Figure 2: Franson interferometer for entangled photons.
From: Equivalence of wave–particle duality to entropic uncertainty
In this interferometer, the ‘quanton’ consists of two photons. That is, a source produces time–energy entangled photons that each head separately towards a MZI that contains a long arm (depicted with extra loops) and a short arm. A simple model considers the 4D Hilbert space associated with the four possible paths: |SS›, |SL›, |LS› and |LL› with S=short path and L=long path. Two of these dimensions are post-selected away by considering only coincidence counts, that is, the photons arriving at the same time is inconsistent with the |SL› and |LS› paths. The remaining paths, |0›=|SS› and |1›=|LL›, are indistinguishable in the special case of perfect visibility and they produce interference fringes as one varies φ=φA+φB. Namely, the intensity of coincidence counts at detector pair 
oscillates with φ. Interaction with an environment system, or making the beam splitters asymmetric, may allow one to partially distinguish between |SS› and |LL›, and our entropic uncertainty framework can be applied to derive a trade-off, for example, of the form of equation (1). This trade-off captures the idea that Alice can either guess which path (|SS› versus |LL›) or which phase (φ=φ0 versus φ=φ0+π), but she cannot do both (even if she extracts information from other systems E1 and E2).
