Symmetric Extension of Two-Qubit States

Abstract

Quantum key distribution uses public discussion protocols to establish shared secret keys. In the exploration of ultimate limits to such protocols, the property of symmetric extendibility of underlying bipartite states ${\rho }_{AB}$ plays an important role. A bipartite state ${\rho }_{AB}$ is symmetric extendible if there exits a tripartite state ${\rho }_{AB{B}^{\prime }}$, such that the $AB$ marginal state is identical to the $A{B}^{\prime }$ marginal state, i.e. ${\rho }_{A{B}^{\prime }}={\rho }_{AB}$. For a symmetric extendible state ${\rho }_{AB}$, the first task of the public discussion protocol is to break this symmetric extendibility. Therefore to characterize all bi-partite quantum states that possess symmetric extensions is of vital importance. We prove a simple analytical formula that a two-qubit state ${\rho }_{AB}$ admits a symmetric extension if and only if $\tr(ρ_B^2)\geq \tr(ρ_{AB}^2)-4\sqrt{\det{ρ_{AB}}}$. Given the intimate relationship between the symmetric extension problem and the quantum marginal problem, our result also provides the first analytical necessary and sufficient condition for the quantum marginal problem with overlapping marginals.

10 pages, no figure. comments are welcome. Version 2: introduction rewritten

References in corpus (9)

• A complete family of separability criteria
• N-representability is QMA-complete
• The Pauli principle revisited
• Detecting multipartite entanglement
• The structure of degradable quantum channels
• Symmetric extension in two-way quantum key distribution
• Spectrum conditions for symmetric extendible states
• Symmetric extension of bipartite quantum states and its use in quantum key distribution with two-way postprocessing
• Symmetric extendibility for qudits and tolerable error rates in quantum cryptography

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• Numerical evidence for bound secrecy from two-way post-processing in quantum key distribution
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• Local density matrices of many-body states in the constant weight subspaces
• Tapping into Permutation Symmetry for Improved Detection of k-Symmetric Extensions
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