[Paper Review] Monogamy of $\alpha$th Power Entanglement Measurement in Qubit Systems

Yu Luo, Yongming Li|arXiv (Cornell University)|Apr 30, 2015

Quantum Information and Cryptography17 references23 citations

TL;DR

This paper establishes monogamy inequalities for the αth power of entanglement measures—negativity and convex-roof extended negativity (CREN)—in N-qubit systems, proving that these measures are monogamous for α ≥ 2 and polygamous for α ≤ 0. It further shows that GHZ and W states can distinguish the αth power of concurrence (0 < α < 2) and entanglement of formation (0 < α ≤ 1/2), revealing distinct monogamy behaviors between concurrence and negativity.

ABSTRACT

In this paper, we study the $\alpha$th power monogamy properties related to the entanglement measure in bipartite states. The monogamy relations related to the $\alpha$th power of negativity and the Convex- Roof Extended Negativity are obtained for N-qubit states. We also give a tighter bound of hierarchical monogamy inequality for the entanglement of formation. We find that the GHZ state and W state can be used to distinguish the $\alpha$th power the concurrence for $0<\alpha<2$. Furthermore, we compare concurrence with negativity in terms of monogamy property and investigate the difference between them.

Motivation & Objective

• To investigate the monogamy properties of αth power entanglement measures in N-qubit systems.
• To determine for which α values the αth power of negativity and CREN satisfy monogamy or polygamy inequalities.
• To compare the monogamy behavior of concurrence and negativity, identifying differences in their α-dependent properties.
• To use GHZ and W states as probes to distinguish the monogamy behavior of αth power entanglement measures.
• To improve the hierarchical monogamy inequality for the αth power of entanglement of formation (EoF), showing it holds for α ≥ √2.

Proposed method

• Derives a relationship between negativity and concurrence in 2⊗m⊗n systems via Schmidt decomposition, showing NA|BC = CA|BC for pure states.
• Applies known monogamy results for αth power of concurrence (Cα) with α ≥ 2 to prove monogamy for Nα and eNα using the equality between concurrence and negativity.
• Uses the convex roof extension to define CREN (eN), and proves monogamy for eNα via the same concurrence-based argument.
• Introduces a generalized 'residual tangle' τX = Xα(A|BC) − ∑Xα(A|Ai) to quantify monogamy, where X is negativity, CREN, or EoF.
• Analyzes τC, τN, and τE for N-qubit GHZ and W states across varying α to determine monogamy/polygamy regimes.
• Employs binary entropy function approximations and calculus to bound τE(|W⟩), proving negativity for α ≤ 1/2.

Experimental results

Research questions

• RQ1 For which values of α does the αth power of negativity satisfy monogamy in N-qubit systems?
• RQ2 Can the GHZ and W states distinguish the monogamy behavior of the αth power of concurrence for 0 < α < 2?
• RQ3 Does the αth power of entanglement of formation (EoF) exhibit hierarchical monogamy, and for which α values?
• RQ4 How do the monogamy properties of concurrence and negativity differ, especially in the range 0 < α < 2?
• RQ5 Is the residual tangle τN for the W state always negative, or can it be positive for certain α ∈ (0,2)?

Key findings

• The αth power of negativity (Nα) satisfies monogamy for α ≥ 2 and polygamy for α ≤ 0 in N-qubit pure states.
• The αth power of CREN (eNα) also satisfies monogamy for α ≥ 2 and polygamy for α ≤ 0.
• For the N-qubit W state, τC(|W⟩) < 0 for 0 < α < 2, confirming polygamy of Cα; however, τN(|W⟩) can be positive or negative depending on α and N.
• The residual tangle τE(|W⟩) for EoF is negative for 0 < α ≤ 1/2 and N ≥ 3, proving hierarchical monogamy of Eα in this regime.
• The residual tangle τE(|W⟩) can be positive for 1/2 < α < √2, indicating that Eα is not monogamous in this range.
• The GHZ state exhibits monogamy for Cα (τC > 0) and Eα (τE > 0) for 0 < α < √2, while the W state shows polygamy for Cα and Eα in specific α intervals, highlighting a key difference between the two measures.