Entanglement versus mutual information in quantum spin chains

Um, Jaegon; Park, Hyunggyu; Hinrichsen, Haye

doi:10.1088/1742-5468/2012/10/p10026

Cited by 23 publications

(27 citation statements)

References 13 publications

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“…Note that the mutual information is symmetric with respect to A and B, and that the leading contributions proportional to L cancel in the definition (3). It has been first studied for the Ising universality class 29,30 , and the scaling argued to be universal. More precisely, it was found that the Shannon mutual information (SMI) is well described by the following formula:…”

Section: I(a B) = S(a) + S(b) − S(a ∪ B) (3)mentioning

confidence: 99%

“…[1] compared to Refs. [29][30][31], for different models and limits belonging to the Ising universality class. On the conceptual level also, the result appears much simpler than previous findings in the Shannon entropy of the full chain 15,18,20 .…”

Section: I(a B) = S(a) + S(b) − S(a ∪ B) (3)mentioning

confidence: 99%

“…Universal terms can also be accessed in higher dimensions [26][27][28] . Inspired by all the results obtained for the onedimensional entanglement entropy, it is natural to study the Shannon entropy of a subsystem 29 . Let us cut our chain in two parts A and B. Subsystem A is described by the reduced density matrix ρ A = Tr B |ψ ψ|, and the subsystem Shannon entropy is…”

Section: Introductionmentioning

confidence: 99%

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Shannon and Rényi mutual information in quantum critical spin chains

Stéphan

2014

Phys. Rev. B

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We study the Shannon mutual information in one-dimensional critical spin chains, following a recent conjecture (Phys. Rev. Lett. 111, 017201 (2013)[1]), as well as Rényi generalizations of it. We combine conformal field theory arguments with numerical computations in lattice discretizations with central charge c = 1 and c = 1/2. For a periodic system of length L cut into two parts of length ℓ and L − ℓ, all our results agree with the general shape-dependence In(ℓ,, where bn is a universal coefficient. For the free boson CFT we show from general arguments that bn = c = 1. At c = 1/2 we conjecture a result for n > 1. We perform extensive numerical computations in Ising chains to confirm this, and also find b1 ≃ 0.4801629(2), a nontrivial number which we do not understand analytically. Open chains at c = 1/2 and n = 1 are even more intriguing, with a shape-dependent logarithmic divergence of the Shannon mutual information.

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Section: I(a B) = S(a) + S(b) − S(a ∪ B) (3)mentioning

confidence: 99%

Section: I(a B) = S(a) + S(b) − S(a ∪ B) (3)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Shannon and Rényi mutual information in quantum critical spin chains

Stéphan

2014

Phys. Rev. B

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“…In the first case we use the ground state of the quantum system as the source of the probabilities for appearing different configurations and in the second case it will be just the Gibbs distribution. The first definition which has recently found many interesting applications in the study of spin chains 38,[40][41][42] can be defined in the context of harmonic oscillators as follows: The Shannon's classical mutual information I(A : B) between two regions A and B is…”

Section: Mutual Informationmentioning

confidence: 99%

Quantum entanglement entropy and classical mutual information in long-range harmonic oscillators

Nezhadhaghighi

Rajabpour

2013

Phys. Rev. B

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We study different aspects of quantum von Neumann and Rényi entanglement entropy of one dimensional long-range harmonic oscillators that can be described by well-defined non-local field theories. We show that the entanglement entropy of one interval with respect to the rest changes logarithmically with the number of oscillators inside the subsystem. This is true also in the presence of different boundary conditions. We show that the coefficients of the logarithms coming from different boundary conditions can be reduced to just two different universal coefficients. We also study the effect of the mass and temperature on the entanglement entropy of the system in different situations. The universality of our results is also confirmed by changing different parameters in the coupled harmonic oscillators. We also show that more general interactions coming from general singular Toeplitz matrices can be decomposed to our long-range harmonic oscillators. Despite the long-range nature of the couplings we show that the area law is valid in two dimensions and the universal logarithmic terms appear if we consider subregions with sharp corners. Finally we study analytically different aspects of the mutual information such as its logarithmic dependence to the subsystem, effect of mass and influence of the boundary. We also generalize our results in this case to general singular Toeplitz matrices and higher dimensions. Contents

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“…They found that although the mutual information of two halves of a cylinder has its maximum value at a temperature higher than the critical temperature, its derivative diverges at the critical temperature. Most recently Um et al [16] studied the mutual information of a subregion with respect to the rest in the periodic transverse Ising model and surprisingly found that it has the same dependence ln(sin(πℓ/L)) as (1) but with a distinct multiplicative constant. In this Letter we study the Shannon mutual information of local observables in different critical spin chains and argue about their possible connections to the central charge of the underlying CFT.…”

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confidence: 99%

Universal Behavior of the Shannon Mutual Information of Critical Quantum Chains

Alcaraz

Rajabpour

2013

Phys. Rev. Lett.

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We consider the Shannon mutual information of subsystems of critical quantum chains in their ground states. Our results indicate a universal leading behavior for large subsystem sizes. Moreover, as happens with the entanglement entropy, its finite-size behavior yields the conformal anomaly c of the underlying conformal field theory governing the long distance physics of the quantum chain. We studied analytically a chain of coupled harmonic oscillators and numerically the Q-state Potts models (Q = 2, 3 and 4), the XXZ quantum chain and the spin-1 Fateev-Zamolodchikov model. The Shannon mutual information is a quantity easily computed, and our results indicate that for relatively small lattice sizes its finite-size behavior already detects the universality class of quantum critical behavior.PACS numbers: 11.25. Hf, 03.67.Bg, 89.70.Cf, 75.10.Pq Entanglement measures have emerged nowadays as powerful tools for the study of quantum many body systems [1,2]. In one dimension, where most quantum critical systems have their long-distance physics ruled by a conformal field theory (CFT), the entanglement entropy has been proved the most important measure of entanglement. It allows one to identify the distinct universality classes of critical behaviors. Let us consider a periodic quantum chain with L sites, and partition the system into subsystems A and B of length ℓ and L − ℓ, respectively. The entanglement entropy is defined as the von Neumann entropy of the reduced density matrix ρ A of the partition A: S ℓ = −T r A ρ A ln ρ A . If the system is critical and in the ground state, in the regime where the subsystems are large compared with the lattice spacing, S ℓ is given by [3,4] where c is the central charge of the underlying CFT and γ s is a non-universal constant. A remarkable fact is that even in the case where the system is in a pure state formed by an excited state, the conformal anomaly dictates the overall behavior of the entanglement, similarly as in (1) [5]. It is worth mentioning that recently many interesting methods were proposed [6-8] to calculate the entanglement entropy and ultimately central charge, however, up to know they have not been implemented experimentally. A natural question concerns the possible existence of other measures of shared information that, similarly as the entanglement entropy, are also able to detect the several universality classes of critical behavior of quantum critical chains. In this Letter we present results that indicate that the Shannon mutual information of local observables is such a measure. The Shannon mutual information of the subsystems A and B, of sizes ℓ and L − ℓ is defined aswhere Sh(X ) = − x p x ln p x is the Shannon entropy of the subsystem X with probabilities p x of being in a configuration x. These probabilities, in the case where A is a subsystem of a quantum chain with wavefunction |Ψ A∪B = n,m c n,m |φ It is important to notice that the Shannon entropy and the Shannon mutual information are basis dependent quantities, reflecting the several kinds of obse...

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Entanglement versus mutual information in quantum spin chains

Cited by 23 publications

References 13 publications

Shannon and Rényi mutual information in quantum critical spin chains

Shannon and Rényi mutual information in quantum critical spin chains

Quantum entanglement entropy and classical mutual information in long-range harmonic oscillators

Universal Behavior of the Shannon Mutual Information of Critical Quantum Chains

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