On Estimation of Fully Entangled Fraction Rui-Juan Gu $^{1}$, Ming Li$^{2}$, Shao-Ming Fei1 and Xianqing Li-Jost 3 , 4 1 School of Mathematical Sciences, Capital Normal University, 100048 Beijing 2 College of Mathematics and Computational Science, China University of Petroleum, 257061 Dongying 3 Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig 4 Department of Mathematics, Hainan Normal University, 571158 Haikou We study the fully entangled fraction (FEF) of arbitrary mixed states. New upper bounds of FEF are derived. These upper bounds make complements on the estimation of the value of FEF. For weakly mixed quantum states, an upper bound is shown to be very tight to the exact value of FEF. PACS numbers: 03.67.-a, 02.20.Hj, 03.65.-w Quantum entanglement plays crucial roles in quantum information processing such as quantum computation [1, 2], quantum teleportation [3, 4], dense coding [5], quantum cryp- tographic schemes [6], entanglement swapping [7] and remote states preparation (RSP) [8– 10]. For instance in terms of a classical communication channel and a quantum resource (a nonlocal entangled state like an EPR-pair of particles), the teleportation protocol gives ways to transmit an unknown quantum state from a sender to a receiver that are spatially separated. When the sender and receiver share a maximally entangled pure state, the state can be perfectly teleported. However when the shared entangled state is an arbitrary mixed state ρ , then the optimal fidelity of teleportation is given by [4, 11], which solely depends on the fully entangled fraction (FEF) F ( ρ ) of ρ . Detailed discussion about FEF can be found in Ref. [12]. In fact the quantity FEF plays essential roles in many other quantum information pro- cessing such as dense coding, entanglement swapping and quantum cryptography (Bell in- equalities). Thus it is very important to compute the FEF of general quantum states. 1 f ( ρ ) = d F ( ρ ) d + 1 + 1 d + 1 , (1) Abstract arXiv:1001.4756v1 [quant-ph] 26 Jan 2010