Note on von Neumann and Rényi entropies of a graph

We investigate the universal inequalities relating the α -Rényi entropies of the marginals of a multi-partite quantum state. This is in analogy to the same question for the Shannon and von Neumann entropies ( α =1), which are known to satisfy several non-trivial inequalities such as strong subadditivity. Somewhat surprisingly, we find for 0< α <1 that the only inequality is non-negativity: in other words, any collection of non-negative numbers assigned to the non-empty subsets of n parties can be arbitrarily well approximated by the α -entropies of the 2 n −1 marginals of a quantum state. For α >1, we show analogously that there are no non-trivial homogeneous (in particular, no linear) inequalities. On the other hand, it is known that there are further, nonlinear and indeed non-homogeneous, inequalities delimiting the α -entropies of a general quantum state. Finally, we also treat the case of Rényi entropies restricted to classical states (i.e. probability distributions), which, in addition to non-negativity, are also subject to monotonicity. For α ≠0,1, we show that this is the only other homogeneous relation.

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Monotonicity of the von Neumann Entropy Expressed as a Function of Rényi Entropies

Open Systems & Information Dynamics ◽

10.1142/s1230161214500061 ◽

2014 ◽

Vol 21 (03) ◽

pp. 1450006 ◽

Cited By ~ 1

Author(s):

Mark Fannes

Keyword(s):

Density Matrix ◽

Von Neumann Entropy ◽

Von Neumann ◽

Integer Order ◽

Odd Order ◽

Rényi Entropies ◽

Even Order ◽

Renyi Entropies

The von Neumann entropy of a density matrix of dimension d, expressed in terms of the first d − 1 integer order Rényi entropies, is monotonically increasing in Rényi entropies of even order and decreasing in those of odd order.

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Phase-space continuity equations for quantum decoherence, purity, von Neumann and Renyi entropies

Journal of Physics Conference Series ◽

10.1088/1742-6596/1275/1/012032 ◽

2019 ◽

Vol 1275 ◽

pp. 012032 ◽

Cited By ~ 4

Author(s):

Alex E. Bernardini ◽

Orfeu Bertolami

Keyword(s):

Phase Space ◽

Quantum Decoherence ◽

Von Neumann ◽

Continuity Equations ◽

Rényi Entropies ◽

Renyi Entropies

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Entanglement dynamics in Rule 54: Exact results and quasiparticle picture

SciPost Physics ◽

10.21468/scipostphys.11.6.107 ◽

2021 ◽

Vol 11 (6) ◽

Author(s):

Katja Klobas ◽

Bruno Bertini

Keyword(s):

Linear Growth ◽

Von Neumann Entropy ◽

Entanglement Dynamics ◽

Von Neumann ◽

Rényi Entropies ◽

Quantum Quenches ◽

Initial States ◽

Local Observables ◽

Quasiparticle Picture ◽

Renyi Entropies

We study the entanglement dynamics generated by quantum quenches in the quantum cellular automaton Rule 54. We consider the evolution from a recently introduced class of solvable initial states. States in this class relax (locally) to a one-parameter family of Gibbs states and the thermalisation dynamics of local observables can be characterised exactly by means of an evolution in space. Here we show that the latter approach also gives access to the entanglement dynamics and derive exact formulas describing the asymptotic linear growth of all Rényi entropies in the thermodynamic limit and their eventual saturation for finite subsystems. While in the case of von Neumann entropy we recover exactly the predictions of the quasiparticle picture, we find no physically meaningful quasiparticle description for other Rényi entropies. Our results apply to both homogeneous and inhomogeneous quenches.

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Rényi Extrapolation of Shannon Entropy

Open Systems & Information Dynamics ◽

10.1023/a:1025128024427 ◽

2003 ◽

Vol 10 (03) ◽

pp. 297-310 ◽

Cited By ~ 95

Author(s):

Karol Życzkowski

Keyword(s):

Probability Distribution ◽

Quantum State ◽

Shannon Entropy ◽

Upper Bound ◽

Von Neumann Entropy ◽

Discrete Probability ◽

Von Neumann ◽

Integer Order ◽

Rényi Entropies ◽

Renyi Entropies

Relations between Shannon entropy and Rényi entropies of integer order are discussed. For any N-point discrete probability distribution for which the Rényi entropies of order two and three are known, we provide a lower and an upper bound for the Shannon entropy. The average of both bounds provide an explicit extrapolation for this quantity. These results imply relations between the von Neumann entropy of a mixed quantum state, its linear entropy and traces.

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On Renyi Entropies and Their Applications to Guessing Attacks in Cryptography

IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences ◽

10.1587/transfun.e97.a.2542 ◽

2014 ◽

Vol E97.A (12) ◽

pp. 2542-2548 ◽

Cited By ~ 2

Author(s):

Serdar BOZTAS

Keyword(s):

Rényi Entropies ◽

Renyi Entropies

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Analytic critical points of charged Rényi entropies from hyperbolic black holes

Journal of High Energy Physics ◽

10.1007/jhep05(2021)080 ◽

2021 ◽

Vol 2021 (5) ◽

Author(s):

Jie Ren

Keyword(s):

Black Holes ◽

Black Hole ◽

Study Phase ◽

Zero Temperature ◽

Charged Scalar ◽

Yang Mills ◽

Gravitational Systems ◽

Rényi Entropies ◽

Zero Entropy ◽

Renyi Entropies

Abstract We analytically study phase transitions of holographic charged Rényi entropies in two gravitational systems dual to the $$ \mathcal{N} $$ N = 4 super-Yang-Mills theory at finite density and zero temperature. The first system is the Reissner-Nordström-AdS5 black hole, which has finite entropy at zero temperature. The second system is a charged dilatonic black hole in AdS5, which has zero entropy at zero temperature. Hyperbolic black holes are employed to calculate the Rényi entropies with the entangling surface being a sphere. We perturb each system by a charged scalar field, and look for a zero mode signaling the instability of the extremal hyperbolic black hole. Zero modes as well as the leading order of the full retarded Green’s function are analytically solved for both systems, in contrast to previous studies in which only the IR (near horizon) instability was analytically treated.

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