[Paper Review] Entangled multi-qubit states without higher-tangle

Heng Fan, Yong-Cheng Ou+1 more|ArXiv.org|Jul 11, 2007

Quantum Computing Algorithms and Architecture1 references3 citations

TL;DR

This paper investigates mixed multi-qubit states composed of a W-class state and a |0⟩⊗(n+1) product state, deriving the optimal pure state decomposition and convex roof for higher-tangle using the Coffman-Kundu-Wootters (CKW) monogamy inequality. It proves that W-class states saturate the generalized CKW inequality for arbitrary bipartite partitions, confirming they have no residual higher-tangle and establishing new monogamy inequalities using negativity and matrix realignment measures.

ABSTRACT

We analyze mixed multi-qubit states composed of a W class state and a product state with all qubit in |0>. We find the optimal pure state decomposition and convex roofs for higher-tangle with bipartite partition between one qubit and the rest qubits for those mixed states. The optimality of the decomposition is ensured by the Coffman-Kundu-Wootters (CKW) inequality which describes the monogamy of quantum entanglement. The generalized monogamy inequality is found to be true for W class states with arbitrary partitions between one qubit and multi-qubit.

Motivation & Objective

• To analyze mixed multi-qubit states formed by a W-class state and a |0⟩⊗(n+1) product state.
• To derive the optimal pure state decomposition and convex roof for tangle in such mixed states.
• To verify whether the generalized CKW inequality holds for arbitrary bipartite partitions beyond the original proof.
• To explore alternative monogamy inequalities using entanglement measures like negativity and matrix realignment.
• To clarify the absence of higher-tangle in W-class states under mixed-state composition.

Proposed method

• The study focuses on mixed states of the form ρ = p|W⟩⟨W| + (1−p)|0⟩⟨0|, where |W⟩ is a W-class state.
• The optimal pure state decomposition is constructed using the structure of W-class states and verified via the CKW inequality.
• The tangle τ(ρ) is computed using the convex roof construction, with the optimal decomposition ensuring minimal entanglement sum across bipartite cuts.
• The generalized CKW inequality τ(ρ_{A:B₁…Bₙ}) ≥ ∑ⱼ τ(ρ_{ABⱼ}) is tested and proven to hold for W-class states under arbitrary partitions.
• Alternative monogamy inequalities are derived using the negativity 𝒩 and matrix realignment measure 𝒓, both shown to satisfy analogous inequalities.
• The proof relies on the fact that 𝒩 and 𝒓 are lower bounds for concurrence, inheriting the CKW inequality's structure.

Experimental results

Research questions

• RQ1 Does the generalized CKW inequality hold for W-class states under arbitrary bipartite partitions between one qubit and the rest?
• RQ2 Can the optimal pure state decomposition be analytically determined for mixed states composed of a W-class state and a |0⟩⊗(n+1) state?
• RQ3 Do W-class states exhibit non-zero higher-tangle in mixed-state form, or is it always zero?
• RQ4 Can alternative entanglement measures like negativity and matrix realignment satisfy monogamy inequalities analogous to the CKW inequality?
• RQ5 Is the convex roof construction for tangle uniquely determined for such mixed W-state systems?

Key findings

• The optimal pure state decomposition for mixed states ρ = p|W⟩⟨W| + (1−p)|0⟩⟨0| is analytically derived and confirmed via the CKW inequality.
• W-class states composed with |0⟩⊗(n+1) saturate the CKW inequality, indicating no residual higher-tangle.
• The generalized CKW inequality τ(ρ_{A:B₁…Bₙ}) ≥ ∑ⱼ τ(ρ_{ABⱼ}) holds for W-class states under arbitrary partitions, even beyond the original proof scope.
• For a five-qubit W state, τ(ρ_{A:B₁B₂}) = 8/25 and τ(ρ_{A:C₁C₂}) = 8/25, with τ(ρ_{A:BC}) = 16/25, satisfying the generalized inequality.
• The matrix realignment measure 𝒓 and negativity 𝒩 satisfy monogamy inequalities analogous to the CKW inequality, with 𝒓²(ρ_{A:BC}) ≥ 𝒓²(ρ_{AB}) + 𝒓²(ρ_{AC}) and similarly for 𝒩.
• The study confirms that W-class states are monogamous under all tested entanglement measures, with no detectable higher-tangle in mixed-state form.