Experimental Bound Entanglement in a Four-Photon State

Bound entanglement is central to many exciting theoretical results in quantum information processing, but has thus far not been experimentally realized. In this work, we consider a one-parameter family of four-qubit Smolin states. We experimentally produce these states in the polarization of four optical photons produced from parametric down-conversion. Within a range of the parameter, we show that our states are entangled and undistillable, and thus bound entangled. Using these bound-entangled states we demonstrate entanglement unlocking.

Figure 1

Experimental setup to generate four-photon Smolin states. (a) We generate a family of Smolin states by randomly applying unitaries, using two pairs of liquid-crystal variable phase retarders (LCRs), to the initial $|{\varphi }^{+}{⟩}_{AB}\otimes |{\varphi }^{+}{⟩}_{CD}$ state produced by two down-conversion sources (SPDC). For each party $A$, $B$, $C$, and $D$, the polarization is analyzed with a combination of a half-wave plate (HWP), quarter-wave plate (QWP) and polarizing beam splitter (PBS). Birefringent crystals ($T$, $T^{\prime }$, and $c$) are used to compensate temporal and transverse walk-off in the sources. (b) A two-photon interferometer is used to project onto $|{\varphi }^{-}{⟩}_{AC}$ for entanglement unlocking between parties $B$ and $D$. The delay $\Delta \tau$ is adjusted for optimum two-photon interference [34], and the QWP is used to set the phase.

Figure 2

Experimental tests for bound entanglement. We measured the expectation value for the entanglement witness (blue diamonds, right-hand axis) and the minimum PT eigenvalue (red circles, left-hand axis) for various levels of white noise. The lines are best fits to the data and the error bars correspond to 1 standard deviation as determined by Monte Carlo simulation. Each eigenvalue shown is the smallest among all those calculated from the set of bipartite cuts $(AB):(CD)$, $(AC):(BD)$, and $(AD):(BC)$. Using our experimental data, we find that our family of generated Smolin states is entangled within the shaded region and PPT in the hatched region. In the overlapping region, the states are both entangled and PPT, i.e., bound entangled. In particular, we successfully produce a bound-entangled state when the noise level is $p=0.49$; the entanglement witness is $-0.159\pm 0.008$, and the minimum PT eigenvalue is $0.0069\pm 0.0008$. Our state without additional noise is entangled but definitely not PPT, with a negative minimum PT eigenvalue similar to Ref. 24. Our experimental results show that a substantial amount of noise is required to turn this value firmly positive and reach bound entanglement.

Figure 3

Experimentally measured density matrix of a noisy Smolin state. The reconstructed density matrix (real part on the left, imaginary part on the right) for a noise level of $p=0.49$. The fidelity with the target state is $(96.83\pm 0.05)\%$, and the measured witness and smallest PT eigenvalue are $-0.159\pm 0.008$ and $0.0069\pm 0.0008$, respectively. From these values we see that this state is both entangled and PPT, i.e., bound entangled.

Figure 4

Unlocking of entanglement from a bound-entangled state. By performing a joint measurement on the qubits of parties $A$ and $C$, specifically by projecting them on the state $|{\varphi }^{-}{⟩}_{AC}$ using the Bell-state measurement shown in Fig. 1b, entanglement is unlocked between parties $B$ and $D$. Here, we use our bound-entangled Smolin state with noise level $p=0.49$. We reconstruct the density matrix of $B$ and $D$ (real part on the left, imaginary part on the right), given a successful Bell-state measurement on $A$ and $C$. This state is entangled with a negative minimum PT eigenvalue of $-0.0160\pm 0.0035$ and a tangle of $0.00105\pm 0.00046$, experimentally demonstrating entanglement unlocking. We achieve a fidelity of $(99.45\pm 0.05)\%$ with the state expected, given the reconstructed four-qubit density matrix of Fig. 3.