01596nas a2200169 4500008004100000245004400041210004300085260001400128490000700142520114800149100001801297700001801315700001701333700002501350700001401375856003701389 2014 eng d00aSymmetric Extension of Two-Qubit States0 aSymmetric Extension of TwoQubit States c2014/9/170 v903 a 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 $\rho_{ABB'}$, such that the $AB$ marginal
state is identical to the $AB'$ marginal state, i.e. $\rho_{AB'}=\rho_{AB}$.
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
$\tr(\rho_B^2)\geq \tr(\rho_{AB}^2)-4\sqrt{\det{\rho_{AB}}}$. 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.
1 aChen, Jianxin1 aJi, Zhengfeng1 aKribs, David1 aLütkenhaus, Norbert1 aZeng, Bei uhttp://arxiv.org/abs/1310.3530v2