In order to examine the simulation of integrable quantum systems using quantum computers, it is crucial to first classify Yang-Baxter operators. Hietarinta was among the first to classify constant Yang-Baxter solutions for a two-dimensional local Hilbert space (qubit representation). Including the one produced by the permutation operator, he was able to construct eleven families of invertible solutions. These techniques are effective for 4 by 4 solutions, but they become difficult to use for representations with more dimensions. To get over this limitation, we use algebraic ansätze to generate the constant Yang-Baxter solutions in a representation independent way. We employ four distinct algebraic structures that, depending on the qubit representation, replicate 10 of the 11 Hietarinta families. Among the techniques are partition algebras, Clifford algebras, Temperley-Lieb algebras, and a collection of commuting operators. Using these techniques, we do not obtain the (2, 2) Hietarinta class.

India

Somnath Maity

United Arab Emirates

Vivek Singh

Pramod Padmanabhan

United States

Vladimir Korepin

Algebraic classification of Hietarinta’s solutions of Yang-Baxter equations: invertible 4 × 4 operators

The entanglement wedge reconstruction paradigm in AdS/CFT states that for a bulk qudit within the entanglement wedge of a boundary subregion   <math> <mover> <mi>A</mi> <mo>¯</mo> </mover> </math>  $$ \overline{A} $$ , operators acting on the bulk qudit can be reconstructed as CFT operators on   <math> <mover> <mi>A</mi> <mo>¯</mo> </mover> </math>  $$ \overline{A} $$ . This naturally fits within the framework of quantum error correction, with the CFT states containing the bulk qudit forming a code protected against the erasure of the boundary subregion A. In this paper, we set up and study a framework for relational bulk reconstruction in holography: given two code subspaces both protected against erasure of the boundary region A, the goal is to relate the operator reconstructions between the two spaces. To accomplish this, we assume that the two code subspaces are smoothly connected by a one-parameter family of codes all protected against the erasure of A, and that the maximally-entangled states on these codes are all full-rank. We argue that such code subspaces can naturally be constructed in holography in a “measurement-based” setting. In this setting, we derive a flow equation for the operator reconstruction of a fixed code subspace operator using modular theory which can, in principle, be integrated to relate the reconstructed operators all along the flow. We observe a striking resemblance between our formulas for relational bulk reconstruction and the infinite-time limit of Connes cocycle flow, and take some steps towards making this connection more rigorous. We also provide alternative derivations of our reconstruction formulas in terms of a canonical reconstruction map we call the modular reflection operator.

Onkar Parrikar

Harshit Rajgadia

Jonathan Sorce

Relational bulk reconstruction from modular flow

We study the canonical purification of pure, bi-partite states (with respect to one of the parties) obtained by turning on sources in the Euclidean path integral. In holographic conformal field theories, the Lorentzian bulk dual of the canonical purification consists of the corresponding entanglement wedge glued to its CRT image at the quantum extremal surface. However, the mismatch in the classical expansions at the QES due to quantum corrections needs to be supported by a shock in the bulk matter stress tensor in order for the bulk to satisfy Einstein’s equations. Working perturbatively to first order in double-trace sources around the thermofield double state, we demonstrate that the state of the bulk matter in the dual to the canonically purified boundary CFT state precisely has this quantum extremal shock in the bulk stress tensor. We interpret our results as the emergence of gravitational physics from the CFT entanglement structure in a context where bulk quantum corrections are important.

Canonical purification and the quantum extremal shock

The entanglement entropy of many quantum systems is difficult to compute in general. They are obtained as a limiting case of the Rényi entropy of index m, which captures the higher moments of the reduced density matrix. In this work, we study pure bipartite states associated with S 3 complements of a two-component link which is a connected sum of a knot   <math> <mi>K</mi> </math>  $$ \mathcal{K} $$  and the Hopf link. For this class of links, the Chern-Simons theory provides the necessary setting to visualise the m-moment of the reduced density matrix as a three-manifold invariant Z(  <math> <msub> <mi>M</mi> <msub> <mi>K</mi> <mi>m</mi> </msub> </msub> </math>  $$ {M}_{{\mathcal{K}}_m} $$ ), which is the partition function of   <math> <msub> <mi>M</mi> <msub> <mi>K</mi> <mi>m</mi> </msub> </msub> </math>  $$ {M}_{{\mathcal{K}}_m} $$ . Here   <math> <msub> <mi>M</mi> <msub> <mi>K</mi> <mi>m</mi> </msub> </msub> </math>  $$ {M}_{{\mathcal{K}}_m} $$  is a closed 3-manifold associated with the knot   <math> <mi>K</mi> </math>  $$ \mathcal{K} $$   m , where   <math> <mi>K</mi> </math>  $$ \mathcal{K} $$   m  is a connected sum of m-copies of   <math> <mi>K</mi> </math>  $$ \mathcal{K} $$ (i.e.,   <math> <mi>K</mi> </math>  $$ \mathcal{K} $$ #  <math> <mi>K</mi> </math>  $$ \mathcal{K} $$  . . . #  <math> <mi>K</mi> </math>  $$ \mathcal{K} $$ ) which mimics the well-known replica method. We analayse the partition functions Z(  <math> <msub> <mi>M</mi> <msub> <mi>K</mi> <mi>m</mi> </msub> </msub> </math>  $$ {M}_{{\mathcal{K}}_m} $$ ) for SU(2) and SO(3) gauge groups, in the limit of the large Chern-Simons coupling k. For SU(2) group, we show that Z(  <math> <msub> <mi>M</mi> <msub> <mi>K</mi> <mi>m</mi> </msub> </msub> </math>  $$ {M}_{{\mathcal{K}}_m} $$ ) can grow at most polynomially in k. On the contrary, we conjecture that Z(  <math> <msub> <mi>M</mi> <msub> <mi>K</mi> <mi>m</mi> </msub> </msub> </math>  $$ {M}_{{\mathcal{K}}_m} $$ ) for SO(3) group shows an exponential growth in k, where the leading term of ln Z(  <math> <msub> <mi>M</mi> <msub> <mi>K</mi> <mi>m</mi> </msub> </msub> </math>  $$ {M}_{{\mathcal{K}}_m} $$ ) is the hyperbolic volume of the knot complement S 3 \   <math> <mi>K</mi> </math>  $$ \mathcal{K} $$   m . We further propose that the Rényi entropies associated with SO(3) group converge to a finite value in the large k limit. We present some examples to validate our conjecture and proposal.

Aditya Dwivedi

China

Siddharth Dwivedi

Bhabani Mandal

Pichai Ramadevi

Topological entanglement and hyperbolic volume

Weaving knots W(p, n) of type (p, n) denote an infinite family of hyperbolic knots which have not been addressed by the knot theorists as yet. Unlike the well known (p, n) torus knots, we do not have a closed-form expression for HOMFLY-PT and the colored HOMFLY-PT for W(p, n). In this paper, we confine to a hybrid generalization of W(3, n) which we denote as   <math> <msub> <mover> <mi>W</mi> <mo>̂</mo> </mover> <mn>3</mn> </msub> </math>  $$ {\hat{W}}_3 $$ (m, n) and obtain closed form expression for HOMFLY-PT using the Reshitikhin and Turaev method involving   <math> <mi>ℛ</mi> </math>  $$ \mathrm{\mathcal{R}} $$ -matrices. Further, we also compute [r]-colored HOMFLY-PT for W(3, n). Surprisingly, we observe that trace of the product of two dimensional   <math> <mover> <mi>ℛ</mi> <mo>̂</mo> </mover> </math>  $$ \hat{\mathrm{\mathcal{R}}} $$ -matrices can be written in terms of infinite family of Laurent polynomials   <math> <msub> <mi>V</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>t</mi> </mrow> </msub> <mfenced> <mi>q</mi> </mfenced> </math>  $$ {\mathcal{V}}_{n,t}\left[q\right] $$  whose absolute coefficients has interesting relation to the Fibonacci numbers   <math> <msub> <mi>ℱ</mi> <mi>n</mi> </msub> </math>  $$ {\mathrm{\mathcal{F}}}_n $$ . We also computed reformulated invariants and the BPS integers in the context of topological strings. From our analysis, we propose that certain refined BPS integers for weaving knot W(3, n) can be explicitly derived from the coefficients of Chebyshev polynomials of first kind.

Rama Mishra

P. Ramadevi

Colored HOMFLY-PT for hybrid weaving knot   <math> <msub> <mover> <mi>W</mi> <mo>̂</mo> </mover> <mn>3</mn> </msub> </math>  $$ {\hat{\mathrm{W}}}_3 $$ (m, n)

We study the multi-boundary entanglement structure of the state associated with the torus link complement S 3 \T  p,q  in the set-up of three-dimensional SU(2) k  Chern-Simons theory. The focal point of this work is the asymptotic behavior of the Rényi entropies, including the entanglement entropy, in the semiclassical limit of k → ∞. We present a detailed analysis of several torus links and observe that the entropies converge to a finite value in the semiclassical limit. We further propose that the large k limiting value of the Rényi entropy of torus links of type T  p,pn  is the sum of two parts: (i) the universal part which is independent of n, and (ii) the non-universal or the linking part which explicitly depends on the linking number n. Using the analytic techniques, we show that the universal part comprises of Riemann zeta functions and can be written in terms of the partition functions of two-dimensional topological Yang-Mills theory. More precisely, it is equal to the Rényi entropy of certain states prepared in topological 2d Yang-Mills theory with SU(2) gauge group. Further, the universal parts appearing in the large k limits of the entanglement entropy and the minimum Rényi entropy for torus links T  p,pn  can be interpreted in terms of the volume of the moduli space of flat connections on certain Riemann surfaces. We also analyze the Rényi entropies of T  p,pn  link in the double scaling limit of k → ∞ and n → ∞ and propose that the entropies converge in the double limit as well.

Abhishek Roy

Semiclassical limit of topological Rényi entropy in 3d Chern-Simons theory

The particle content of the Singh-Hagen model (SH) in <math id="M2"><mi>D</mi></math> dimensions is revisited. We suggest a complete set of spin-projection operators acting on totally symmetric rank-3 fields. We give a general expression for the propagator and determine the coefficients of the SH model confirming previous results of the literature. Adding source terms, we provide a unitarity analysis in <math id="M3"><mi>D</mi></math> dimensions. In addition, we have also analyzed the positivity of the massless Hamiltonian.

Brazil

Elias L. Mendonça

R. Schimidt Bittencourt

Unitarity of Singh-Hagen Model in Dimensions

Topological entanglement structure amongst disjoint torus boundaries of three manifolds have already been studied within the context of Chern-Simons theory. In this work, we study the topological entanglement due to interaction between the quasiparticles inside three-manifolds with one or more disjoint S 2 boundaries in SU(N) Chern-Simons theory. We focus on the world-lines of quasiparticles (Wilson lines), carrying SU(N) representations, creating four punctures on every S 2. We compute the entanglement entropy by partial tracing some of the boundaries. In fact, the entanglement entropy depends on the SU(N) representations on these four-punctured S 2 boundaries. Further, we observe interesting features on the GHZ-like and W-like entanglement structures. Such a distinction crucially depends on the multiplicity of the irreducible representations in the tensor product of SU(N) representations.

Poland

Yang Zhou

Saswati Dhara

Entanglement on multiple S 2 boundaries in Chern-Simons theory

We study the entanglement for a state on linked torus boundaries in 3 d Chern-Simons theory with a generic gauge group and present the asymptotic bounds of Rényi entropy at two different limits: (i) large Chern-Simons coupling k , and (ii) large rank r of the gauge group. These results show that the Rényi entropies cannot diverge faster than ln k and ln r , respectively. We focus on torus links T (2 , 2 n ) with topological linking number n . The Rényi entropy for these links shows a periodic structure in n and vanishes whenever n = 0 (mod p), where the integer p is a function of coupling k and rank r . We highlight that the refined Chern-Simons link invariants can remove such a periodic structure in n .