Transverse entanglement migration in Hilbert space

Entanglement is one of the truly central features of the quantum world and it forms the core of many applications based on quantum theory. The observation of entanglement is generally achieved through the measurement of correlations between entangled subsystems. Correlation in quantum systems takes many forms and is open to observation in a variety of ways. Therefore the determination of the amount of entanglement of quantum states depends on the measurement of the correlations where entanglement resides. This is of paramount importance, since in some experimental configurations one registers types of correlation that might not be appropriate to quantify the entangled nature of the quantum state. In this Rapid Communication, we show that the measurement of correlation between paired photons can miss the detection of entanglement entirely. The underlying reason is an interesting migration of entanglement that occurs in Hilbert space, but that depends on coordinate location in real space. This is manifest in photon correlations that show a rich and complex structure that evolves during propagation, although the amount of entanglement is constant. We focus here on entanglement that can become partly or entirely identified with the phase of the state, in which case the measurement of intensity correlations partially or completely misses the existing entanglement. This is an observable manifestation of the “phase entanglement” previously noted 1 for massive particle breakup in an Einstein-Podolsky-Rosen EPR scenario. Entangled photons generated in spontaneous parametric down-conversion SPDC are particularly open to the observation of this phenomenon. The generated two-photon states have been shown to exhibit entanglement in transverse momentum 2 and in orbital angular momentum 3,4. Moreover, one can enlarge the Hilbert space of the two-photon state by using several degrees of freedom 5. The spatial transverse degrees of freedom of photon pairs produced in SPDC have attracted great attention because of the vast Hilbert space involved 6,7 and the availability of techniques to implement the d-dimensional quantum channel 8‐10. Observations of SPDC entanglement have usually been made either in the near zone or the far zone 11. Interestingly, in the course of photon propagation from the near field zone to the far field zone, the entanglement embedded in the two-photon positional amplitude migrates out of the positional wave function’s modulus into its phase, and then back again. In the region between near and far zones, the entanglement not obtained through the measurement of intensity correlations can be recovered by measuring the phase information of the joint wave function. Here we propose an experimental setup to accomplish this by exploiting the symmetries of the wave function. We consider a nonlinear optical crystal of length L, illuminated by a quasimonochromatic laser pump beam, propagating in the z direction. The signal and idler photons generated propagate from the output face of the nonlinear crystal under the sole effect of diffraction. The quantum state of the two-photon pair generated in SPDC, at a distance z from the output face of the nonlinear crystal z=0, reads in wave number space as z=dpdqp,q,zas pai q0,0,