Research ArticleQUANTUM INFORMATION

Superadditivity of two quantum information resources

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Science Advances  22 Sep 2017:
Vol. 3, no. 9, e1602485
DOI: 10.1126/sciadv.1602485

Figures

  • Fig. 1 Two families of qubit states in a six-qubit free entangled state.

    The red (green) circles symbolize the fact that PPT with respect to the subsystems they mark is satisfied (violated). Only between the pairs with two green circles (marked by green dashed lines) can pure entanglement be distilled if some extra conditions are also satisfied [see the study by Dür and Cirac (9) for details].

  • Fig. 2 The two resource states used in the protocol.

    (A) The special bound entangled three-qubit state ρbound designed by Dür and Cirac (9) (see the formula in the main text) is depicted symbolically on the left-hand side. Because there is always at least one qubit guaranteeing a PPT property in each pair of qubits, with the original three-qubit state, there is no chance to distill any pure entanglement out of the state. Thus, distillable entanglement vanishes D(ρbound)=0. In particular, no singlet can be distilled between B and C, which we write as DB:C(ρbound)=0. However, there is still some entanglement in the state because the PPT test is violated with respect to subsystem C. Thus, the state is entangled, and because it is nondistillable, it is therefore bound entangled. (B) The second, free entangled state ρfree corresponds to two-qubit singlet and the virtual (vacuum) part. There is no chance to distill entanglement between B′ and C′ from ρfree. Summarizing the two pictures, no pure entanglement between Bob and Charlie parts can be created from an arbitrary number of copies of any of the state ρbound or ρfree. In that sense, any of the two states alone is weak because some important quantum entanglement ingredient is completely absent in any of them.

  • Fig. 3 Superadditivity protocol.

    The weakness of the two resources shown in the previous picture in Fig. 2 disappears when we allow them to interact through LOCC and the bound entanglement of the first state is activated and creates free entanglement between Bob’s and Charlie’s part (BB′ versus CC). This is the result of Alice’s local measurement M (projection onto singlet) followed by classical communication to Bob and Charlie about whether the projection was successful. Here, Alice is teleporting to Bob. This emergence of absence before the interaction of free entanglement between Bob and Charlie represents the extreme form of the superadditivity of the two quantum resources. To make a complete description of the consequences of the effect, observe that it implies that given many copies of the two states, one can distill pure singlets among the two parts. Note that because we already have the singlet resource between Alice and Bob, the creation of either AC singlet or just full three-partite GHZ is possible (by teleportation from the Bob station provided that he has also some extra copies of particles ρfree at his disposal).

  • Fig. 4 Experimental results: The density matrix.

    Density matrix of the mixed three photon bound entangled state ρboundexp in the computational base {|H〉, |V〉}.

  • Fig. 5 Experimental results: The density matrix after LOCC operation.

    Density matrix of the mixed three photon entangled state ρ(BB)Cexp in the computational base {|H〉, |V〉}.

  • Fig. 6 Experimental setup for the generation of three-qubit polarization bound entangled state.

    The colored area represents the state preparation. See Methods for more details. QWP, quarter–wave plate; HWP, half–wave plate; PBS, polarizing beam splitter; BBO, β-barium borate; UV, ultraviolet.

  • Fig. 7 Experimental setup for the superadditivity protocol.

    See Methods for more details.

Tables

  • Table 1 Table of eigenvalues of the partially transposed density matrix of the bound entangled state around A/BC, B/AC, and C/AC cuts.

    The theoretical eigenvalues are {1/3; 1/6; 1/6; 1/6; 1/6; 0; 0; 0}, {1/3; 1/6; 1/6; 1/6; 1/6; 0; 0; 0}, and {1/6; 1/6; 1/6; 1/6; 1/6; 1/6; 1/6; −1/6} for these cuts, respectively.

    A/BCB/ACC/AB
    EigenvalueErrorEigenvalueErrorEigenvalueError
    0.290980.002940.290560.002870.172650.00291
    0.170860.001670.170590.001400.170420.00125
    0.167870.001290.169120.001800.169520.00096
    0.159680.001120.158340.001480.165680.00269
    0.155370.001500.156560.001840.158230.00089
    0.047640.002610.047260.002670.156000.00187
    0.004850.001440.005990.001550.125280.00281
    0.002750.001490.001590.00149−0.117780.00258
  • Table 2 Table of eigenvalues of the partially transposed density matrix of the bound entangled state after activation around the C/BB′ cut.

    The theoretical eigenvalues are {1/6; 1/6; 1/6; 1/6; 1/6; 1/6; 1/6; −1/6} for this cut.

    EigenvalueError
    0.182920.00417
    0.176480.00158
    0.171220.00174
    0.161680.00355
    0.147140.00169
    0.145030.00177
    0.106570.00252
    −0.091090.00274

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