Exact Mapping from Singular-Value Spectrum of Fractal Images to Entanglement Spectrum of One-Dimensional Quantum Systems

Abstract

We examine the snapshot entropy of general fractal images defined by their singular values. Remarkably, the singular values for a large class of fractals are in exact correspondence with the entanglement spectrum of free fermions in one dimension. These fermions allow for a holographic interpretation of the logarithmic scaling of the snapshot entropy, which is in agreement with the Calabrese-Cardy formula. However, the coarse-grained entropy exhibits a linear scaling due to the degeneracy of the spectrum, in contrast with the logarithmic scaling behavior in one-dimensional quantum near-critical systems.PACS numbers: 05.10.Cc, 07.05. Pj, 11.25.Hf, 11.25.Tq, 89.70.Cf The study of quantum entanglement has continuously attracted enormous attention. The most important aspect concerns the scaling relation of the entanglement entropy in various quantum systems. Well-known scaling relations are the area law, the Calabrese-Cardy formula [1,2], and the finite-entanglement scaling (we call this as finite-χ scaling for simplicity) [3,4]. Since the latter two formulae enable us to estimate the central charge, the entropy is a poweful tool for the study of critical phenomena.On the other hand, quantum entanglement is also deeply intertwined with the theme of holography. Quantum entanglement holds the key to surprising holographic correspondence between completely different physical systems. Two important manifestations are the anti-de Sitter space / conformal field theory correspondence in string theory and the multiscale entanglement renormalization ansatz in statistical physics. In order to compare between different systems, a crucial factor is the amount of information behind these systems, not their detailed physical properties. Thus, the entropy plays a central role in the comparison. Since any classical system does not have entanglement, we must reconsider the meaning of entanglement entropy in the classical side, if there exists possible classical representation of entanglement. In the conformal field theory language, the entanglement entropy is logarithm of two point correlation function of scaling operators, and this indicates the entropy contains the information from the physics at different length scales. Then, it should be possible to encode these degrees of freedom in an emergent space with an additional dimension. One of the solutions for the encoding is the so-called Ryu-Takayanagi formula [5].Furthermore, the Suzuki-Trotter decomposition is also a well-known quantum-classical correspondence. A typical example is transformation of the transverse-field Ising chain into the anisotropic two-dimensional (2D) classical spin model. One of the authors has found that the entropy of the spin snapshot in the classical system corresponds to the holographic entanglemenet entropy of the original quantum 1D system [6]. Remarkably, the snapshot entropy contains an equivalent amount of information as the Calabrese-Cardy and finite-χ scaling formulae combined. There, the singular value decomposition (SVD) of the snapshot dat...

Scale-invariant feature extraction of neural network and renormalization group flow

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“…Information of the weight matrix W can be inferred by using the method of the singular value decomposition (See, e.g., [26,27]). Suppose that the matrix W W T has eigenvalues λ a Figure 13: Averaged values of the off-diagonal components of W W T (normalized by the diagonal components).…”

Exact holographic mapping in free fermion systems

“…First, we recall the following well-known result for the entanglement entropy of free fermions [37][38][39]: (8) where S X = −Tr (ρ X log ρ X ) is the entanglement entropy for the region X, and C X is the projector (correlator) onto region X. With the help of Eq.…”

Unconventional scaling at non-Hermitian critical points

“…Critical phase transitions have mesmerized generations of physicists, with their profound implications in conformal and statistical field theory [1][2][3][4][5][6][7], entanglement entropy [8][9][10][11][12][13][14][15] and thermodynamics [7,[16][17][18][19][20][21]. Placed in a non-Hermitian context, critical transitions become even more intriguing, involving the closure of various line or point gaps and exceptional points [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37].…”