[Paper Review] Holographic Entropy Cone Measures

Ning Bao, Samuel Blitz+1 more|arXiv (Cornell University)|Jan 12, 2017

Black Holes and Theoretical Physics8 references3 citations

TL;DR

This paper investigates the fraction of quantum states that satisfy holographic entanglement entropy inequalities using multiple measures. It finds a tension: while state-based measures suggest most random quantum states lie within the holographic entropy cone, volume-based cone measures show the holographic cone is vastly smaller than the quantum cone, highlighting a fundamental discrepancy between measurement approaches.

ABSTRACT

We investigate numerically several proxy measures for the number of states contained within the holographic entropy cone, compared to the number contained within the quantum entropy cone, for states with $3$ and $4$ parties. We find an interesting tension: while measures focused on calculating the volume ratios between the two cones indicate that the quantum cone is much larger than the holographic one, measures based on the generation of random states and then calculating the entropies thereof imply that almost all such randomly generated states are also contained within the holographic entropy cone. Also interestingly, the volume measures strongly indicate a difference in the number of states within the quantum or stabiliser cones versus the number in the holographic cone, which is not reproduced by the other classes of measures. We comment on the difference between the two classes of measures, and why each may be preferable.

Motivation & Objective

• To assess the fraction of quantum states that can have holographic duals using multiple entropy cone measures.
• To resolve the discrepancy between state-based and volume-based measures in estimating the size of the holographic entropy cone relative to the quantum entropy cone.
• To investigate whether holographic entanglement inequalities are typical for generic quantum states or rare.
• To compare the relative sizes of the quantum, stabiliser, and holographic entropy cones using both state-generation and entropy-vector-generation methods.
• To evaluate the strengths and limitations of different measures for assessing the typicality of holographic states.

Proposed method

• The authors compute entropy vectors for 3- and 4-party quantum states using random pure states with uniformly distributed coefficients (uniform state measure).
• They generate random entropy vectors constrained to lie within the quantum entropy cone (cone measure), independent of Hilbert space dimension.
• They compare the volume ratios of the holographic entropy cone (HEC), quantum entropy cone (QEC), and stabiliser entropy cone (SEC) using Monte Carlo integration.
• They use the 4-party stabiliser cone as a lower bound for the 4-party quantum cone, given its known containment in the QEC.
• They analyze the distribution of entropy vectors from small-qubit systems and assess their proximity to the HEC, particularly focusing on star-configuration states.
• They evaluate the consistency of results across multiple measures: uniform state, finite family, exhaustive n-qubit, and cone measures.

Experimental results

Research questions

• RQ1 What fraction of randomly generated quantum states satisfy all holographic entanglement entropy inequalities?
• RQ2 How do different measures—state-based versus volume-based—compare in estimating the relative size of the holographic entropy cone?
• RQ3 To what extent do typical quantum states lie within the holographic entropy cone, and is this fraction sensitive to the choice of measure?
• RQ4 Why do state-based measures suggest most quantum states are holographic, while volume-based measures indicate the holographic cone is exponentially smaller?
• RQ5 How do the quantum, stabiliser, and holographic entropy cones compare in size, and what does this imply about the typicality of holographic states?

Key findings

• The cone measure shows that the holographic entropy cone (HEC) is significantly smaller than the quantum entropy cone (QEC), with a volume ratio of v(HEC)/v(QEC) ≈ 0.078 for 4-party systems.
• For 4-party systems, the stabiliser entropy cone (SEC) is very close to the quantum cone, with v(SEC)/v(QEC) = 0.98 ± 0.03, indicating that most quantum states are stabiliser-like.
• In contrast, the uniform state measure and related state-based methods suggest that almost all randomly generated quantum states satisfy holographic inequalities, implying a high fraction of holographic states.
• The cone measure strongly differentiates between the HEC and SEC, while state-based measures do not, indicating a fundamental inconsistency between the two classes of measures.
• The authors find that state-based measures are biased toward star-configuration entropy vectors, which are more likely to lie in the HEC, explaining the discrepancy with volume-based results.
• The study concludes that no single measure is conclusive, and the choice of measure significantly affects the inferred typicality of holographic states, highlighting a critical open question in quantum gravity and entanglement structure.