Experimental Validation of Quantum Steering Ellipsoids and Tests of Volume Monogamy Relations

The set of all qubit states that can be steered to by measurements on a correlated qubit is predicted to form an ellipsoid—called the quantum steering ellipsoid—in the Bloch ball. This ellipsoid provides a simple visual characterization of the initial two-qubit state, and various aspects of entanglement are reflected in its geometric properties. We experimentally verify these properties via measurements on many different polarization-entangled photonic qubit states. Moreover, for pure three-qubit states, the volumes of the two quantum steering ellipsoids generated by measurements on the first qubit are predicted to satisfy a tight monogamy relation, which is strictly stronger than the well-known monogamy of entanglement for concurrence. We experimentally verify these predictions, using polarization and path entanglement. We also show experimentally that this monogamy relation can be violated by a mixed entangled state, which nevertheless satisfies a weaker monogamy relation.

Figure 1

Experimental setup. An ultraviolet laser pulse (centered at 390 nm) pumps a type-I SPDC source to generate a pair of polarization-entangled photons. Photon 1, a polarization qubit, is sent straight to Alice to perform projection measurements. Photon 2 passes through a series of beam-displacers (BD). The BDs cause the horizontally polarized ($H$) components (blue) to be walked off, while the vertically polarized ($V$) ones (orange) are transmitted undeviated, where BD1 and BD2 introduce a pair of path qubits, while BD3 eliminates the path qubit generated by BD1. We label the generated quantum state in each stage of state preparation setup (normalization coefficients are omitted), see Supplemental Material [27] for further details. Bob’s qubit, the polarization of photon 2, is first measured by using BD4 and the rotatable wave plates. Meanwhile Charlie’s qubit, encoded in the path degree of photon 2 (modes u and d), is transferred to the polarization degree of photon 2, which is then measured via the standard polarization analysis. The rotatable wave plates before BD4 are within a Mach-Zehnder interferometer, and they introduce a rotation-setting-dependent phase shift between its arms. Thus we implement another unitary operation ($\mathcal{U}$), to undo the effect of this phase shift at each setting. Each rotatable wave plate is mounted on a motorized rotation stage. Symbols used in the figure are HWP, half wave plate; QWP, quarter wave plate; RHWP, rotatable HWP; RQWP, rotatable QWP; FC, fiber-coupled detector; PBS, polarization beam splitter.

Figure 2

The experimentally determined set of steered states (red points) for the family of pure three-qubit states [Eq. (4)] and the mixed state in Eq. (5). In the above blue box, $\alpha$ is fixed at $\pi /2$ and $\beta$ varies from 0 to $\pi /2$—each entangled state belongs to the W class. In the green box, we fix $\beta =\pi /4$ and vary $\alpha$ from $\pi /4$ to $\pi /12$—the entangled states belong to the GHZ class. In the red box, ${\rho }_{ABC}$ in Eq. (5) is prepared. In each box, the upper figure refers to Bob’s steered states while the lower one corresponds to Charlie’s. For each state, we choose 1000 directions at random on the Bloch sphere [33] for Alice’s measurements. Each red point, corresponding to Bob’s or Charlie’s steered state, is reconstructed from $5.0\times {10}^4$ detection events via quantum state tomography. The bottom right inset shows the error bars (one line for each component) of measured red points, which has an average value of 0.007 for each component.

Figure 3

Volume monogamy relations. The $x$ axis and $y$ axis refer to the normalized volumes $V_{B|A}$ and $V_{C|A}$, respectively. The blue solid curve describes $\sqrt{V_{B|A}}+\sqrt{V_{C|A}}=1$, and the orange dashed curve represents $(V_{B|A}{)}^{2/3}+(V_{C|A}{)}^{2/3}=1$. Blue (green) points represent our experimental results for the $W$- (GHZ-)class states in the blue (green) box of Fig. 2. These blue points are almost located on the blue solid line, implying these states saturate the monogamy relation [Eq. (2)] as predicted. The red point characterizes the measurement outcome for the mixed three-qubit state [Eq. (5)]. It is sandwiched by the blue line and orange dashed line, indicating this state violates the monogamy relation [Eq. (2)], but still satisfies the weaker one [Eq. (3)]. The error bars of the experimental data are of the order of ${10}^{-4}$, which is much smaller than the marker size.