One-Photon Measurement of Two-Photon Entanglement

Measuring entanglement is an essential step in a wide range of applied and foundational quantum experiments. When a two-particle quantum state is not pure, standard methods to measure the entanglement require detection of both particles. We realize a conceptually new method for verifying and measuring entanglement in a class of two-part (bipartite) mixed states. Contrary to the approaches known to date, in our experiment we verify and measure entanglement in mixed quantum bipartite states by detecting only one subsystem, the other remains undetected. Only one copy of the mixed or pure state is used but that state is in a superposition of having been created in two identical sources. We show that information shared in entangled systems can be accessed through single-particle interference patterns. Our experiment enables entanglement characterization even when one of the subsystems cannot be detected, for example, when suitable detectors are not available.

Figure 1

Entanglement verification scheme. Two identical sources, $Q_1$ and $Q_2$, individually generate the same two-photon state ($\widehat{\rho }$). Source $Q_1$ can emit a photon pair ($\alpha$, $\beta$) into propagation modes ${\alpha }_1$ and ${\beta }_1$. Source $Q_2$ is restricted to emit photon $\alpha$ also in the mode ${\alpha }_1$. Photon $\alpha$, which is never detected, interacts with a device, $O$, between $Q_1$ and $Q_2$. Source $Q_2$ can emit photon $\beta$ in propagation mode ${\beta }_2$. Modes ${\beta }_1$ and ${\beta }_2$ are combined by a beam splitter ($BS$) and an output of $BS$ is collected by a photodetector ($PD$). Another device ($\mathrm{\Gamma }$), placed before $PD$, allows us to choose the measurement basis. Sources $Q_1$ and $Q_2$ never emit simultaneously and stimulated emission in $Q_2$ due to the insertion of the ${\alpha }_1$ mode is negligible. When it is impossible to know the source of a detected photon, single-photon interference is observed at $PD$. Information about the entanglement is retrieved from the single-photon interference patterns. In the Supplemental Material [20] we present an alternative setup that does not require two identical sources because the laser passes twice the same pair of crystals.

Figure 2

(a) Source of the polarization-entangled photon pair: Each source ($Q_j$) is composed of two nonlinear crystals, $C_H^j$ and $C_V^j$, which produce the horizontally polarized ($|H_{\alpha }{H}_{\beta }⟩$) and vertically polarized ($|V_{\alpha }{V}_{\beta }⟩$) parts of the entangled state, respectively. The relative intensity of emissions $|H_{\alpha }{H}_{\beta }⟩$ at $C_H^j$ and $|V_{\alpha }{V}_{\beta }⟩$ at $C_V^j$ are $I_H$ and $I_V$, respectively. The coherence between these two emissions is $\mathcal{I}$. (b) $Q_1$ and $Q_2$ are illuminated by mutually coherent laser propagation modes (not shown) such that the horizontal ($H$) components of the possible emissions at the separate sources are coherent. Highest interference visibility is observed at $PD$ if $H$ polarized photons are detected. (c) For $\theta =\pi /4$, we probe the indistinguishability between emissions at $C_H^1$ and $C_V^2$ and also between emissions at $C_V^1$ and $C_H^2$ (not shown) by detecting diagonally ($D$) linearly polarized $\beta$ photons. The visibility of the resulting interference pattern depends on the entanglement in the two-photon state.

Figure 3

Signature of entanglement in single-photon interference visibility. The data show single-photon interference patterns recorded in projective measurements onto state $|D_{\beta }⟩=(|H_{\beta }⟩+|V_{\beta }⟩)/\sqrt{2}$ for five mixed states, $\widehat{\rho }$, when HWP angle $\theta =\pi /4$. The interferometer phase is varied using a system of wave plates in the pump. The relevant parameters of $\widehat{\rho }$ and the visibilities are (a) $I_H=0.96$, $\mathcal{I}=0.25$, ${\mathcal{V}}_D{|}_{\theta =(\pi /4)}=0.04$; (b) $I_H=0.51$, $\mathcal{I}=0.042$, ${\mathcal{V}}_D{|}_{\theta =(\pi /4)}=0.04$; (c) $I_H=0.50$, $\mathcal{I}=0.22$, ${\mathcal{V}}_D{|}_{\theta =(\pi /4)}=0.15$; (d) $I_H=0.65$, $\mathcal{I}=0.38$, ${\mathcal{V}}_D{|}_{\theta =(\pi /4)}=0.26$ (e) $I_H=0.47$, $\mathcal{I}=0.94$, ${\mathcal{V}}_D{|}_{\theta =(\pi /4)}=0.70$. The lowest visibilities (a and b) correspond to the (almost) separable states (${\rho }_1$ and ${\rho }_2$ in Fig. 4). The intermediate values of visibilities (c and d) correspond to a state that is not maximally entangled (${\rho }_4$ in Fig. 4). The highest visibility (e) corresponds to the (almost) maximally entangled state (${\rho }_5$ in Fig. 4). Note that the reduced polarization density matrix for photon $\beta$ in cases (b) and (e) represents the same unpolarized state, yet the single-photon interference visibilities are radically different, allowing us to distinguish between the two cases. The error bars are smaller than the data points in the plots.

Figure 4

Experimentally measured concurrence. The blue bars show the results, obtained by our scheme where only subsystem $\beta$ is detected (singles counts). In order to generate the five states ${\widehat{\rho }}_1,\dots ,{\widehat{\rho }}_5$, the parameters $I_H$ and $\mathcal{I}$ [Eq. (1)] were varied (Supplemental Material [20]). We compare our results with the values of concurrence obtained from the full two-qubit tomography (red bars). State ${\widehat{\rho }}_1$ is approximately pure and separable; state ${\widehat{\rho }}_2$ is almost maximally mixed (separable); states ${\widehat{\rho }}_3$ and ${\widehat{\rho }}_4$ are mixed and entangled; state ${\widehat{\rho }}_5$ is almost pure and maximally entangled.