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Sequences of resource monotones from modular Hamiltonian polynomials

Raúl Arias, Jan de Boer, Giuseppe Di Giulio, Esko Keski-Vakkuri, Erik Tonni

TL;DR

This work constructs two infinite families of entanglement monotones from the cumulants of the modular Hamiltonian, $M^{(n)}(\rho;b_n)$ and the extremal polynomials $P_E^{(n)}(\rho)$, providing a rich hierarchy of majorization inequalities beyond the von Neumann entropy. By connecting these monotones to Rényi entropies through generating functions, the authors derive finite-time Landauer bounds and finite-size corrections to Clausius inequalities, and extend the framework to relative quantifiers for commuting states. They further explore applications to information erasure and marginal entropy production, and initiate a study of majorization in discretized quantum field theories by analyzing pairs of states in 1+1D CFTs and their periodic chains. In parallel, they develop a relative-quantifier formalism for commuting pairs, improving our understanding of resource theories in quantum thermodynamics. The results provide a systematic route to sharper bounds in single-shot thermodynamics and offer a window into majorization-like structure in field theories, with concrete tests in CFT-like models.

Abstract

We introduce two infinite sequences of entanglement monotones, which are constructed from expectation values of polynomials in the modular Hamiltonian. These monotones yield infinite sequences of inequalities that must be satisfied in majorizing state transitions. We demonstrate this for information erasure, deriving an infinite sequence of "Landauer inequalities" for the work cost, bounded by linear combinations of expectation values of powers of the modular Hamiltonian. These inequalities give improved lower bounds for the work cost in finite dimensional systems, and depend on more details of the erased state than just on its entropy and variance of modular Hamiltonian. Similarly one can derive lower bounds for marginal entropy production for a system coupled to an environment. These infinite sequences of entanglement monotones also give rise to relative quantifiers that are monotonic in more general processes, namely those involving so-called $σ$-majorization with respect to a fixed point full rank state $σ$; such quantifiers are called resource monotones. As an application to thermodynamics, one can use them to derive finite-dimension corrections to the Clausius inequality. Finally, in order to gain some intuition for what (if anything) plays the role of majorization in field theory, we compare pairs of states in discretized theories at criticality and study how majorization depends on the size of the bipartition with respect to the size of the entire chain.

Sequences of resource monotones from modular Hamiltonian polynomials

TL;DR

This work constructs two infinite families of entanglement monotones from the cumulants of the modular Hamiltonian, and the extremal polynomials , providing a rich hierarchy of majorization inequalities beyond the von Neumann entropy. By connecting these monotones to Rényi entropies through generating functions, the authors derive finite-time Landauer bounds and finite-size corrections to Clausius inequalities, and extend the framework to relative quantifiers for commuting states. They further explore applications to information erasure and marginal entropy production, and initiate a study of majorization in discretized quantum field theories by analyzing pairs of states in 1+1D CFTs and their periodic chains. In parallel, they develop a relative-quantifier formalism for commuting pairs, improving our understanding of resource theories in quantum thermodynamics. The results provide a systematic route to sharper bounds in single-shot thermodynamics and offer a window into majorization-like structure in field theories, with concrete tests in CFT-like models.

Abstract

We introduce two infinite sequences of entanglement monotones, which are constructed from expectation values of polynomials in the modular Hamiltonian. These monotones yield infinite sequences of inequalities that must be satisfied in majorizing state transitions. We demonstrate this for information erasure, deriving an infinite sequence of "Landauer inequalities" for the work cost, bounded by linear combinations of expectation values of powers of the modular Hamiltonian. These inequalities give improved lower bounds for the work cost in finite dimensional systems, and depend on more details of the erased state than just on its entropy and variance of modular Hamiltonian. Similarly one can derive lower bounds for marginal entropy production for a system coupled to an environment. These infinite sequences of entanglement monotones also give rise to relative quantifiers that are monotonic in more general processes, namely those involving so-called -majorization with respect to a fixed point full rank state ; such quantifiers are called resource monotones. As an application to thermodynamics, one can use them to derive finite-dimension corrections to the Clausius inequality. Finally, in order to gain some intuition for what (if anything) plays the role of majorization in field theory, we compare pairs of states in discretized theories at criticality and study how majorization depends on the size of the bipartition with respect to the size of the entire chain.
Paper Structure (23 sections, 2 theorems, 180 equations, 2 figures)

This paper contains 23 sections, 2 theorems, 180 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Some concepts of quantum information theory
  3. Resource monotones and majorizing state transitions
  4. An infinite sequence of entanglement monotones
  5. Extremal polynomial monotones and inequalities for state transitions
  6. Other resource monotones from pure state entanglement monotones
  7. Finite-size correction to Clausius inequality
  8. Applications
  9. Information erasure
  10. Lower bound on marginal entropy production
  11. Majorization and states of periodic chains
  12. Pairs of bipartite pure states in 1+1 CFT and periodic chains
  13. Some known CFT results
  14. Excited states in CFT
  15. Massless compact boson
  16. ...and 8 more sections

Key Result

Theorem 1

All positive semidefinite polynomials $G(y)$ on the negative half-line $y\in (-\infty ,0]$ have the following form. For polynomials $G(y)$ of degree $2d$ (with $d\geq 1$), they are linear combinations with non-negative coefficients of polynomials of the form $G_{\vec{a}}(y)= \prod^d_{i=1} (y+a_i)^2$

Figures (2)

  • Figure 1: Difference $S_{\mathcal{O} ,A} - C_{\mathcal{O} ,A}$ as function of $\ell/L$. The data in the top curve have been obtained by considering the excited state of the XX chain corresponding to $\mathcal{O}=\textrm{i} \partial\phi$ in the continuum limit, while the data in the bottom curve have been obtained for the excited state of the Ising chain corresponding to $\mathcal{O}=\psi$ in the continuum limit. The red and the blue curves are obtained from (\ref{['S-minus-C-current']}) with $\gamma=1$ and $\gamma=1/2$ respectively. The additive constants for the two curves are reported in Sec. \ref{['subsec:compactboson']} and in Sec. \ref{['subsec:freefermion']}.
  • Figure 2: Changes in the monotones as function of $\ell/L$ for moving from the ground state to the excited state of the Ising chain corresponding to $\mathcal{O}=\psi$ in the continuum limit. The black, the blue and the red solid curves are obtained from (\ref{['Renyi-exc']}), (\ref{['entropy-cap-exc']}), (\ref{['F_O-gamma']}) and (\ref{['F-function-def']}), while the dashed curves are obtained using (\ref{['capacityequalentropy']}) and (\ref{['entropy-cap-exc']}), (\ref{['F_O-gamma']}) and (\ref{['F-function-def']}) into (\ref{['Mdefinition']}) with $\ell=100$ (black dashed curve) and $\ell=200$ (red dashed curve). The non universal constants have been fixed as specified in in Sec. \ref{['subsec:compactboson']} and in Sec. \ref{['subsec:freefermion']}.

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • proof
  • proof