Geometrically induced singular behavior of entanglement

We show that the geometry of the set of quantum states plays a crucial role in the behavior of entanglement in different physical systems. More specifically, it is shown that singular points at the border of the set of unentangled states originate singularities in the dynamics of entanglement of smoothly varying quantum states. We illustrate this result by implementing a photonic parametric down-conversion experiment. Moreover, this effect is connected to recently discovered singularities in condensed matter models.

Figure 1

State space. The dotted line represents the path $\rho \left(q\right)$ followed by $\rho$ when the parameter $q$ is changed. It is worth noting that $S$ can present singular points in its shape and to remember that the “true” picture is much subtler, given the large dimensionality of even the simplest example [9].

Figure 2

Experimental setup: The state source is composed by a 2-mm-thick $\left(\beta -{\text{BaB}}_2{\text{O}}_4\right)$ (BBO) nonlinear crystal $\left(C_1\right)$ pumped by a cw krypton laser operating at 413 nm, generating photon pairs at 826 nm by type-II spontaneous parametric down-conversion. Crystal $C_1$ is cut and oriented to generate either one of the polarization-entangled Bell states $|{\Psi }_{-}⟩$ or $|{\Psi }_{+}⟩$. Walk-off and phase compensation is provided by the half-wave plate $H_0$ followed by a 1-mm-thick BBO crystal $\left(C_2\right)$ [23], together with two 1-mm-thick crystalline quartz plates $\left(z\right)$ inserted in one of the down-converted photon paths. The unconverted laser beam transmitted by crystal $C_1$ is discarded by means of a dichroic mirror $\left(u\right)$. The detection stages are composed by photon counting diode modules $D_1$ and $D_2$, preceded by 8 nm full width at half maximum interference filters $F_1$ and $F_2$ centered at 825 nm, and by circular apertures $A_1$ of $1.6\text{mm}\varnothing$ and $A_2$ of $3.0\text{mm}\varnothing$. Single and coincidence counts with 5 ns resolving time are registered by a computer-controlled electronic module (CC). Polarization analyzers are composed of quarter-wave plates $Q_1$ and $Q_2$, half-wave plates $H_1$ and $H_2$, followed by polarizing cubes $P_1$ and $P_2$. The state source produces state $|{\Psi }_{-}⟩$. For each pair, the photon emerging in the upper path goes straight to the polarization analyzer and to the detection stage 1. The lower-path photon is directed by mirror $M_3$ through the circular aperture $A_3$ into the state mixer (an unbalanced Michelson interferometer), composed by the beam splitter BS, mirrors $M_4$ and $M_5$, quarter-wave plates $Q_4$ and $Q_5$, variable circular apertures $A_4$ and $A_5$, and by the half-wave plate $H_3$, whose purpose is to compensate for an unwanted slight polarization rotation caused by the beam splitter. The quarter-wave plate $Q_4$ is switched off, which means that if the lower photon follows the path labeled 4, there is no change to its polarization and the half-wave plate $H_3$ changes the state to $|{\Psi }_{+}⟩$. On the other hand, if the lower photon follows the path labeled 5, $Q_5$ is oriented with the fast axis at 45° in order to flip its polarization. The path length difference, 130 mm, is much larger than the coherence length of the down-converted fields, ensuring an incoherent recombination at BS. The pair detected by CC is in state $q|{\Psi }_{+}⟩⟨{\Psi }_{+}|+\left(1-q\right)|{\Phi }_{+}⟩⟨{\Phi }_{+}|$, where $q$ is defined by the relative sizes of apertures $A_4$ and $A_5$.

Figure 3

(Color online) Reconstructed density matrices corresponding (ideally) to the states $|{\Phi }_{+}⟩$, (a) real and (b) imaginary parts, and $|{\Psi }_{+}⟩$, (c) real and (d) imaginary parts. The attained fidelities for these states are, respectively, $F_{{\Phi }_{+}}\equiv ⟨{\Phi }_{+}|\rho |{\Phi }_{+}⟩\approx \left(92\pm 4\right)\%$ and $F_{{\Psi }_{+}}\equiv ⟨{\Psi }_{+}|\rho |{\Psi }_{+}⟩\approx \left(96\pm 4\right)\%$.

Figure 4

(Color online) Measurement of the mean value of both operators described in (7) for the full range $0\le q\le 1$. Each $W$ is expanded as a linear combination of products of local operators which are then measured independently. The blue continuous line corresponds to the theoretical value of $\mathcal{N}\mathbf{\left(}\rho \left(q\right)\mathbf{\right)}$ for the state $\rho \left(q\right)=q|{\Psi }_{+}⟩⟨{\Psi }_{+}|+\left(1-q\right)|{\Phi }_{+}⟩⟨{\Phi }_{+}|$. Note that each $W$ witnesses entanglement for only a restricted range of $q$ values as predicted by the theory. The local singularity of $\partial S$ is evidenced by the abrupt change of optimal $W$. Experimental errors are within the dots’ sizes.