Quantum entropies

Chehade, Sarah; Vershynina, Anna

doi:10.4249/scholarpedia.53131

Cited by 13 publications

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“…The trace distance also bounds other interesting quantities such as the entanglement entropy, Rényi entropies, RDM moments, and the correlation functions. The interested reader can find all necessary details in the review [52]. The Fannes-Audenaert inequality provides an upper bound for the entanglement entropy difference [53,54]…”

Section: Trace Distance and Its Propertiesmentioning

confidence: 99%

Lattice Bisognano-Wichmann modular Hamiltonian in critical quantum spin chains

et al. 2020

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We carry out a comprehensive comparison between the exact modular Hamiltonian and the lattice version of the Bisognano-Wichmann (BW) one in one-dimensional critical quantum spin chains. As a warm-up, we first illustrate how the trace distance provides a more informative mean of comparison between reduced density matrices when compared to any other Schatten nn-distance, normalized or not. In particular, as noticed in earlier works, it provides a way to bound other correlation functions in a precise manner, i.e., providing both lower and upper bounds. Additionally, we show that two close reduced density matrices, i.e. with zero trace distance for large sizes, can have very different modular Hamiltonians. This means that, in terms of describing how two states are close to each other, it is more informative to compare their reduced density matrices rather than the corresponding modular Hamiltonians. After setting this framework, we consider the ground states for infinite and periodic XX spin chain and critical Ising chain. We provide robust numerical evidence that the trace distance between the lattice BW reduced density matrix and the exact one goes to zero as \ell^{-2}ℓ−2 for large length of the interval \ellℓ. This provides strong constraints on the difference between the corresponding entanglement entropies and correlation functions. Our results indicate that discretized BW reduced density matrices reproduce exact entanglement entropies and correlation functions of local operators in the limit of large subsystem sizes. Finally, we show that the BW reduced density matrices fall short of reproducing the exact behavior of the logarithmic emptiness formation probability in the ground state of the XX spin chain.

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Section: Trace Distance and Its Propertiesmentioning

confidence: 99%

Lattice Bisognano-Wichmann modular Hamiltonian in critical quantum spin chains

et al. 2020

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“…It is important to note that this entropy measure is relative to a pure state, and is, therefore, essentially different from the much better known quantum entropy measure introduced by Von Neumann [23] and the subsequent measures inspired by it (Rényi, Tsallis, etc. ), which are instead null for a pure state [24,25].…”

Section: Schrödinger Equationmentioning

confidence: 99%

Wave Function and Information

Chiatti

2024

Quantum Reports

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Two distinct measures of information, connected respectively to the amplitude and phase of the wave function of a particle, are proposed. There are relations between the time derivatives of these two measures and their gradients on the configuration space, which are equivalent to the wave equation. The information related to the amplitude measures the strength of the potential coupling of the particle (which is itself aspatial) with each volume of its configuration space, i.e., its tendency to participate in an interaction localized in a region of ordinary physical space corresponding to that volume. The information connected to the phase is that required to obtain the time evolution of the particle as a persistent entity starting from a random succession of bits. It can be considered as the information provided by conservation principles. The meaning of the so-called “quantum potential” in this context is briefly discussed.

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“…which, for n i=1 p i = 1, Arimoto [15] discussed as a member of this entropy in (7). From Theorem 13, the entropy in (36) is not concave since the condition (α − 1)δ ≥ 1 is not met, but it is Schur-concave. However, when n i=1 p i = 1, this entropy is concave from Theorem 7.…”

mentioning

confidence: 99%

“…This entropy is typically identified as the limit of Rényi's entropy H Rα (P n ) in (4a) as α → ∞ (see, e.g., [36], [37], [38]). Since, for β = 1 (or for any 0 ≤ β ≤ 1), lim α→∞ G α1 (P n ) = p max , it follows from ( 14) that lim α→∞…”

mentioning

confidence: 99%

“…The equivalent quantum entropies could also be formulated by substituting traces of density matrices for the probability summations in the entropies discussed above as done, for example, by Hu and Ye [6] when expressing the quantum equivalent of the Rathie-Taneja entropy in (12). See also [36] for the quantum equivalents of some other entropies. Thus, if ρ is a density matrix of a system of interest involving a finite dimensional Hilbert space, one could define the quantum equivalent to the most general entropy in (14) as follows:…”

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confidence: 99%

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Entropies and Their Concavity and Schur-Concavity Conditions

Kvålseth

2022

IEEE Access

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Concavity and Schur-concavity are two of the important properties of any entropy. Since Shannon's classical entropy formulation, a number of generalized entropies have been proposed as parameterized generalizations of Shannon's entropy. For such generalized entropies, the conditions under which they are concave and/or Schur-concave have not always been determined or have been incompletely and incorrectly reported in a variety of publications. This paper provides proofs of those two properties for the various proposed generalized entropies using a unifying approach. First, a new three-parameter entropy is introduced of which other proposed generalized entropies are particular members. Second, a proof is derived for the concavity and Schur-concavity of the new entropy and the underlying conditions. Those results are then applied to the particular one-parameter and two-parameter members. Some new such members are also discussed as are some related inequalities. The various derivations are based on so-called generalized probability distributions when the sum of component probabilities may be less than 1.

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Quantum entropies

Cited by 13 publications

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Lattice Bisognano-Wichmann modular Hamiltonian in critical quantum spin chains

Lattice Bisognano-Wichmann modular Hamiltonian in critical quantum spin chains

Wave Function and Information

Entropies and Their Concavity and Schur-Concavity Conditions

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