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The randomized quantum marginal problem asks about the joint distribution of the partial traces (“marginals”) of a uniform random Hermitian operator with fixed spectrum acting on a space of tensors. We introduce a new approach to this problem based on studying the mixed moments of the entries of the marginals. For randomized quantum marginal problems that describe systems of distinguishable particles, bosons, or fermions, we prove formulae for these mixed moments, which determine the joint distribution of the marginals completely. Our main tool is Weingarten calculus, which provides a method for computing integrals of polynomial functions with respect to Haar measure on the unitary group. As an application, in the case of two distinguishable particles, we prove some results on the asymptotic behavior of the marginals as the dimension of one or both Hilbert spaces goes to infinity. 

Abstract The current studies of microlensing planets are limited by small number statistics. Follow-up observations of high-magnification microlensing events can efficiently form a statistical planetary sample. Since 2020, the Korea Microlensing Telescope Network (KMTNet) and the Las Cumbres Observatory (LCO) global network have been conducting a follow-up program for high-magnification KMTNet events. Here, we report the detection and analysis of a microlensing planetary event, KMT-2023-BLG-1431, for which the subtle (0.05 mag) and short-lived (5 hr) planetary signature was characterized by the follow-up from KMTNet and LCO. A binary-lens single-source (2L1S) analysis reveals a planet/host mass ratio ofq= (0.72 ± 0.07) × 10⁻⁴, and the single-lens binary-source (1L2S) model is excluded by Δχ²= 80. A Bayesian analysis using a Galactic model yields estimates of the host star mass of $M_{\mathrm{host}}={0.57}_{-0.29}^{+0.33}{M}_{\odot }$ , the planetary mass of $M_{\mathrm{planet}}={13.5}_{-6.8}^{+8.1}{M}_{\oplus }$ , and the lens distance of $D_{\mathrm{L}}={6.9}_{-1.7}^{+0.8}$ kpc. The projected planet-host separation of $a_{\perp }={2.3}_{-0.5}^{+0.5}$ au or $a_{\perp }={3.2}_{-0.8}^{+0.7}$ au, subject to the close/wide degeneracy. We also find that without the follow-up data, the survey-only data cannot break the degeneracy of central/resonant caustics and the degeneracy of 2L1S/1L2S models, showing the importance of follow-up observations for current microlensing surveys. 

Aims.We inspect the microlensing data of the KMTNet survey collected during the 2018-2020 seasons in order to find lensing events produced by binaries with brown dwarf (BD) companions. Methods.In order to pick out binary-lens events with candidate BD lens companions, we conducted systematic analyses of all anomalous lensing events observed during the seasons from 2018 to 2020. By applying a selection criterion of mass ratio between the lens components of 0.03 ≲q≲ 0.1, we identify four binary-lens events with candidate BD companions, namely KMT-2018-BLG-0321, KMT-2018-BLG-0885, KMT-2019-BLG-0297, and KMT-2019-BLG-0335. For the individual events, we present interpretations of the lens systems and measure the observables that can be used to constrain the physical lens parameters. Results.The masses of the lens companions estimated from the Bayesian analyses based on the measured observables indicate high probabilities that the lens companions are in the BD mass regime; that is, 59%, 68%, 66%, and 66% for the four respective events. 

Aims.Recently, there have been reports of various types of degeneracies in the interpretation of planetary signals induced by planetary caustics. In this work we check whether such degeneracies persist in the case of well-covered signals by analyzing the lensing event KMT-2021-BLG-1150, the light curve of which exhibits a densely and continuously covered short-term anomaly. Methods.In order to identify degenerate solutions, we thoroughly investigated the parameter space by conducting dense grid searches for the lensing parameters. We then checked the severity of the degeneracy among the identified solutions. Results.We identify a pair of planetary solutions resulting from the well-known inner-outer degeneracy, and find that interpreting the anomaly is not subject to any degeneracy other than the inner-outer degeneracy. The measured parameters of the planet separation (normalized to the Einstein radius) and mass ratio between the lens components are (s,q)_in~ (1.297, 1.10 × 10⁻³) for the inner solution and (s,q)_out~ (1.242, 1.15 × 10⁻³) for the outer solution. According to a Bayesian estimation, the lens is a planetary system consisting of a planet with a massM_p= 0.88_−0.36^+0.38M_jand its host with a massM_h= 0.73_−0.30^+0.32M_⊙lying toward the Galactic center at a distanceD_L= 3.8_−1.2^+1.3kpc. By conducting analyses using mock data sets prepared to mimic those obtained with data gaps and under various observational cadences, we find that gaps in data can result in various degenerate solutions, while the observational cadence does not pose a serious degeneracy problem as long as the anomaly feature can be delineated. 

Aims. We investigate the data collected by the high-cadence microlensing surveys during the 2022 season in search of planetary signals appearing in the light curves of microlensing events. From this search, we find that the lensing event MOA-2022-BLG-249 exhibits a brief positive anomaly that lasted for about one day, with a maximum deviation of ~0.2 mag from a single-source, single-lens model. Methods. We analyzed the light curve under the two interpretations of the anomaly: one originated by a low-mass companion to the lens (planetary model) and the other originated by a faint companion to the source (binary-source model). Results. We find that the anomaly is better explained by the planetary model than the binary-source model. We identified two solutions rooted in the inner-outer degeneracy and for both of them, the estimated planet-to-host mass ratio, q ~ 8 × 10 −5 , is very small. With the constraints provided by the microlens parallax and the lower limit on the Einstein radius, as well as the blend-flux constraint, we find that the lens is a planetary system, in which a super-Earth planet, with a mass of (4.83 ± 1.44) Μ ⊕ , orbits a low-mass host star, with a mass of (0.18 ± 0.05) M ⊙ , lying in the Galactic disk at a distance of (2.00 ± 0.42) kpc. The planet detection demonstrates the elevated microlensing sensitivity of the current high-cadence lensing surveys to low-mass planets. 

Abstract We present the analysis of five black hole candidates identified from gravitational microlensing surveys. Hubble Space Telescope astrometric data and densely sampled light curves from ground-based microlensing surveys are fit with a single-source, single-lens microlensing model in order to measure the mass and luminosity of each lens and determine if it is a black hole. One of the five targets (OGLE-2011-BLG-0462/MOA-2011-BLG-191 or OB110462 for short) shows a significant >1 mas coherent astrometric shift, little to no lens flux, and has an inferred lens mass of 1.6–4.4 M ⊙ . This makes OB110462 the first definitive discovery of a compact object through astrometric microlensing and it is most likely either a neutron star or a low-mass black hole. This compact-object lens is relatively nearby (0.70–1.92 kpc) and has a slow transverse motion of <30 km s −1 . OB110462 shows significant tension between models well fit to photometry versus astrometry, making it currently difficult to distinguish between a neutron star and a black hole. Additional observations and modeling with more complex system geometries, such as binary sources, are needed to resolve the puzzling nature of this object. For the remaining four candidates, the lens masses are <2 M ⊙ , and they are unlikely to be black holes; two of the four are likely white dwarfs or neutron stars. We compare the full sample of five candidates to theoretical expectations on the number of black holes in the Milky Way (∼10 8 ) and find reasonable agreement given the small sample size. 

Abstract We present the analysis of three more planets from the KMTNet 2021 microlensing season. KMT-2021-BLG-0119Lb is a ∼6 M Jup planet orbiting an early M dwarf or a K dwarf, KMT-2021-BLG-0192Lb is a ∼2 M Nep planet orbiting an M dwarf, and KMT-2021-BLG-2294Lb is a ∼1.25 M Nep planet orbiting a very-low-mass M dwarf or a brown dwarf. These by-eye planet detections provide an important comparison sample to the sample selected with the AnomalyFinder algorithm, and in particular, KMT-2021-BLG-2294 is a case of a planet detected by eye but not by algorithm. KMT-2021-BLG-2294Lb is part of a population of microlensing planets around very-low-mass host stars that spans the full range of planet masses, in contrast to the planet population at ≲0.1 au, which shows a strong preference for small planets. 

Abstract This supplement provides supporting material for Lam et al. We briefly summarize past gravitational microlensing searches for black holes (BHs) and present details of the observations, analysis, and modeling of five BH candidates observed with both ground-based photometric microlensing surveys and Hubble Space Telescope astrometry and photometry. We present detailed results for four of the five candidates that show no or low probability for the lens to be a BH. In these cases, the lens masses are <2 M ⊙ , and two of the four are likely white dwarfs or neutron stars. We also present detailed methods for comparing the full sample of five candidates to theoretical expectations of the number of BHs in the Milky Way (∼10 8 ). 

ABSTRACT We present the observations and analysis of a high-magnification microlensing planetary event, KMT-2022-BLG-0440, for which the weak and short-lived planetary signal was covered by both the KMTNet survey and follow-up observations. The binary-lens models with a central caustic provide the best fits, with a planet/host mass ratio, q = 0.75–1.00 × 10−4 at 1σ. The binary-lens models with a resonant caustic and a brown-dwarf mass ratio are both excluded by Δχ2 > 70. The binary-source model can fit the anomaly well but is rejected by the ‘colour argument’ on the second source. From Bayesian analyses, it is estimated that the host star is likely a K or M dwarf located in the Galactic disc, the planet probably has a Neptune-mass, and the projected planet-host separation is $$1.9^{+0.6}_{-0.7}$$ or $$4.6^{+1.4}_{-1.7}$$  au, subject to the close/wide degeneracy. This is the third q < 10−4 planet from a high-magnification planetary signal (A ≳ 65). Together with another such planet, KMT-2021-BLG-0171Lb, the ongoing follow-up program for the KMTNet high-magnification events has demonstrated its ability to detect high-magnification planetary signals for q < 10−4 planets, which are challenging for the current microlensing surveys.