A New Inequality for the von Neumann Entropy

Linden, Noah; Winter, Andreas

doi:10.1007/s00220-005-1361-2

Cited by 57 publications

(84 citation statements)

References 11 publications

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“…There are many more possible inequalities that entanglement entropies for n ≥ 3 intervals have to satisfy [92,93], which can be seen to follow from (6.12) and the other inequalities discussed so far for n ≤ 3 [88]. Hence these inequalities are also proven to hold in holography, assuming appropriate energy conditions.…”

Section: Jhep10(2017)034mentioning

confidence: 78%

Time evolution of entanglement for holographic steady state formation

Erdmenger

Fernández

Flory

et al. 2017

J. High Energ. Phys.

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Within gauge/gravity duality, we consider the local quench-like time evolution obtained by joining two 1+1-dimensional heat baths at different temperatures at time t = 0. A steady state forms and expands in space. For the 2+1-dimensional gravity dual, we find that the "shockwaves" expanding the steady-state region are of spacelike nature in the bulk despite being null at the boundary. However, they do not transport information. Moreover, by adapting the time-dependent Hubeny-Rangamani-Takayanagi prescription, we holographically calculate the entanglement entropy and also the mutual information for different entangling regions. For general temperatures, we find that the entanglement entropy increase rate satisfies the same bound as in the 'entanglement tsunami' setups. For small temperatures of the two baths, we derive an analytical formula for the time dependence of the entanglement entropy. This replaces the entanglement tsunami-like behaviour seen for high temperatures. Finally, we check that strong subadditivity holds in this time-dependent system, as well as further more general entanglement inequalities for five or more regions recently derived for the static case.

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Section: Jhep10(2017)034mentioning

confidence: 78%

Time evolution of entanglement for holographic steady state formation

Erdmenger

Fernández

Flory

et al. 2017

J. High Energ. Phys.

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“…In network coding theory, they give rise to tighter capacity bounds [16]. The quantum case has so far remained elusive, though there has been partial progress [17][18][19][20]. Understanding the phase space of entanglement entropies for specific subclasses of quantum systems is also of general interest for information theorists.…”

Section: Jhep09(2015)130mentioning

confidence: 99%

The holographic entropy cone

Bao

Nezami

Ooguri

et al. 2015

J. High Energ. Phys.

173

510

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We initiate a systematic enumeration and classification of entropy inequalities satisfied by the Ryu-Takayanagi formula for conformal field theory states with smooth holographic dual geometries. For 2, 3, and 4 regions, we prove that the strong subadditivity and the monogamy of mutual information give the complete set of inequalities. This is in contrast to the situation for generic quantum systems, where a complete set of entropy inequalities is not known for 4 or more regions. We also find an infinite new family of inequalities applicable to 5 or more regions. The set of all holographic entropy inequalities bounds the phase space of Ryu-Takayanagi entropies, defining the holographic entropy cone. We characterize this entropy cone by reducing geometries to minimal graph models that encode the possible cutting and gluing relations of minimal surfaces. We find that, for a fixed number of regions, there are only finitely many independent entropy inequalities. To establish new holographic entropy inequalities, we introduce a combinatorial proof technique that may also be of independent interest in Riemannian geometry and graph theory.

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“…Since we know the relative entropy terms in eq. (18) are zero if we use Q as the minimising Markov chain:…”

Section: Classical Casementioning

confidence: 99%

Robustness of Quantum Markov Chains

Ibinson

Linden

Winter

2007

Commun. Math. Phys.

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If the conditional information of a classical probability distribution of three random variables is zero, then it obeys a Markov chain condition. If the conditional information is close to zero, then it is known that the distance (minimum relative entropy) of the distribution to the nearest Markov chain distribution is precisely the conditional information. We prove here that this simple situation does not obtain for quantum conditional information. We show that for tri-partite quantum states the quantum conditional information is always a lower bound for the minimum relative entropy distance to a quantum Markov chain state, but the distance can be much greater; indeed the two quantities can be of different asymptotic order and may even differ by a dimensional factor.

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A New Inequality for the von Neumann Entropy

Cited by 57 publications

References 11 publications

Time evolution of entanglement for holographic steady state formation

Time evolution of entanglement for holographic steady state formation

The holographic entropy cone

Robustness of Quantum Markov Chains

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