,6]. The decoherence of entanglement [7,8] is a problem for the operation of **quantum** computers and so understanding the mechanism behind this is an important contribution to the development of future machines. One particular measurement of entanglement and **quantum** information is the von Neumann **entropy** (see, for instance, Sec. 11.3 in Ref. [1]). This is well understood in the standard **quantum** mechanical setting, however, to date there has only been a small amount of work done concerning the proper treatment of **entropy** in non-Hermitian, parity- time (PT )-symmetric systems [9–12]. These differ from open **quantum** systems as the energy eigenvalues are real or appear as complex conjugate pairs and do not describe decay.

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Furthermore we present new bounds for **entropy** and mutual information. Corollary 3.5. Let X be a random variable whose range has |X | elements and has the probability mass function p(x) > 0, with m(x) = min{p(x), 1/|X |} and M (x) = max{p(x), 1/|X|}, x ∈ X. If r(x) = p(x) − 1/|X|, then

IV. Choice of the Measures of **Entropy**: For the investigation at hand, we have chosen a few general measures of **entropy** such as Rényi, Tsallis, Abe, Kaniadakis and Sharma-Mittal measures. It may be noted that in the present context (**entropy** of Pena’s synthetic indicators) we cannot presume stability. More particularly, since the weights assigned by the Pena method depend on the order in which the constituent variables enter into the formula, the weight w j (associated with d ij ) is contingent upon the

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This article proposes a method for estimating the local number of signals components. It is based on the short- term Rényi **entropy** of signals in the time-frequency plane. Using the Rényi **entropy** of a short-term segment of a TFD of a multicomponent nonstationary signal, **relative** to the short-term Rényi **entropy** of a reference signal, the number of components present in the signal can be accurately estimated. The proposed method does not require any a priori information about the analyzed signal, nor the knowledge of the Rényi **entropy** of one of the signal components. The method was tested on var- ious synthetic signals, including signals embedded in additive white Gaussian noise, and its use in practice was illustrated on a real-life signal. The method is sensi- tive to the selection of the TFD. The presented results indicate that the MBD, being an example of Separable kernel TFDs, is a good choice of a TFD when the pro- posed method is applied in practical situations [4]. These results show that the proposed algorithm can be useful in many applications that require component count and component separation; and, it can be a pre- ferred alternative to other methods such as the Empiri- cal Mode Decomposition [15].

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maximum value (HM ) - HR = H/HM.The criterion used to choose the number of categories in each country is related with the number of parties most voted, in the ﬁ rst considered election. The objective was that, at the beginning of the sample, the proportion of the category "Others" shouldn’t exceed 10%. It allows to see, for example, the relevance and the evolution of this category. An increase in its value means that exists more dispersion on votes, being a clear indicator of global uncer- tainty by electors in relation to their political opinions, which will increase **entropy**. While there exist countries where this category decreased (the case of Spain, that could be related with the particular event of terrorist attacks in the days before the elections in 2004 and 2008), others are where that category is, inclusively, the most important one (the case, for example, of Belgium, where in the ﬁrst election it had a residual weight of 1% and, in 2007, after considered the ﬁrst six parties, the value of "Others" corresponded to about 24% of the votes). Excluding the cases of Germany, Spain and Portugal, all other countries show a trend, more or less evident, of increase in that category, in spite of the cases of Italy, Switzerland and Denmark where that value had decreased in the last elections. The behaviour of the category "Others" could be seen, for the diﬀerent countries, in Figure 2.

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between A and the minimal surface ending on ∂A, plus a boundary term, and can be interpreted as a form of quasilocal energy associated to region A. Because **relative** **entropy** obeys certain universal properties, this apparently innocuous statement turns out to have some significant consequences in the bulk, in that it imposes constraints that any geometry dual to a well-defined CFT state must satisfy. For small regions these constraints turn into integrated positive energy conditions, and it is possible to turn them back into the CFT and obtain an additional constraint that **relative** **entropy** must satisfy (in theories with geometric duals). Furthermore, using the inverse Radon transform near the boundary it is possible to reconstruct the bulk stress energy tensor from **relative** **entropy**, up to a certain order which will be made precise. The formula we will derive also indicates a possible path to reconstructing the bulk action from **relative** **entropy** at points deep in the bulk, although more mathematical advances will be required in order to make this work.

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