Witnesses of coherence and dimension from multiphoton indistinguishability tests

Quantum coherence marks a deviation from classical physics, and has been studied as a resource for metrology and quantum computation. Finding reliable and effective methods for assessing its presence is then highly desirable. Coherence witnesses rely on measuring observables whose outcomes can guarantee that a state is not diagonal in a known reference basis. Here, we experimentally measure a type of coherence witness that uses pairwise state comparisons to identify superpositions in a basis-independent way. Our experiment uses a single interferometric setup to simultaneously measure the three pairwise overlaps among three single-photon states via Hong-Ou-Mandel tests. Aside from coherence witnesses, we show the measurements also serve as a Hilbert-space dimension witness. Our results attest to the effectiveness of pooling many two-state comparison tests to ascertain various relational properties of a set of quantum states.

Figure 1

Polyhedron $C$ of overlaps compatible with diagonal states, and quantum violations. (a) Five-vertex polyhedron $C$ in the space of three overlaps $\vec{r}=(r_{AB},r_{BC},r_{AC})$, corresponding to states which are diagonal in a single basis. The convex volumes outside of $C$ correspond to coherent, nondiagonal states. (b) The convex body of overlaps corresponding to quantum states (pink) includes the classical polyhedron $C$ (cyan). (c) The convex body $Q_b$ (blue) corresponds to overlaps obtainable from three states spanning a two-dimensional Hilbert space, and is a strict subset of $Q$. The part of $Q$ outside of $Q_b$ (pink) corresponds to overlaps among states spanning a Hilbert space of dimension $d=3$.

Figure 2

Conceptual scheme for measuring coherence and dimension witnesses. (a) Triangle graph $C_3$ represents states (vertices) and overlap measurements (edges). Overlaps $r_{ij}=\text{tr}\left({\rho }_i{\rho }_j\right)$ are measured via pairwise HOM tests. (b) All overlaps in $C_3$ can be simultaneously measured using this six-mode linear-optical network. Each overlap is estimated via the probability of bunching at the indicated output modes. (c) Bloch-sphere representation of the one-qubit states that maximally violate inequalities (1, 2, 3). Experimentally, qubits are encoded in the polarization degree of freedom of single-photon states. (d) Pictorial model of the time-domain degree of freedom of wave packets associated with the three photons. Tuning of the relative delays allows for a violation of the dimension witness described in the main text. (e) Three-photon states are produced via two-pair generation by spontaneous parametric down conversion (SPDC) in a beta-barium borate (BBO) crystal pumped at 392.5 nm. One photon serves as the trigger signal and the others, labeled $A$, $B$, and $C$, are prepared for the test. The first interferometer layer in (b) is realized with bulk beam splitters made by a half-wave plate ($\lambda /2$) and a polarizing beam splitter (PBS) (see the legend below). Then, photons are coupled into single-mode fibers (SMF), while both polarization and temporal relative delays can be manipulated via additional half-wave plates, polarization controllers (PC), and delay lines. In the end, three in-fiber beam splitters (FBS) connect photon pairs $AB$, $AC$, and $BC$ so as to estimate the overlaps. Each output of FBS is connected to an approximate number-resolving detector (ANRD), consisting of a further FBS and two avalanche-photodiode detectors (APD).

Figure 3

Coherence witness of three-photon polarization states. (a) Amount of violation $W_c(\beta ,\gamma )$ of inequality (1), expressed as the Euclidean distance from the closest face of polyhedron $C$, as a function of the angles $\beta$ ($\gamma$) between states $A\text{−}B$ ($B\text{−}C$) along a great circle of the Bloch sphere. The purple area corresponds to outcome preparations for which we can witness coherence ($W_c>0$). The surface was calculated numerically from the shapes and visibility of the three Hong-Ou-Mandel dips measured during the preparation of the setup. Points $\{S_1,S_2,S_3,S_4\}$ result from suitably prepared polarization states. (b) Projection of the four experimental points on the plane perpendicular to the face of $C$ that saturates inequality (1) (see inset). The maximum Euclidean distance $W_c$ to $C$ is reached by $S_1$. All the error bars have been retrieved from propagating the Poissonian uncertainties associated to single-photon counts.

Figure 4

Output probability distribution for set $S_1$ of polarization states. Above: experimental data, where brighter colors indicate bunched outcomes. The $x$ axis reports the three-photon outcomes at the output of the six-port interferometer. Outcomes associated with photon pairs $\left\{\right(A,B),(B,C),(A,C\left)\right\}$ are marked, respectively, with red, blue, and green. The gray part of the distribution is insensitive to two-photon interference and so it is irrelevant for the estimation of overlaps. Below: expected distribution according to our model based on preliminary measurements of Hong-Ou-Mandel dips. We consider the total variation distance (TVD) as a quantifier of the agreement with the theory. Error bars are computed propagating, via Monte Carlo methods, the Poissonian uncertainties associated to photon counts. We collected $N\sim {10}^4$ fourfold coincidence and we obtained a $\mathrm{TVD}=0.138\pm 0.005$. In the inset, the three single-photon states on the equator of the Bloch sphere.

Figure 5

Dimension witness $W_d$. $W_d$ is the minimum Euclidean distance from an overlap triple $(r_{AB},r_{BC},r_{AC})$ to the body $Q_b$ of triples compatible with states spanning only a two-dimensional Hilbert space. (a) $W_d$ as a function of the two relative time delays between the three photons, in the presence of experimental imperfections. The purple area corresponds to $W_d>0$, i.e., triples $\notin Q_b$. (b) Geometrical representation of the set $Q_b$ (blue), and the set of overlap triples only reachable by states spanning a three-dimensional Hilbert space (magenta). The plane in light blue includes the experimental datum and the green line segment (of length $W_d$) connecting it to the closest point in $Q_b$.

Figure 6

Scheme of the three-photon source. In the experiment, we exploit two-photon pairs emission through spontaneous parametric down-conversion by a nonlinear crystal. Two photons in the pump beam (in purple) are annihilated. Each of the two photons generate a pair, marked with red and orange, respectively. The photons are then split into four auxiliary arms to postselect the input state of the experiment and perform the preliminary characterization of pairwise indistinguishability.

Figure 7

Probability distributions of outputs for different preparations of polarization and time delays. Here, we report the three-photon distributions at the output of the device. Above the experimental data, where brighter colors indicate bunched outcomes. Outcomes associated with photon pairs $\left\{\right(A,B),(B,C),(A,C\left)\right\}$ are marked, respectively, with red, blue, and green. The gray part of the distribution is insensitive to two-photon interference and so it is irrelevant for the estimation of overlaps. Below the expected distribution according to our model based on preliminary measurements of Hong-Ou-Mandel dips. For each distribution we calculate the total variation distance (TVD) with respect to the theory. Error bars are computed propagating, via Monte Carlo methods, the Poissonian uncertainties associated to photon counts. For each preparation we collected $N\sim {10}^4$ fourfold coincidences. States with zero relative delays and different polarization preparation: (a) zero relative angles between qubit states, $S_4$ in the main text ($\mathrm{TVD}=0.122\pm 0.005$); (b), (c) sets of polarization states $S_2$ and $S_3$ of the main text ($\mathrm{TVD}=0.152\pm 0.004$, $\mathrm{TVD}=0.120\pm 0.004$, respectively); (d) distribution for single-photon states with the same polarization state but different time delays for violating the dimension witness ($\mathrm{TVD}=0.126\pm 0.005$).