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arXiv:2205.08915 · Quantum

Correlations in typicality and an affirmative solution to the exact catalytic entropy conjecture

Henrik Wilming

Published Nov 10, 2022• 16 citations
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Abstract

I show that if a finite-dimensional density matrix has strictly smaller von Neumann entropy than a second one of the same dimension (and the rank is not bigger), then sufficiently (but finitely) many tensor-copies of the first density matrix majorize a density matrix whose single-body marginals are all exactly equal to the second density matrix. This implies an affirmative solution of the exact catalytic entropy conjecture (CEC) introduced by Boes et al. [PRL 122, 210402 (2019)]. Both the Lemma and the solution to the CEC transfer to the classical setting of finite-dimensional probability vectors (with permutations of entries instead of unitary transformations for the CEC).

DOI
10.22331/q-2022-11-10-858

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I show that if a finite-dimensional density matrix has strictly smaller von Neumann entropy than a second one of the same dimension (and the rank is not bigger), then sufficiently (but finitely) many tensor-copies of the first density matrix majorize a density matrix whose single-body marginals are all exactly equal to the second density matrix. This implies an affirmative solution of the exact catalytic entropy conjecture (CEC) introduced by Boes et al. [PRL 122, 210402 (2019)]. Both the Lemma and the solution to the CEC transfer to the classical setting of finite-dimensional probability vectors (with permutations of entries instead of unitary transformations for the CEC).
One tuple of probability vectors is more informative than another tuple when there exists a single stochastic matrix transforming the probability vectors of the first tuple into the probability vectors of the other. This is called matrix majorization. Solving an open problem raised by Muet al, we show that if certain monotones—namely multivariate extensions of Rényi divergences—are strictly ordered between the two tuples, then for sufficiently largen, there exists a stochastic matrix taking then-fold Kronecker power of each input distribution to then-fold Kronecker power of the corresponding output distribution. The same conditions, with non-strict ordering for the monotones, are also necessary for such matrix majorization in large samples. Our result also gives conditions for the existence of a sequence of statistical maps that asymptotically (with vanishing error) convert a single copy of each input distribution to the corresponding output distribution with the help of a catalyst that is returned unchanged. Allowing for transformation with arbitrarily small error, we find conditions that are both necessary and sufficient for such catalytic matrix majorization. We derive our results by building on a general algebraic theory of preordered semirings recently developed by one of the authors. This also allows us to recover various existing results on majorization in large samples and in the catalytic regime as well as relative majorization in a unified manner.
Density matrices for systems with a finite number of states are considered as elements in a vector space. A density matrix that is a convex combination of density matrices that are unitary transformations of some initial density matrix will be closer to, or possibly at the same distance from, the density matrix for the ensemble in which all states are equally occupied, compared with the initial density matrix. This distance is correlated, for few-state systems, with a function simply related to the entropy. This function increases (entropy decreases) with distance from the equal-occupancy density matrix. Other properties of the function are also established.
Quantum many-body states that frequently appear in physics often obey an entropy scaling law, meaning that an entanglement entropy of a subsystem can be expressed as a sum of terms that scale linearly with its volume and area, plus a correction term that is independent of its size.We conjecture that these states have an efficient dual description in terms of a set of marginal density matrices on bounded regions, obeying the same entropy scaling law locally.We prove a restricted version of this conjecture for translationally invariant systems in two spatial dimensions.Specifically, we prove that a translationally invariant marginal obeying three nonlinear constraints-all of which follow from the entropy scaling law straightforwardlymust be consistent with some global state on an infinite lattice.Moreover, we derive a closed-form expression for the maximum-entropy density compatible with those marginals, deriving a variational upper bound on the thermodynamic free energy.The main assumptions of our construction are satisfied exactly by solvable models of topological order and approximately by finite-temperature Gibbs states of certain quantum spin Hamiltonians.
Given probability vectors x and y, necessary and sufficient conditions are given for the existence of a probability vector z such that xominusz is majorized by yominusz. This is of interest because the "catalytic majorization" relation is known to be useful in classifying which transformations of jointly held pure quantum states are possible using local operations and classical communication when an additional jointly held state may be specified to facilitate the transformation without being consumed
Entanglement of purification is a fundamental measure of total correlations whose additivity remains unresolved. We study its additivity for classical states on two qubits at different Rényi orders. For every α∈[0,1)α\in[0,1)α∈[0,1), we prove nonadditivity within this family, witnessed by two copies of a single state. We first solve the one-copy optimization exactly for the entire family at every Rényi order. We then restrict the two-copy optimization to a natural finite set of purifications and exhibit one whose entropy is strictly below the product value. In contrast, for α∈[2,∞]α\in[2,\infty]α∈[2,∞] we prove additivity under tensor products within this family. The interval α∈[1,2)α\in[1,2)α∈[1,2), including the von Neumann case α=1α=1α=1, remains open, and we conjecture additivity there throughout the same family.
Spectral properties of the reduced density matrix (RDM) of permutational invariant quantum many-body systems are investigated. The RDM block diagonalization which accounts for all symmetries of the Hamiltonian is achieved. The analytical expression of the RDM spectrum is provided for arbitrary parameters and rigorously proved in the thermodynamical limit. The existence of several sum rules and recurrence relations among RDM eigenvalues is also demonstrated and the distribution function of RDM eigenvalues (including degeneracies) characterized. In particular, we prove that the distribution function approaches a two-dimensional Gaussian in the limit of large subsystem sizes n>>1. As a physical application we discuss the von Neumann entropy (VNE) of a block of size n for a system of hard-core bosons on a complete graph, as a function of n and of the temperature T. The occurrence of a crossover of VNE from purely logarithmic behavior at T=0 to a purely linear behavior in n for T≥Tc, is demonstrated.