Nyi Entropies Research Articles

X-exponential-G Family of Distributions With Applications

A new family of continuous distributions called the X-exponential-G (XE-G) family is proposed. Explicit expressions are derived for the ordinary and incomplete moments, generating functions, mean deviation about the mean and median, Shannon and R\'{e}nyi entropies, and order statistics of this new family. Estimation of the parameters of the new family is done using the method of maximum likelihood. Assessment of the performance of the maximum likelihood estimates is carried out through a simulation study using the quantile function of the XE-G distribution. The usefulness of this new family is illustrated by modeling two real datasets.

A simpler security proof for 6-state quantum key distribution

Six-state Quantum Key Distribution (QKD) achieves the highest key rate in the class of qubit-based QKD schemes. The standard security proof, which has been developed since 2005, invokes complicated theorems involving smooth R\'{e}nyi entropies. In this paper we present a simpler security proof for 6-state QKD that entirely avoids R\'{e}nyi entropies. This is achieved by applying state smoothing directly in the Bell basis. We obtain the well known asymptotic rate, but with slightly more favorable finite-size terms. We furthermore show that the same proof technique can be used for 6-state quantum key recycling.

Quantum-information theory of a Dirichlet ring with Aharonov–Bohm field

Shannon quantum information entropies $S_{\rho,\gamma}$, Fisher informations $I_{\rho,\gamma}$, Onicescu energies $O_{\rho,\gamma}$ and R\'{e}nyi entropies $R_{\rho,\gamma}(\alpha)$ are calculated both in the position (subscript $\rho$) and momentum ($\gamma$) spaces as functions of the inner radius $r_0$ for the two-dimensional Dirichlet unit-width annulus threaded by the Aharonov-Bohm (AB) flux $\phi_{AB}$. Discussion is based on the analysis of the corresponding position and momentum waveforms. Position Shannon entropy (Onicescu energy) grows logarithmically (decreases as $1/r_0$) with large $r_0$ tending to the same asymptote $S_\rho^{asym}=\ln(4\pi r_0)-1$ [$O_\rho^{asym}=3/(4\pi r_0)$] for all orbitals whereas their Fisher counterpart $I_{\rho_{nm}}(\phi_{AB},r_0$) approaches in the same regime the $m$-independent limit mimicking in this way the energy spectrum variation with $r_0$, which for the thin structures exhibits quadratic dependence on the principal index. Frequency of the fading oscillations of the radial parts of the wave vector functions increases with the inner radius what results in the identical $r_0\gg1$ asymptote for all momentum Shannon entropies $S_{\gamma_{nm}}(\phi_{AB};r_0)$ with the alike $n$ and different $m$. The same limit causes the Fisher momentum components $I_\gamma(\phi_{AB},r_0)$ to grow exponentially with $r_0$. It is proved that the lower limit $\alpha_{TH}$ of the semi-infinite range of the dimensionless coefficient $\alpha$, where the momentum component of this one-parameter entropy exists, is \textit{not} influenced by the radius; in particular, the change of the topology from the simply, $r_0=0$, to the doubly, $r_0>0$, connected domain is \textit{un}able to change $\alpha_{TH}=2/5$. AB field influence on the measures is calculated too.

Measurement-induced entanglement transitions in many-body localized systems

The resilience of quantum entanglement to a classicality-inducing environment is tied to fundamental aspects of quantum many-body systems. The dynamics of entanglement has recently been studied in the context of measurement-induced entanglement transitions, where the steady-state entanglement collapses from a volume-law to an area-law at a critical measurement probability $p_{c}$. Interestingly, there is a distinction in the value of $p_{c}$ depending on how well the underlying unitary dynamics scramble quantum information. For strongly chaotic systems, $p_{c} > 0$, whereas for weakly chaotic systems, such as integrable models, $p_{c} = 0$. In this work, we investigate these measurement-induced entanglement transitions in a system where the underlying unitary dynamics are many-body localized (MBL). We demonstrate that the emergent integrability in an MBL system implies a qualitative difference in the nature of the measurement-induced transition depending on the measurement basis, with $p_{c} > 0$ when the measurement basis is scrambled and $p_{c} = 0$ when it is not. This feature is not found in Haar-random circuit models, where all local operators are scrambled in time. When the transition occurs at $p_{c} > 0$, we use finite-size scaling to obtain the critical exponent $\nu = 1.3(2)$, close to the value for 2+0D percolation. We also find a dynamical critical exponent of $z = 0.98(4)$ and logarithmic scaling of the R\'{e}nyi entropies at criticality, suggesting an underlying conformal symmetry at the critical point. This work further demonstrates how the nature of the measurement-induced entanglement transition depends on the scrambling nature of the underlying unitary dynamics. This leads to further questions on the control and simulation of entangled quantum states by measurements in open quantum systems.

Measurement-Induced Phase Transitions in the Dynamics of Entanglement

We define dynamical universality classes for many-body systems whose unitary evolution is punctuated by projective measurements. In cases where such measurements occur randomly at a finite rate $p$ for each degree of freedom, we show that the system has two dynamical phases: `entangling' and `disentangling'. The former occurs for $p$ smaller than a critical rate $p_c$, and is characterized by volume-law entanglement in the steady-state and `ballistic' entanglement growth after a quench. By contrast, for $p > p_c$ the system can sustain only area-law entanglement. At $p = p_c$ the steady state is scale-invariant and, in 1+1D, the entanglement grows logarithmically after a quench. To obtain a simple heuristic picture for the entangling-disentangling transition, we first construct a toy model that describes the zeroth R\'{e}nyi entropy in discrete time. We solve this model exactly by mapping it to an optimization problem in classical percolation. The generic entangling-disentangling transition can be diagnosed using the von Neumann entropy and higher R\'{e}nyi entropies, and it shares many qualitative features with the toy problem. We study the generic transition numerically in quantum spin chains, and show that the phenomenology of the two phases is similar to that of the toy model, but with distinct `quantum' critical exponents, which we calculate numerically in $1+1$D. We examine two different cases for the unitary dynamics: Floquet dynamics for a nonintegrable Ising model, and random circuit dynamics. We obtain compatible universal properties in each case, indicating that the entangling-disentangling phase transition is generic for projectively measured many-body systems. We discuss the significance of this transition for numerical calculations of quantum observables in many-body systems.

Comparison of incoherent operations and measures of coherence

Considerable work has recently been directed toward developing resource theories of quantum coherence. In most approaches, a state is said to possess quantum coherence if it is not diagonal in some specified basis. In this letter we establish a criterion of physical consistency for any resource theory in terms of physical implementation of the free operations, and we show that all currently proposed basis-dependent theories of coherence fail to satisfy this criterion. We further characterize the physically consistent resource theory of coherence and find its operational power to be quite limited. After relaxing the condition of physical consistency, we introduce the class of dephasing-covariant incoherent operations, present a number of new coherent monotones based on relative R\'{e}nyi entropies, and study incoherent state transformations under different operational classes. In particular, we derive necessary and sufficient conditions for qubit state transformations and show these conditions hold for all classes of incoherent operations.

Entanglement of spatial regions vs. entanglement of particles

Consider an arbitrary local quantum field theory with a gap or an arbitrary gapless free theory. We consider states in such a theory, that describe two entangled particles localized in disjoint regions of space. We show that in such a state, to leading order, R\'{e}nyi entropies of spatial regions, containing only one of the particles are same as their quantum mechanical counterparts, after subtraction of vacuum contribution. Subleading corrections depend on overlap of wave functions. These results suggest that Von Neumann entropy of a spatial region, after subtraction of vacuum contribution, can serve as a measure of entanglement of indistinguishable particles in pure states.

Thermal corrections to Rényi entropies for free fermions

We calculate thermal corrections to R\'{e}nyi entropies for free massless fermions on a sphere. More specifically, we take a free fermion on $\mathbb{R}\times\mathbb{S}^{d-1}$ and calculate the leading thermal correction to the R\'{e}nyi entropies for a cap like region with opening angle $2\theta$. By expanding the density matrix in a Boltzmann sum, the problem of finding the R\'{e}nyi entropies can be mapped to the problem of calculating a two point function on an $n$ sheeted cover of the sphere. We follow previous work for conformal field theories to map the problem on the sphere to a conical region in Euclidean space. By using the method of images, we calculate the two point function and recover the R\'{e}nyi entropies.

Category Theoretic Properties of the A. Rényi and C. Tsallis Entropies

The problem of embedding the Tsallis and R\'{e}nyi entropies in the framework of category theory and their axiomatic foundation is studied. To this end, we construct a special category MES related to measured spaces. We prove that both of the R\'{e}nyi and Tsallis entropies can be imbedded in the formalism of category theory by proving that the same basic functional that appears in their definitions, as well as in the associated Lebesgue space norms, has good algebraic compatibility properties. We prove that this functional is both additive and multiplicative with respect to the direct product and the disjoint sum (the coproduct) in the category MES, so it is a natural candidate for the measure of information or uncertainty. We prove that the category MES can be extended to monoidal category, both with respect to the direct product as well as to the coproduct. The basic axioms of the original R\'{e}nyi entropy theory are generalized and reformulated in the framework of category MES and we prove that these axioms foresee the existence of an universal exponent having the same values for all the objects of the category MES. In addition, this universal exponent is the parameter, which appears in the definition of the Tsallis and R\'{e}nyi entropies.

Investigating properties of a family of quantum Rényi divergences

Audenaert and Datta recently introduced a two-parameter family of relative R\'{e}nyi entropies, known as the $\alpha$-$z$-relative R\'{e}nyi entropies. The definition of the $\alpha$-$z$-relative R\'{e}nyi entropy unifies all previously proposed definitions of the quantum R\'{e}nyi divergence of order $\alpha$ under a common framework. Here we will prove that the $\alpha$-$z$-relative R\'{e}nyi entropies are a proper generalization of the quantum relative entropy by computing the limit of the $\alpha$-$z$ divergence as $\alpha$ approaches one and $z$ is an arbitrary function of $\alpha$. We also show that certain operationally relevant families of R\'enyi divergences are differentiable at $\alpha = 1$. Finally, our analysis reveals that the derivative at $\alpha = 1$ evaluates to half the relative entropy variance, a quantity that has attained operational significance in second-order quantum hypothesis testing.

Uncertainty relations for MUBs and SIC-POVMs in terms of generalized entropies

We formulate uncertainty relations for mutually unbiased bases and symmetric informationally complete measurements in terms of the R\'{e}nyi and Tsallis entropies. For arbitrary number of mutually unbiased bases in a finite-dimensional Hilbert space, we give a family of Tsallis $\alpha$-entropic bounds for $\alpha\in(0;2]$. Relations in a model of detection inefficiences are obtained. In terms of R\'{e}nyi's entropies, lower bounds are given for $\alpha\in[2;\infty)$. State-dependent and state-independent forms of such bounds are both given. Uncertainty relations in terms of the min-entropy are separately considered. We also obtain lower bounds in term of the so-called symmetrized entropies. The presented results for mutually unbiased bases are extensions of some bounds previously derived in the literature. We further formulate new properties of symmetric informationally complete measurements in a finite-dimensional Hilbert space. For a given state and any SIC-POVM, the index of coincidence of generated probability distribution is exactly calculated. Short notes are made on potential use of this result in entanglement detection. Further, we obtain state-dependent entropic uncertainty relations for a single SIC-POVM. Entropic bounds are derived in terms of the R\'{e}nyi $\alpha$-entropies for $\alpha\in[2;\infty)$ and the Tsallis $\alpha$-entropies for $\alpha\in(0;2]$. In the Tsallis formulation, a case of detection inefficiences is briefly mentioned. For a pair of symmetric informationally complete measurements, we also obtain an entropic bound of Maassen-Uffink type.

Entanglement entropy of the two-dimensional Heisenberg antiferromagnet

We compute the von Neumann and generalized R\'{e}nyi entanglement entropies in the ground-state of the spin-1/2 antiferromagnetic Heisenberg model on the square lattice using the modified spin-wave theory for finite lattices. The addition of a staggered magnetic field to regularize the Goldstone modes associated with symmetry-breaking is shown to be essential for obtaining well-behaved values for the entanglement entropy. The von Neumann and R\'{e}nyi entropies obey an area law with additive logarithmic corrections, and are in good quantitative agreement with numerical results from valence bond quantum Monte Carlo and density matrix renormalization group calculations. We also compute the spin fluctuations and observe a multiplicative logarithmic correction to the area law in excellent agreement with quantum Monte Carlo calculations.

Some General Properties of Unified Entropies

Basic properties of the unified entropies are examined. The consideration is mainly restricted to the finite-dimensional quantum case. Bounds in terms of ensembles of quantum states are given. Both the continuity in Fannes' sense and stability in Lesche's sense are shown for wide ranges of parameters. In particular, uniform estimates are obtained for the quantum R\'{e}nyi entropies. Stability properties in the thermodynamic limit are discussed as well. It is shown that the unified entropies enjoy both the subadditivity and triangle inequality for a certain range of parameters. Non-decreasing of all the unified entropies under projective measurement is proved.

Additivity properties of a Gaussian channel

The Amosov-Holevo-Werner conjecture implies the additivity of the minimum Re'nyi entropies at the output of a channel. The conjecture is proven true for all Re'nyi entropies of integer order greater than two in a class of Gaussian bosonic channel where the input signal is randomly displaced or where it is coupled linearly to an external environment.