Classification of quantum states of light using random measurements through a multimode fiber

TL;DR

The work tackles the resource-intensive challenge of quantum-state tomography by introducing a random-measurement protocol implemented with a spatial light modulator and a multimode fiber to map unknown states into high-dimensional output spaces. By analyzing the first two moments and full distributions of photocurrents, two-fold coincidences, and normalized second-order correlations across many random projections, the authors infer state properties such as the number of occupied modes $d$, purity $\\mathcal{P}$, and entanglement dimensionality $D$, achieving state classification without tomography. The approach is demonstrated with a range of ground-truth states, including spectrally entangled biphotons in both non-dispersive and dispersive regimes, revealing distinct signatures of indistinguishability and dispersion in the statistics of $g^{(2)}$ and $C$, and enabling effective classification even when $C$ alone is insufficient. The results motivate a resource-efficient pathway for high-dimensional quantum-state characterization and point toward integration with reconfigurable linear-optical circuits and multi-outcome detectors for near-term tomography and device benchmarking in quantum technologies.

Abstract

Extracting meaningful information about unknown quantum states without performing a full tomography is an important task. Low-dimensional projections and random measurements can provide such insight but typically require careful crafting. In this paper, we present an optical scheme based on sending unknown input states through a multimode fiber and performing two-point intensity and coincidence measurements. A short multimode fiber implements effectively a random projection in the spatial domain, while a long-dispersive multimode fiber performs a spatial and spectral projection. We experimentally show that useful properties -- i.e., the purity, dimensionality, and degree of indistinguishability -- of various states of light including spectrally entangled biphoton states, can be obtained by measuring statistical properties of photocurrents and their correlation between two outputs over many realizations of unknown random projections. Moreover, we show that this information can then be used for state classification.

Figures (10)

• Figure 1: Concept and experimental scheme: An unknown state of light $\rho_{\text{unknown}}$ is evolved through a random interferometer and mapped onto a high-dimensional output that is subjected to a measurement step. The statistical features of the outcomes are used to infer properties of the state. In the experiment, ground-truth states of light, as listed in Sec.\ref{['sec:Ground-truth states']}, have been generated through Spontaneous Parametric Down-Conversion (SPDC) and Superluminescent Diode (SLD). Such states are randomly launched into a multimode fiber (MMF) using the spatial light modulators $\text{SLM}_1$ and $\text{SLM}_2$ that are placed on the Fourier plane of two input orthogonal polarization channels of the MMF. At each setting of random measurement, randomly generated holograms are displayed on the SLMs, and a state is controlled and evolves through the MMF. We probed the output states on randomly selected two-mode subspaces assigned at two different diffraction-limited spots on the near-field plane of the MMF and associated with orthogonal polarizations, labeled $\left|\right.\!{H_i}\!\left.\right\rangle$ and $\left|\right.\!{V_j}\!\left.\right\rangle$. Photocurrents ($I$), two-fold coincidence counts ($C$), and normalized second-order intensity correlation ($g^{(2)}$) are measured by avalanche photodiodes (APDs) and a coincidence electronic circuit with the coincidence window of 2.5 ns. (L: lenses, HWP: half-wave plate, PBS: polarizing beamsplitter).
• Figure 2: Statistics of two-fold coincidences: (a-c) Statistical distributions of experimental normalized second-order correlation $P_2(g^{(2)})$ for (a) heralded single-photon state (light green), incoherent source (dark blue) and the two-photon Fock state $\left|\right.\!{2}\!\left.\right\rangle$ (magenta) in the 55-cm long MMF. The first two states have the means at the accidental coincidence (the red dashed line, $\overline{g^{(2)}}=1$), while the two-photon Fock state shows $\overline{g^{(2)}}>1$. (b)$P_2(g^{(2)})$ for the group of two-mode states in a 55-cm long MMF: Indistinguishable biphoton state (blue), N=2 N00N state (orange), and distinguishable biphoton state (red). Their distributions are different owing to the presence of quantum interference. The histogram of indistinguishable biphotons shows the flat distribution as predicted by the probability density functions (PDF) $P_2(g^{(2)})$, represented by the blue solid line. The red curve represents the prediction for distinguishable biphotons, Eq. \ref{['eq:PDFg2Dismain']}. (c)$P_2(g^{(2)})$ for the group of states evolving through the 25-m long MMF: Incoherent source (light yellow), indistinguishable (green) and distinguishable (light magenta) biphoton states. The incoherent state has the means at $\overline{g^{(2)}}=1$, while for the biphoton states ($g^{(2)}>1$) the width of the indistinguishable case is broader than that of the distinguishable case. (d-f) Correlation of two-fold coincidences $C/\overline{C}$ and normalized second-order correlation $g^{(2)}$ for the ground-truth states as previously labelled. Each circle on the scatter plots displays each outcome of random measurements. (g-i) Statistical distributions of two-fold coincidences $P_2(C/\overline{C})$ for the corresponding ground-truth states. The solid curves indicate the PDF of two-fold coincidences $P_2(C/\overline{C})$ for pure $d$-dimensional spatially maximally entangled biphoton states, Eq. \ref{['eq:PDF_Cmain']}. The dashed curves represent the PDF of accidental coincidences $P_2(R/\overline{R})$, Eq. \ref{['eq:PDF_ACCmain']}. (g)$P_2(C/\overline{C})$ for the group of states propagating through 25-m MMF: Indistinguishable and distinguishable biphoton states and incoherent state. The first two biphoton states show the same distribution which cannot be described by the PDF of expected accidental coincidences $P_2(R/\overline{R})$ for $d\approx14$ predicted from the visibility of intensity. This is in contrast to the distribution of the incoherent source that is classically well predicted from the visibility of intensity with $d=20$. (h)$P_2(C/\overline{C})$ for the group of two-mode states propagating through the 55-cm long MMF: Indistinguishable biphoton state, N=2 N00N state, distinguishable biphoton state, and incoherent source, presents no statistical difference between their distributions. (i)$P_2(C/\overline{C})$ for the group of single-mode states: single-photon and two-photon Fock states show the same distribution of $C/\overline{C}$ with $d=1$.
• Figure 3: Classification of input states from intensity and second-order correlations Visibility $\mathcal{V}_I$ and second-order correlations $\mathcal{V}_{g^{(2)}}$ for the different input states. The clustering patterns clearly show that the different classes of input states can be separated using only these two statistical features. In particular, it is possible to discriminate both the distinguishability and the number of occupied modes of the input states.
• Figure S1: Simulated statistical distributions of two-fold coincidences: (a) Two-mode states. The indistinguishable N00N state experiences the dephasing effect at the input of a random interferometer. (b) Two-photon Fock state. $P_{2}(R/\overline{R})$ has the same form as $P_{2}(C/\overline{C})$ when $d=1$.
• Figure S2: Simulated statistical distributions of normalized second-order correlation: Indistinguishable two-photon state follows the uniform distribution, whereas fully distinguishable states follow Eq.\ref{['eq:PDFg2Dis']}. The effect of dephasing in the case of the $N=2$ N00N state causes the reduction in $\mathcal{V}_{g^{(2)}}$. The simulation includes the presence of accidental coincidences.