Research ArticlePHYSICS

Manipulation of eight-dimensional Bell-like states

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Science Advances  14 Jun 2019:
Vol. 5, no. 6, eaat9206
DOI: 10.1126/sciadv.aat9206
  • Fig. 1 Transformations and theoretical results of coincidence measurement for eight Bell-like states.

    (A) Preparation of Bell-like states from ∣Θ1〉 by manipulating unitary operation UOUS on photon-A. Details on UOUS are shown on the right side in (A). Here, YES (NO) means that the DP is (is not) in the optical path. @0° (@45° or @90°) means that the wave plate is oriented at 0° (45° or 90°, respectively) from the horizontal polarization direction. (B) Simulated coincidence measurement results of eight Bell-like states with projecting Bell-like states into {∣ψiA∣ψjB} (i, j = 1, 2, ..., 8). The colored small squares (empties) mean that there are (are no) coincidence counts.

  • Fig. 2 Experimental setup for the generation and projection of Bell-like states.

    (A) An ultraviolet femtosecond laser pumps two 0.6-mm-thick BBO crystals to produce HD entanglement source. A spatial filter (SF) is used to obtain the maximum entangled state. Detail on function of SF has been given in the inset under the setup (see also the Supplementary Materials). (B) Projection of Bell-like states. The optical elements from left to right are QWP1, q-plate1, QWP2, HWP, PBS, QWP3, q-plate2, QWP4, lens, and single-mode fiber (SMF) in turn (as shown in row 3). Row 1 shows the evolution of the state ∣φ1〉 = ∣+3〉 ∣H〉 by performing the operations (row 2) of the wave plates. Row 5 shows the evolution of the state ∣ψ1〉 ∝∣+1〉 ∣H〉 + ∣+3〉 ∣V〉 by performing the operations (row 4) of the wave plates.

  • Fig. 3 Experimental coincidence counts of ∣ΘSPDC〉.

    The coincidence counts under the projective basis set ∣φi〉(i = 1, 2, ..., 8) for eight-dimensional entangled state produced from the SPDC process directly. Inset shows the result after the distillation using the SF.

  • Fig. 4 Experimental coincidence measurement results for the eight Bell-like states.

    The projective basis is ∣ψi〉 (i = 1, 2, ..., 8). Vertical axis represents the coincidence counts in 10 s.

  • Fig. 5 The real part of reconstructed density matrices and the corresponding fidelities in every subspace for the eight Bell-like states.

    Taking ∣Θ1〉 as an example, fidelities are calculated as Fspin=tr(ρspinΦspin+Φspin+) in the spin subspace, F1=tr(ρ1Ψ1+Ψ1+) in the first-order OAM subspace, and F3=tr(ρ3Ψ3+Ψ3+) in the third-order OAM subspace. Here, ρspin = [ρspin(+3) + ρspin(+1) + ρspin(−1) + ρspin(−3)]/4, ρ1 = [ρ1(H) + ρ1(V)]/2, and ρ3 = [ρ3(H) + ρ3(V)]/2. ρspin(m) is the measured density matrix of Φspin+mAmB. ρm(H) and ρm(V) are the measured density matrices of Ψm+HAHB and Ψm+VAVB, respectively. It should be pointed out that the imaginary parts of reconstructed density matrices in every subspace are almost zero for any Bell-like state.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/6/eaat9206/DC1

    Section S1. Distillation of the HD entangled states

    Section S2. Projection of the basis set {∣ψ1〉, ∣ψ2〉, ..., ∣ψ8〉}

    Section S3. Eight-outcome Bell-like state analyzer

    Section S4. Sixty-four Bell-like states in the eight-dimensional Hilbert space

    Section S5. Symmetry and antisymmetry of Bell-like states

    Fig. S1. Distillation of the HD hyperentanglement source.

    Fig. S2. Experimental coincidence measurement results under the projective basis set {∣φi〉} (i = 1, 2, ..., 8).

    Fig. S3. Eight-outcome Bell-like state analyzer.

    Fig. S4. Scheme for dense coding with eight Bell-like states.

    Fig. S5. Theoretical results of coincidence measurement for 64 Bell-like states.

    Fig. S6. Schematic diagram of coherence among three subspaces.

    Fig. S7. Verification of Bell-like states.

    Fig. S8. Coefficient distribution pattern(s) of Bell-like state(s).

    Table S1. Scheme for projecting the basis set ∣φi〉 into ∣0〉.

    Table S2. Scheme for projecting the basis set ∣ψi〉 into ∣0〉.

  • Supplementary Materials

    This PDF file includes:

    • Section S1. Distillation of the HD entangled states
    • Section S2. Projection of the basis set {∣ψ1〉, ∣ψ2〉, ..., ∣ψ8〉}
    • Section S3. Eight-outcome Bell-like state analyzer
    • Section S4. Sixty-four Bell-like states in the eight-dimensional Hilbert space
    • Section S5. Symmetry and antisymmetry of Bell-like states
    • Fig. S1. Distillation of the HD hyperentanglement source.
    • Fig. S2. Experimental coincidence measurement results under the projective basis set {∣φi〉} (i = 1, 2, ..., 8).
    • Fig. S3. Eight-outcome Bell-like state analyzer.
    • Fig. S4. Scheme for dense coding with eight Bell-like states.
    • Fig. S5. Theoretical results of coincidence measurement for 64 Bell-like states.
    • Fig. S6. Schematic diagram of coherence among three subspaces.
    • Fig. S7. Verification of Bell-like states.
    • Fig. S8. Coefficient distribution pattern(s) of Bell-like state(s).
    • Table S1. Scheme for projecting the basis set ∣φi〉 into ∣0〉.
    • Table S2. Scheme for projecting the basis set ∣ψi〉 into ∣0〉.

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