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Multi-partite entanglement monotones

Abhijit Gadde, Shraiyance Jain, Harshal Kulkarni

Abstract

If we want to transform the quantum state of a system to another using local measurement processes, what is the probability of success? This probability is bounded by quantifying entanglement in both the states. In this paper, we construct a family of local unitary invariants of multipartite states that are monotonic under local operations and classical communication on average. These monotones are constructed from local unitary invariant polynomials of the state and its conjugate, and hence are easy to compute for pure states. Using these measures we bound the success probability of transforming a given state into another state using local quantum operations and classical communication.

Multi-partite entanglement monotones

Abstract

If we want to transform the quantum state of a system to another using local measurement processes, what is the probability of success? This probability is bounded by quantifying entanglement in both the states. In this paper, we construct a family of local unitary invariants of multipartite states that are monotonic under local operations and classical communication on average. These monotones are constructed from local unitary invariant polynomials of the state and its conjugate, and hence are easy to compute for pure states. Using these measures we bound the success probability of transforming a given state into another state using local quantum operations and classical communication.
Paper Structure (22 sections, 20 theorems, 88 equations, 15 figures)

This paper contains 22 sections, 20 theorems, 88 equations, 15 figures.

Table of Contents

  1. Introduction and summary
  2. Main results
  3. Pure state entanglement monotones
  4. For bi-partite states
  5. For multi-partite states
  6. PSEMs from polynomial invariants
  7. Reflection symmetry and positivity
  8. Strategy
  9. Some simple examples
  10. Example 1: Bi-partite
  11. Example 2: Tri-partite
  12. Edge-convexity
  13. A look back at the examples
  14. A new example
  15. Strengthening convexity
  16. ...and 7 more sections

Key Result

Theorem 1.1

The maximal success probability $p_{\rho\to \rho_*}$ of converting a multipartite state $\rho$ to $\rho_*$ using LOCC is where the minimization is performed over all entanglement monotones $\mu$.

Figures (15)

  • Figure 1: White (black) vertex denoting $\psi$ ($\bar{\psi}$). The parties are labeled by colored edges.
  • Figure 2: Example of a $\psi$-graph constructed from three copies of $\psi$ and $\bar{\psi}$ each by connecting edges of identical colors.
  • Figure 3: The reflecting cut is shown by a straight black line passing through the graph. It is easy to see that the graph is symmetric under the reflection across the reflecting cut, after vertex color flip.
  • Figure 4: Connecting only the $A$-edge (denoted by red color) of a $\psi$-$\bar{\psi}$ pair to get $\rho_{\bar{A}}$. The dotted line represents insertion of $\delta\rho_{\bar{A}}$ instead of $\rho_{\bar{A}}$.
  • Figure 5: The first line denotes $\delta {\mathcal{Z}}$ for ${\mathcal{Z}}$ given in figure \ref{['example']}. Each of the $A$-edges has been replaced by $\delta\rho_{\bar{A}}$ and the terms are added up. The second line denotes $\delta^2 {\mathcal{Z}}/2$ for same ${\mathcal{Z}}$. A pair of $A$-edges have been replaced by $\delta\rho_{\bar{A}}$ and the terms are added up.
  • ...and 10 more figures

Theorems & Definitions (48)

  • Definition 1.1: Fully separable state
  • Definition 1.2: Entanglement monotone
  • Remark 1.1: SzalaySzalay_2015
  • Theorem 1.1: Vidal Vidal_2000
  • Theorem 1.2: Vidal Vidal_2000
  • Definition 1.3
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • ...and 38 more