Two methods for measuring Bell nonlocality via local unitary invariants of two-qubit systems in Hong-Ou-Mandel interferometers

We describe a direct method to experimentally determine local two-qubit invariants by performing interferometric measurements on multiple copies of a given two-qubit state. We use this framework to analyze two different kinds of two-qubit invariants of Makhlin and Jing et al. These invariants allow us to fully reconstruct any two-qubit state up to local unitaries. We demonstrate that measuring three invariants is sufficient to find, e.g., the optimal Bell inequality violation. These invariants can be measured with local or nonlocal measurements. We show that the nonlocal strategy that follows from Makhlin's invariants is more resource efficient than local strategy following from the invariants of Jing et al. To measure all of the Makhlin's invariants directly one needs to use both two-qubit singlets and three-qubit $W$-state projections on multiple copies of the two-qubit state. This problem is equivalent to a coordinate system handedness measurement. We demonstrate that these three-qubit measurements can be performed by utilizing Hong–Ou–Mandel interference, which gives significant speedup in comparison to the classical handedness measurement. Finally, we point to potential applications of our results in quantum secret sharing.

Figure 1

Local chained singlet projection measurements. With each label we associated a probability of projecting the multicopy system onto singlets (red curves), i.e., HOM anticoalescence. This notation is used for expressing both Mahklin's and Jing's invariants in terms of multicopy singlet projections. Note that every low-rank composite singlet projection can be observed as an invent in a more complex interferometer designed to measure a higher-rank chained singlet projection.

Figure 2

The same as in Fig. 1 but for looped singlet measurement sequences. Note that all measurements corresponding to diagrams from $c_1$ to $c_8$ can be implemented by an interferometer designed for measuring one of the looped diagrams $l_n$ for $n=1,2,3$.

Figure 3

Nonlocal chained and looped singlet projection operations. With each label we associated a probability of projecting the multicopy system onto singlets (red curves). This notation is used for expressing both Mahklin's invariants in terms of multicopy singlet projections. Note that measurements ${\overline{c}}_1$ and ${\overline{c}}_2$ can can be performed in interferometers designed for measuring ${\overline{l}}_1$ (only ${\overline{c}}_1$) or ${\overline{l}}_2$.

Figure 4

Optical circuit for implementing linear-optical measurement of observable ${\widehat{w}}_{k,l,m}$ given in Eq. (6). The circuit implement is composed of a $\pi /2$ phase shift corresponding to a phase factor $i$, a balanced beam splitter (BS) described by Eq. (2), polarizing beam splitters (PBSs), and detectors ${\mathrm{D}}_{p,n}$ that count photons of polarization $p=H,V$ in spatial modes $n=k,l,m$. This circuit registers the outcome $+1$ if the following triples of detectors register a photon each, i.e., $({\mathrm{D}}_{V,k},{\mathrm{D}}_{H,k},{\mathrm{D}}_{V,m})$, $({\mathrm{D}}_{V,l},{\mathrm{D}}_{H,l},{\mathrm{D}}_{V,m})$, $({\mathrm{D}}_{V,k},{\mathrm{D}}_{H,l},{\mathrm{D}}_{H,m})$, and $({\mathrm{D}}_{H,k},{\mathrm{D}}_{V,l},{\mathrm{D}}_{V,m}).$ Similarly, it registers $-1$ for the triples $({\mathrm{D}}_{V,k},{\mathrm{D}}_{H,k},{\mathrm{D}}_{H,m})$, $({\mathrm{D}}_{V,l},{\mathrm{D}}_{H,l},{\mathrm{D}}_{H,m})$, $({\mathrm{D}}_{V,k},{\mathrm{D}}_{H,l},{\mathrm{D}}_{V,m})$, and $({\mathrm{D}}_{H,k},{\mathrm{D}}_{V,l},{\mathrm{D}}_{H,m}).$ The other possible detection events are associated with value 0.

Figure 5

Interferometric configurations for measuring two independent invariants $I_2$ and $I_3$ (or $J_1$ and $J_2$) associated with looped singlet projections $l_1$ and $l_2$ with two and four copies of polarization-encoded $\widehat{\rho }$. Subsystems of a single copy are depicted as black and white discs connected with dashed lines. Photons interfere on beam splitters BS and their coalescence or anticoalescence is detected by detector modules ${\mathrm{D}}_{a,n}$ (Alice) and ${\mathrm{D}}_{b,n}$ (Bob) for $n=1,2,3$. For a detailed analysis of all the possible detection events, see Table 1. To measure these invariants one needs to access at most two or four copies of the investigated state for $I_2$ and $I_3$ (or $J_1$ and $J_2$), respectively. All the measurements are local.

Figure 6

Same as in Fig. 5, but for an interferometer measuring $J_3$. To measure this invariant one needs to access six copies of the investigated state. For a detailed analysis of all the possible detection events, see Table 2.

Figure 7

Same as in Fig. 5, but for an interferometer measuring $I_1$ in terms of (from top) $l_0$, ${\overline{l}}_1$, and ${\overline{l}}_2$. All the measurements are nonlocal (singlet projections are performed between subsystems of Alice and Bob, i.e., black and white dots). To measure this invariant one needs to access at most three copies of the investigated state. For a detailed analysis of all the possible detection events, see Table 3.

Figure 8

The same as in Figs. 1 and 2, but for modified singlet projections (triangular markers) and ${\widehat{\sigma }}_3$ measurements (square markers) involving $W$-state projections [see Eq. (6)] needed for determining the handedness invariants $I_n$ for $n=10,11,15,16,17,18$.