Holographic entropy inequalities pass the majorization test

Quantities computed by minimal cuts, such as entanglement entropies achievable by the Ryu-Takayanagi proposal in the AdS/CFT correspondence, are constrained by linear inequalities. We prove a previously conjectured property of all such constraints: Any $k$ systems on the “greater-than” side of the inequality whose overlap is nonempty are subsumed in some $k$ systems on its “less-than” side (accounting for multiplicity). This finding adds evidence that the same inequalities also constrain the entropies under time-dependent conditions because it preempts a large class of potential counterexamples. We prove several other properties of holographic entropy inequalities and comment on their relation to quantum erasure correction and the Renormalization Group.

💡 Research Summary

The paper addresses a fundamental question in holographic quantum information theory: whether the linear entropy inequalities that follow from the Ryu‑Takayanagi (RT) prescription (and its covariant generalization, the Hubeny‑Ryu‑Takayanagi (HRT) proposal) satisfy a purely combinatorial “majorization test”. The authors prove that every holographic entropy inequality—whether balanced or almost‑balanced—passes this test, thereby establishing a new structural property of these inequalities and providing strong evidence that they remain valid in time‑dependent (non‑static) settings.

Background and Motivation
Entanglement entropy in quantum field theory obeys universal constraints such as strong subadditivity. In holographic CFTs, the RT formula computes entropies as minimal-area surfaces in the bulk, leading to an enlarged set of linear constraints beyond the standard quantum information inequalities. To date, there are infinitely many families of holographic inequalities together with roughly two thousand sporadic examples. All known holographic inequalities are “balanced” (the total number of appearances of each elementary region cancels between the two sides) and, except for a few trivial cases, are also “superbalanced” (pairwise appearances cancel). These properties guarantee UV finiteness of the inequalities.

Contraction Proof Framework
Any holographic inequality can be encoded by two binary indicator matrices, X for the left‑hand side (LHS) and Y for the right‑hand side (RHS). Rows correspond to elementary subsystems (A_p) (p = 1,…,N) and columns to the individual entropy terms. The inequality reads (\sum_{i} S(\cup_{p:x_{pi}=1} A_p) \ge \sum_{j} S(\cup_{p:y_{pj}=1} A_p)). The contraction method, introduced in earlier works, states that the inequality holds if there exists a map (f:{0,1}^L \to {0,1}^R) satisfying:

1. Contraction condition: the Hamming distance never increases, (|x-x’| \ge |f(x)-f(x’)|) for all binary vectors.
2. Boundary conditions: (f(x_p)=y_p) for each row vector (x_p) (the indicator of subsystem (A_p)) and also for the empty set.

These conditions are equivalent to the balance and superbalance relations: the 1‑norms of corresponding rows are equal, and the inner products (\langle x_p,x_q\rangle = \langle y_p,y_q\rangle) for all p,q.

Null Reduction and the Majorization Test
Given a balanced inequality, one can perform a “null reduction” with respect to a particular subsystem (A_p) by discarding all terms that do not contain (A_p). Because of balance, the numbers of remaining LHS and RHS terms remain equal (denoted (L_p = R_p)). The majorization test is then defined on the reduced inequality as follows: assign positive variables (a_q) to each elementary region, form vectors \