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Alternatives of entanglement depth and metrological entanglement criteria

Szilárd Szalay, Géza Tóth

TL;DR

The article develops a unified framework for one-parameter families of multipartite entanglement, introducing f-entanglement properties and f-entanglement depth to generalize concepts like partitionability, producibility, and stretchability. It shows how quantum Fisher information bounds can be expressed through these depths, with squareability emerging as the natural metrological resource and the formation-based depths providing stronger, convex bounds. By incorporating entropy-based generator functions and dominance-monotone properties, the authors derive metrological criteria that bound not only the maximal entangled subsystem size but also the average size of entangled subsystems in a mixture. The work connects metrology with a rich lattice of entanglement classifications, offering tools for tighter experimental bounds and a deeper understanding of how entanglement structure influences precision in quantum sensing.

Abstract

We work out the general theory of one-parameter families of partial entanglement properties and the resulting entanglement depth-like quantities. Special cases of these are the depth of partitionability, the depth of producibility (or simply entanglement depth) and the depth of stretchability, which are based on one-parameter families of partial entanglement properties known earlier. We also construct some further physically meaningful properties, for instance the squareability, the toughness, the degree of freedom, and also several ones of entropic motivation. Metrological multipartite entanglement criteria with the quantum Fisher information fit naturally into this framework. Here we formulate these for the depth of squareability, which therefore turns out to be the natural choice, leading to stronger bounds than the usual entanglement depth. Namely, the quantum Fisher information turns out to provide a lower bound not only on the maximal size of entangled subsystems, but also on the average size of entangled subsystems for a random choice of elementary subsystems. We also formulate criteria with convex quantities for both cases, which are much stronger than the original ones. In particular, the quantum Fisher information puts a lower bound on the average size of entangled subsystems. We also argue that one-parameter partial entanglement properties, which carry entropic meaning, are more suitable for the purpose of defining metrological bounds.

Alternatives of entanglement depth and metrological entanglement criteria

TL;DR

The article develops a unified framework for one-parameter families of multipartite entanglement, introducing f-entanglement properties and f-entanglement depth to generalize concepts like partitionability, producibility, and stretchability. It shows how quantum Fisher information bounds can be expressed through these depths, with squareability emerging as the natural metrological resource and the formation-based depths providing stronger, convex bounds. By incorporating entropy-based generator functions and dominance-monotone properties, the authors derive metrological criteria that bound not only the maximal entangled subsystem size but also the average size of entangled subsystems in a mixture. The work connects metrology with a rich lattice of entanglement classifications, offering tools for tighter experimental bounds and a deeper understanding of how entanglement structure influences precision in quantum sensing.

Abstract

We work out the general theory of one-parameter families of partial entanglement properties and the resulting entanglement depth-like quantities. Special cases of these are the depth of partitionability, the depth of producibility (or simply entanglement depth) and the depth of stretchability, which are based on one-parameter families of partial entanglement properties known earlier. We also construct some further physically meaningful properties, for instance the squareability, the toughness, the degree of freedom, and also several ones of entropic motivation. Metrological multipartite entanglement criteria with the quantum Fisher information fit naturally into this framework. Here we formulate these for the depth of squareability, which therefore turns out to be the natural choice, leading to stronger bounds than the usual entanglement depth. Namely, the quantum Fisher information turns out to provide a lower bound not only on the maximal size of entangled subsystems, but also on the average size of entangled subsystems for a random choice of elementary subsystems. We also formulate criteria with convex quantities for both cases, which are much stronger than the original ones. In particular, the quantum Fisher information puts a lower bound on the average size of entangled subsystems. We also argue that one-parameter partial entanglement properties, which carry entropic meaning, are more suitable for the purpose of defining metrological bounds.
Paper Structure (61 sections, 228 equations, 10 figures)

This paper contains 61 sections, 228 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Permutation invariant partial separability and one-parameter entanglement properties
  3. State spaces and classes
  4. Examples of f-entanglement properties
  5. f-entanglement properties in general
  6. Remarks
  7. f-entanglement depth
  8. Examples of f-entanglement depth
  9. f-entanglement depth in general
  10. Examples of f-entanglement depth of formation
  11. f-entanglement depth of formation in general
  12. Remarks
  13. Some notable cases of f-en-tan-gle-ment
  14. Max and min generator functions, toughness
  15. Power sums and power means
  16. ...and 46 more sections

Figures (10)

  • Figure 1: The down-sets \ref{['eq:vxik']} corresponding to partitionability, producibility and stretchability, illustrated on the posets $\hat{P}_\text{I}$ for $n=2,3,4,5,6$. (The integer partitions $\hat{\xi}\in\hat{P}_\text{I}$ are represented by their Young diagrams, and the refinement order $\preceq$ is denoted by consecutive arrows, which are pointing from the finer partition to the coarser one. The refinement is a partial order, that is, not every pair of partitions are connected by arrows. The integer partitions describing trivial separability \ref{['eq:topbot.t']} and full separability \ref{['eq:topbot.b']} are the $\top$ and $\bot$ elements of the poset $\hat{P}_\text{I}$, drawn in the upper right and lower left corners of these plots, respectively. The three kinds of down-sets \ref{['eq:vxik']} contain the Young diagrams below the green, to the left from the yellow, and below the gray dashed lines. These are down-sets, that is, the borders of any of them are crossed by arrows in only one direction. These three kinds of down-sets \ref{['eq:vxik']} form chains \ref{['eq:vximonk']} for the indexing $k$, that is, the dashed lines (borders) of the same color do not cross one another.)
  • Figure 2: The nested state spaces $\mathcal{D}_{k,f}$ given in \ref{['eq:DsepIIpf']} and disjoint classes $\mathcal{C}_{k,f}$ given in \ref{['eq:CsepIIIpf']} as differences of the state spaces. The numbering $k\in f(\hat{P}_\text{I})$ increases/decreases outwards in the case of increasing/decreasing generator function \ref{['eq:genf']}. LOCC maps cannot bring outwards \ref{['eq:CsepIIIpfmonk']}. Different generator functions lead to different such onion like structures, intersecting nontrivially in general.
  • Figure 3: The height \ref{['eq:hwr.h']} vs. width \ref{['eq:hwr.w']}, rank \ref{['eq:hwr.r']} and toughness \ref{['eq:t']} plots of $\hat{P}_\text{I}$ for $n=8$. See also figure \ref{['fig:PpI23456PPS']} for the width for $n\leq6$.
  • Figure 4: The height \ref{['eq:hwr.h']} vs. power-sum \ref{['eq:sq']}, $q$-sum \ref{['eq:Nq']} and $q$-mean \ref{['eq:Mq']} plots of $\hat{P}_\text{I}$ for $q=2$, $n=8$.
  • Figure 5: The height \ref{['eq:hwr.h']} vs. Tsallis function \ref{['eq:Tq']}, Rényi function \ref{['eq:Rq']}, Shannon function \ref{['eq:S']} and $P_q$ function \ref{['eq:Pq']} plots of $\hat{P}_\text{I}$ for $q=2$, $n=8$.
  • ...and 5 more figures