Maximally Entangled Mixed States: Creation and Concentration

Using correlated photons from parametric down-conversion, we extend the boundaries of experimentally accessible two-qubit Hilbert space. Specifically, we have created and characterized maximally entangled mixed states that lie above the Werner boundary in the linear entropy-tangle plane. In addition, we demonstrate that such states can be efficiently concentrated, simultaneously increasing both the purity and the degree of entanglement. We investigate a previously unsuspected sensitivity imbalance in common state measures, i.e., the tangle, linear entropy, and fidelity.

Figure 1

Experimental arrangement to create and concentrate MEMS. A half-wave plate (${\mathrm{HWP}}_1$) sets the initial entanglement of the pure state. The $\varphi$ plate sets the relative phase between $|HH⟩$ and $|VV⟩$ in the initial state. ${\mathrm{HWP}}_2$ and ${\mathrm{HWP}}_3$ rotate the state into the active bases of the decoherers to adjust the amount of entropy. The tomography system uses a quarter-wave plate (QWP), HWP, and a polarizer in each arm to analyze in arbitrary polarization bases; the transmitted photons are counted in coincidence via avalanche photodiodes. The dashed box contains ${\mathrm{HWP}}_4$ (oriented to rotate $|H⟩\leftrightarrow |V⟩$ in the first arm of the experiment) and concentrating elements (a variable number of glass pieces oriented at Brewster’s angle to completely transmit $|H⟩$ but only partially transmit $|V⟩$).

Figure 2

MEMS data. (a) Density matrix plots of the real components of a MEMS I ($r=\frac{2}{3}$) and a MEMS II ($r=0.3651$). The imaginary components are negligible (on average less than 0.02) and not shown. (b) Linear entropy-tangle plane. Shown are theoretical curves for MEMS I (solid line), MEMS II (dashed line), and Werner states (dotted line). Four target MEMS are indicated by stars; experimental realizations are shown as squares with error bars. The shaded patches around each target state show the tangle ($T$) and linear entropy ($S_L$) for 5000 numerically generated density matrices that have at least 0.99 fidelity [31] with the target state. $T=0$ (1) corresponds to a product (maximally entangled) state. $S_L=0$ (1) corresponds to a pure (completely mixed) state.

Figure 3

Concentration data. Shown are concentrations for three initial states, A (triangles) and B (filled squares) as in Fig. 2, and C (open squares), along with the number of partial polarizing glass pieces in each arm. The expected concentrated state path, calculated using [7], is shown with stars. The concentrated states agree with theory for small numbers of glass pieces, but as more slips are used, the state concentrates better than expected. We believe this is due to extreme sensitivity of the trajectory to small changes in the initial state. However, even in theory, excessive filtration will eventually produce a pure product state (shown as an extension of A’s theory curve), due to small errors in the initial MEMS.