Synthesizing arbitrary two-photon polarization mixed states

Two methods for creating arbitrary two-photon polarization pure states are introduced. Based on these, four schemes for creating two-photon polarization mixed states are proposed and analyzed. The first two schemes can synthesize completely arbitrary two-qubit mixed states, i.e., control all 15 free parameters: scheme I requires several sets of crystals, while scheme II requires only a single set, but relies on decohering the pump beam. Additionally, we describe two further schemes which are much easier to implement. Although the total capability of these is still being studied, we show that they can synthesize all two-qubit Werner states, maximally entangled mixed states, Collins-Gisin states, and arbitrary Bell-diagonal states.

Figure 1

(Color online) Arbitrary pure states via Schmidt decomposition. PBS, polarizing beam splitter; HWP, half-wave plate; QWP, quarter-wave plate; NLC, nonlinear crystal.

Figure 2

(Color online) Arbitrary pure states via inteferometry.

Figure 3

(Color online) Scheme I employs four sets of nonlinear crystals. The two-photon state created at the $i\mathrm{th}$ set of crystals is chosen such that it is the correct state $\mid {\psi }_i⟩$ after propagation through the subsequent elements. The necessary local unitary transformations at each down-conversion location can be readily calculated [22]. PBS, polarizing beam splitter; HWP, half-wave plate; QWP, quarter-wave plate; NLC, nonlinear crystal.

Figure 4

(Color online) Arbitrary two-qubit mixed-state synthesis scheme II. The transmission probabilities of the various beam splitters depend on the desired final state. The variable beam splitters immediately preceding the unitary rotations could also be realized by polarizing beam splitters with suitable polarization rotations before and after.

Figure 5

(Color online) Scheme III employs decoherence. Down-conversion photon pairs can be decohered, as well as pump photons. The decoherers are thick birefringent crystals, which separate different polarizations and decrease the coherence between them. Also shown is a possible decoherence on the pump beam: the vertical polarization component experiences an adjustable extra delay.

Figure 6

(Color online) Scheme IV is a hybrid technique. (a) Mixing a pure state with a mixed state. The local unitary transformations immediately after the decoherers are used to precompensate the effect of local unitary transformations used afterward to rotate the pure part, and also to undo any effects of passing through the nonlinear crystals (cf. Fig. 3). (b) A reduced setup of the method in (a), using only one set of nonlinear crystals, and retroreflecting the pump back through the nonlinear crystals and the first photon pair back into the spatial modes of the second pair. Setup (b) is less general than (a), as the pure-state part cannot be chosen arbitrarily.