High-Dimensional Pixel Entanglement: Efficient Generation and Certification

Photons offer the potential to carry large amounts of information in their spectral, spatial, and polarisation degrees of freedom. While state-of-the-art classical communication systems routinely aim to maximize this information-carrying capacity via wavelength and spatial-mode division multiplexing, quantum systems based on multi-mode entanglement usually suffer from low state quality, long measurement times, and limited encoding capacity. At the same time, entanglement certification methods often rely on assumptions that compromise security. Here we show the assumption-free certification of photonic high-dimensional entanglement in the transverse position-momentum degree-of-freedom with a record quality, measurement speed, and entanglement dimensionality. Using a tailored macro-pixel basis, precise spatial-mode measurements, and a modified entanglement witness, we demonstrate state fidelities of up to 94.4% in a 19-dimensional state-space, entanglement in up to 55 local dimensions, and an entanglement-of-formation of up to 4 ebits. Furthermore, we demonstrate the certification of 18-dimensional entanglement in a record 3.6 minutes, showing an improvement in measurement times of more than two orders of magnitude. Our results pave the way for noise-robust quantum networks that saturate the information-carrying capacity of single photons.


I. Introduction
Quantum entanglement plays a pivotal role in the development of quantum technologies, resulting in revolutionary concepts in quantum communication such as superdense coding [1] and device-independent security [2, 3], as well as enabling fundamental tests of the very nature of reality [4][5][6]. While many initial demonstrations have relied on entanglement between qubits, recent advances in technology and theory now allow us to fully exploit high-dimensional quantum systems. In particular, the large dimensionality offered by photonic quantum systems has provided the means for quantum communication with record capacities [7,8], noise-resistant entanglement distribution [9, 10], robust loophole-free tests of local realism [11,12], and scalable methods for quantum computation [13].
In order to make full use of the potential of highdimensional entanglement, it is of key importance to achieve the certification of entanglement with as few measurements as possible. The characterisation of a bipartite state with local dimension d through full state tomography requires O(d 4 ) single-outcome projective measurements [14,15], making this task extremely impractical in high dimensions. More efficient tools for quantifying high-dimensional entanglement involve entanglement witnesses that use semi-definite programming [16], matrix completion techniques [17], or compressed sensing [18]. In this context, it is crucial to certify entan-glement without compromising the security and validity of the applications by introducing assumptions on the state, e.g., purity of the generated state [19] or conservation of quantities [20]. Recent work has shown how measurements in mutually unbiased bases (MUBs) allow the efficient and assumption-free certification of highdimensional entanglement [21,22]. Here we employ and extend this method to improve upon the quality and speed of high-dimensional entanglement certification.
High-dimensional entanglement has been demonstrated in multiple photonic platforms, with encodings in the orbital angular momentum (OAM) [22,23], time-frequency [16], path [24], and transverse positionmomentum degrees of freedom (DoF) [18]. While timefrequency encoding offers the potential of accessing spaces with very large dimensions, the difficulty of measuring coherent superpositions of multiple time-bins hinders the scalability of the technique, and in turn necessitates certification methods that require unwanted assumptions on the reconstruction of the state in question [16]. Path-encoding in integrated photonic circuits offers yet another promising avenue for realising highdimensional entanglement [24]. However, the precise fabrication and control of d(d−1) 2 Mach-Zehnder interferometers required for universal operations in d dimensions poses significant practical challenges as the dimension is increased [25].
Meanwhile, techniques for the creation, manipulation and detection of entanglement in photonic OAM bases have seen rapid progress in recent years [26,27], where devices such as spatial light modulators (SLMs) enable generalized measurements of complex amplitude modes. However, such measurements necessarily suffer from loss [28] and have limited quality [29,30], resulting in long measurement times and reduced entanglement quality in large dimensions [22]. An alternative choice of basis is the discretised transverse position-momentum, or "pixel" basis [31], where the lack of efficient single-photon detector arrays necessitates scanning through localised position or momentum modes, or subtracting a large noise background [32]. Such measurements are also subject to extreme loss, and as a result require long measurement times and strong assumptions on detector noise, such as background or accidental count subtraction.
In this work, we report on significant progress towards overcoming the challenges of scalability, speed, and quality in the characterisation of high-dimensional entanglement with a strategy that combines three distinct improvements in the generation and measurement of spatially entangled modes. Working in the discretized transverse position-momentum DoF, we first tailor our spatialmode basis by adapting it to the characteristics of the two-photon state generated and measured in our experimental setup. Second, we implement a recently developed spatial-mode measurement technique [29] that ensures precise projective measurements in any mode basis of our choice. Third, we generalise a recently developed entanglement dimensionality witness [22] to certify high-dimensional entanglement using any two highdimensional MUBs. Crucially, this allows us to bypass lossy localised mode measurements in the transverse position or momentum bases.
The combination of these improvements in basis optimisation, spatial-mode measurement, and certification tools allow us to certify assumption-free high-dimensional entanglement with a record quality, speed, and dimension, reaching state fidelities up to 98%, certified entanglement dimensionality up to 55 (in local dimension 97), and an entanglement-of-formation of up to 4 ebits. Below we introduce our theoretical framework and elaborate on our improved techniques.

II. Theory
A two-photon state entangled in transverse positionmomentum and produced via the process of spontaneous parametric down-conversion (SPDC) is characterised by its joint-transverse-momentum-amplitude or JTMA (Fig. 1 a), which is well approximated by the function [33,34] where ⃗ k s and ⃗ k i are the transverse momentum components of the signal and idler photons. The parameters σ P and σ S are the widths of the minor and major axis of the JTMA, which are dependent on the pump transverse momentum bandwidth and the crystal characteristics, respectively. In effect, σ P indicates the degree of momen-tum correlation, while σ S is a measure of the number of correlated modes in the state.
Experimental studies of spatial-mode entanglement usually involve projective measurements made with a combination of a hologram and single-mode fiber (SMF), which together act as spatial-mode filter. The addition of collection optics for coupling to an SMF results in the collected bi-photon JTMA of where σ C denotes the collection bandwidth, that is, the size of the back-projected detection mode. The effect of the collection on the correlations is illustrated in Fig. 1, where the Gaussian functions attributed to the SMF modes decrease the probability of detecting higher-order modes associated with the edges of the JTMA. Finally, we include the hologram functions Φ s ( ⃗ k s ) and Φ i ( ⃗ k i ) used for performing projective measurements on the signal and idler photons, respectively, which leads to the two-photon coincidence probability (3) Maximizing the quality and dimensionality of an experimentally generated entangled state involves a complex interplay of optimizing the generation and measurement parameters introduced above. Recent work in this direction has focused on pump shaping [35,36] and entanglement witnesses that adapt to the non-maximally entangled nature of the state [22]. Below, we introduce a new approach to engineering high-quality spatial-mode entanglement based on three distinct improvements: tailoring the spatial-mode measurement basis, precise twophoton measurements via intensity-flattening, and a modified witness for high-dimensional entanglement.

A. Pixel basis design
In our experiment, we choose to work in the discretized transverse-momentum, or macro-pixel basis. This basis provides several advantages over other spatial mode bases. First, projective measurements in the pixel basis or MUB require phase-only measurements. In contrast to the lossy amplitude and phase measurements required for Laguerre-Gauss modes [29,30], phase-only measurements are lossless in theory and produce a MUBindependent count rate. As a result, they maximize photon flux, are resistant to detector noise, and allow us to minimize the total number of measurements required. Second, as the distribution of macro-pixels is determined by circle packing formulas, this basis is compatible with state-of-the-art quantum communication technologies based on multi-core fibres [37,38] and was recently employed for high-dimensional entanglement transport }. The square regions indicate values of ksy and kiy used for generating an optimised pixel basis mask. As such, integrals over the square regions are closely related to the coincidence rate for projections on to the given pair of pixels (up to the additional 2-d integral over ksx and kix), demonstrating the necessity to increase the pixel radii for those positioned over the less intense regions of the JTMA. b) An example 31-dimensional tailored pixel mask. The color function displays the marginal JTMA intensity of the signal photon corresponding to ∫ d 2 ⃗ kiF ( ⃗ ks, ⃗ ki). The line at xs = 0 intersects 5 pixels having 3 unique radii, and its intersections with their boundaries are marked by red dots. These points are mapped through the optical system onto the corresponding boundaries of the regions in momentum space at the crystal shown in the preceding figure. through a commercial multi-mode fibre [39]. Third and most significantly, informed by knowledge of the JTMA, we can optimise this basis to approach a maximally entangled state of the form Φ ⟩ = 1 √ d ∑ m mm ⟩. This in turn enables us to use powerful theoretical techniques that rely on mutually unbiased bases for entanglement certification. In contrast with Procrustean filtering techniques that achieve a similar goal by adding modedependent loss [23,40], this optimisation can be done in a relatively lossless manner by tailoring the size and spacing of individual pixel modes.
As illustrated in Fig. 1, the JTMA indicates that outer macro-pixel modes exhibit the strongest correlation albeit with lower amplitudes, while the inner pixel modes show weaker correlation with higher amplitudes. In order to obtain the highest fidelity to the maximally entangled state, the spacing and size of pixels can be optimised to minimise cross-talk arising from non-vanishing pump transverse bandwidth σ P , while simultaneously equalizing pixel probability amplitudes and maximizing photon count rates. This proceeds as follows: the diameter of the circular region containing the macro-pixel basis is determined by the width σ S of the JTMA major axis, such that the outermost pixel modes for a chosen dimension have sufficient amplitude. The maximum coincidence rate for the outermost pixels is obtained by giving them the maximum radius allowed for a given dimension and the chosen spacing. Taking into account the decreasing correlation strength, we then proceed to choose the radius for the inner pixels such that the photon count rate is equal for all pixels, thereby approximating a state with equal Schmidt coefficients extremely well.

B. Intensity-flattening telescopes
As shown in Fig. 2, the detection of our entangled state depends on a spatial-mode filtering scheme composed of holographic mode projectors implemented on spatial light modulators (SLMs) and single-mode fibers (SMFs). As Eq. (2) indicates, the collection bandwidth of the entangled state is limited by the SMF Gaussian mode width σ C that depends on the specific characteristics of the fiber and coupling optics used. As a result, higher order modes are relatively suppressed, which has an especially adverse effect when measuring coherent high-dimensional superpositions of spatial modes. In recent work, we demonstrated an "intensity-flattening" technique that dramatically improves the quality of general mode-projective measurements at the expense of adding loss [29].
This technique can be extended to the two-photon case, as depicted by the illustration of the collection modes superposed on the JTMA (represented by the purple curves in Fig. 1 a). If the Gaussian back-propagated modes are made wider, higher order modes associated with the edges of the JTMA will couple more efficiently to the SMF, while lower order modes are relatively suppressed. This effectively allows us to make the approx- (2), equating the collected JTMA with the generated one. We imple- ment this technique by using intensity-flattening telescopes (IFTs) that afocally magnify the back-propagated collection modes to a size that optimally increases the two-photon collection bandwidth while keeping loss at a tolerable level. While our previous work demonstrated the IFT technique with classical light from a laser, this work extends it to an entangled state for the first time, demonstrating a marked improvement in reconstructed two-photon state fidelity [22].

C. High-dimensional entanglement witness
Our entanglement certification technique builds on an entanglement-dimensionality witness that uses measurements in mutually unbiased bases (MUBs) in order to efficiently certify high-dimensional entanglement [22]. Here, we briefly introduce this witness, identify its limitations when applied to experiment, and discuss our modifications that significantly improve its utility. The method described in Ref.
[22] allows one to certify high-dimensional entanglement by lower bounding the fidelity F (ρ, Φ) of a measured state ρ to a chosen target state Φ ⟩ = 1 √ d ∑ m mm ⟩. By making two-photon measurements in the standard basis { mn ⟩} and a second basis mutually unbiased (MUB) with respect to it { m kñ * k ⟩}, one can obtain a lower bound to the fidelity F (ρ, Φ), allowing one to use it as a Schmidt number witness. That is, the certified entanglement dimensionality is the max- The MUBs can be constructed in a standard manner by following the prescription by Wootters and Fields [41]: where ω = exp( 2π i d ) are the complex d-th roots of unity and k ∈ {0, . . . , d − 1}.
The standard basis of a position-momentum entangled state normally consists of localised transverse spatial (or momentum) modes. When performing single-outcome projective measurements (as described above) in such a basis, one runs into a limitation. Measurements in a spatially localised basis are subnormalized owing to their limited projection onto the collection mode, i.e., they contain significantly fewer counts than a MUB measurement, which addresses several localised modes at once. As a result, MUB measurements take O(d) less time individually, or O(d 2 ) less time in the bipartite case, than localised standard basis single-outcome projective measurements for a given photon flux. Thus, it is extremely beneficial to have a witness that allows one to certify entanglement with measurements in any two MUBs, without resorting to the standard basis.
In order to do so, we use a property of maximally entangled states. Such states are invariant under transformations (U ⊗ U * ) for any unitary operator U , and thus have the same form in any global mutually unbiased basis, allowing us to express our target state as: The fidelity of our experimental state to this target state F (ρ, Φ) can then be expressed in terms of MUBs and be split into two contributions, While the first term F 1 (ρ, Φ) can be calculated from measurements in one MUB { m kñ * k ⟩}, we show in the supplementary material [42] how one can determine a lower bound for F 2 (ρ, Φ) with measurements in a second MUB { m k ′ñ * k ′ ⟩}, allowing us to calculate a lower bound for the fidelity to the maximally entangled state given by: where the termγ mnm ′ n ′ vanishes whenever (m−m ′ −n+n ′ ) mod (d) ≠ 0, and is equal to 1 d otherwise. This result shows that measurements in any two MUBs can be used to lower bound the fidelity to a target state, and hence to certify the entanglement dimensionality, allowing us to bypass the problematic standard basis and giving significant flexibility in the measurement settings that must be used in experiment. Further note that the witness does not require any assumptions about the state ρ, but is not fully device-independent. While entanglement witnesses based on MUBs can in principle be constructed in a device-independent way [43], deviceindependent certification of the Schmidt number is generally not possible [44].

III. Results
We implement the above improvements in basis optimisation, spatial-mode measurements, and entanglement certification in a two-photon entanglement experiment. Figure 2 a) depicts a tailored diffractive hologram used for performing spatial-mode projective measurements in a 19-dimensional mutually unbiased pixel-mode basis. The pixel orientations are determined by circle packing formulas and the pixel sizes and spacings are optimised according to the JTMA as described above in sec. II A. The experimental setup is depicted in Fig. 2 b). While not discussed in the previous section, an additional consideration involves the use of Gaussian beam simulations to ensure that mode waists and crystal conjugate planes are optimally located in the setup [42]. A grating-stabilised CW laser at 405 nm is loosely focused onto a 5 mm ppKTP crystal to produce photon pairs at 810 nm entangled in their transverse positionmomentum via type-II spontaneous parametric downconversion (SPDC). Single-outcome projective measurements in the optimized macro-pixel basis (or any of its  MUBs) are performed on the photon pairs using diffractive holograms implemented on spatial light modulators (SLMs), which perform transformations such that only modes of interest couple efficiently to single-mode fibers (SMF). Two intensity-flattening telescopes (IFT) are used for ensuring precise, generalized spatial-mode measurements across the entire modal bandwidth of interest [42].
Our modified entanglement witness allows characterizing the generated state by lower bounding its fidelity to the maximally entangled state through measurements in any two mutually unbiased bases. However, if one uses the standard basis and all of its MUBs, the fidelity bound becomes tight and one can estimate the exact fidelity to the maximally entangled state [22]. To showcase the effect of our improvements on the resulting pixel entanglement quality, we measure correlations in all MUBs for subspaces of prime dimension up to d = 19. From these measurements, we obtain record fidelities to the maximally entangled state of ≈ 94% or greater in all subspaces, certifying entanglement dimensions of d ent = d for d = 3, ..., 17 (see Table I). In d = 19, we obtain an estimate of exact fidelity of 94.4 ± 0.4%, certifying an entanglement dimensionality of d ent = 18 (and just below the fidelity bound of 94.7% for d ent = 19). We also calculate the entanglement-of-formation (E oF ), which is an entropic measure of the amount of entanglement needed to create our state [45,46] (please see Appendix for more details). Our E oF reaches a value of 3.8 ± 0.1 ebits in d = 19, which is already higher than previous values reported in the literature [16,18]. All our measures are calculated without subtracting any accidental or background counts. Measured correlation data in all 20 mutually unbiased bases of the d = 19 pixel basis are shown in Fig. 3. The uncertainty in fidelity is calculated assuming Poisson counting statistics and error propagation via a Monte-Carlo simulation of the experiment.
We now demonstrate the speed of our measurement technique, which is enabled by the use of phase-only pixel-basis holograms and our entanglement witness that allows the use of any two pixel MUBs for certifying entanglement. We perform measurements in two MUBs for Using these measurements, we obtain a fidelity bound ofF (ρ, Φ + ) = 56 ± 1% that is above the bound B54 = 54 97 = 0.5567, thus certifying an entanglement dimensionality of dent = 55 ± 1. prime dimensions ranging from d = 19 to 97. In d = 19, we are able to certify a fidelity lower bound of (93±2)% using a total data acquisition time of 3.6 minutes (corresponding to 722 single-outcome measurements), certifying an entanglement dimensionality of d ent = 18 ± 1. In comparison, prior experiments on OAM entanglement required a measurement time two orders of magnitude larger for certifying just d ent = 9 in 11 dimensions [22]. A summary of results for dimensions 19 and higher is shown in Table II. As can be seen, the entanglement quality starts dropping above d ≈ 30, which is a result of the limited resolution of our devices and the SPDC generation bandwidth itself. Nevertheless, we are able to certify an entanglement dimensionality of d ent = 55±1 (See Fig. 4) in a 97-dimensional space and an entanglement-of-formation of E oF = 4.0 ± 0.1 ebits in a 31-dimensional space, which both constitute a record-breaking entanglement dimensionality and entanglement-of-formation certified without any assumptions on the state. A very recent experiment demonstrated comparable state fidelities and E oF values in d = 32 using multi-path downconversion [47]. The required measurement time in the 97-dimensional pixel space was on the order of a day, while using standard basis measurements would require ∼14 years of measurement time for data of similar quality.

IV. Conclusion
We have demonstrated the assumption-free certification of photonic high-dimensional entanglement in the transverse position-momentum degree-of-freedom with a record quality, measurement time, and entanglement dimensionality. These results are made possible through the combination of three new methods: tailored design of the spatial-mode basis, precise two-photon spatialmode measurements, and a versatile entanglement witness based on measurements in mutually unbiased bases (MUBs). As a demonstration of the quality of our entanglement, we achieve state fidelities of ≈ 94% or above in local dimensions of 19 or below, certifying up to 18dimensional entanglement. In addition, the use of any two MUBs enables the measurement of 18-dimensional entanglement in 3.6 minutes, a reduction in measurement time by more than two orders of magnitude over previous demonstrations [22]. Finally, we are able to certify an entanglement dimensionality of at least 55 local dimensions and an entanglement-of-formation of 4 ebits, which to our knowledge is the highest amount certified without any assumptions on the state. While we have used projective single-outcome measurements in our experiment, recent progress on generalized multi-outcome measurement devices [48][49][50] and superconducting detector arrays [51] promises to increase measurement speeds even further. Our results show that high-dimensional entanglement can indeed break out of the confines of an experimental laboratory and enable noise-resistant entanglement-based quantum networks that saturate the information-carrying potential of a photon.

Acknowledgments
This work was made possible by financial support from the QuantERA ERA-NET Co-fund (FWF Project I3773-N36) and the UK Engineering and Physical Sciences Research Council (EPSRC) (EP/P024114/1). NF acknowledges support from the Austrian Science Fund (FWF) through the project P 31339-N27. MH acknowledges funding from the Austrian Science Fund (FWF) through the START project Y879-N27. MP acknowledges fund-ing from VEGA project 2/0136/19 and GAMU project MUNI/G/1596/2019.

APPENDIX
We have demonstrated photonic high-dimensional entanglement in the discretized transverse position-momentum degree of freedom that allows the assumption-free certification of entanglement with a record quality, measurement time, and entanglement dimensionality. In this appendix we provide additional information on the experimental setup used for engineering our entangled state, give a detailed proof of the fidelity bound that we use for certifying high-dimensional entanglement, and provide a description of the entanglement of formation bound we have used in the main text for quantifying the amount of entanglement of the produced state.

A.I. Experimental Setup
A nonlinear ppKTP crystal (1 mm × 2 mm × 5 mm) is pumped with a continuous wave grating-stabilized 405 nm laser in order to generate a pair of photons at 810 nm entangled in their transverse position-momentum via the process of type-II spontaneous parametric down conversion (SPDC). The crystal is temperature-tuned and phase-matching conditions are met by housing it in a custom-built oven that keeps it at 30 ○ C. With the purpose of increasing the dimensionality of the generated state, we consider Gaussian beam propagation and the transformation by two lenses to determine how to shape our beam and loosely focus it on the nonlinear crystal. As shown in Fig. A.1, the pump laser goes through a telescope system where the focal lengths and position of the lenses are chosen such that the final beam waist is located at the crystal with a large enough size to increase the number of generated modes, but without clipping the beam by the crystal aperture.
After the pump is removed with a dichroic mirror (DM), the entangled pair of photons is separated with a polarising beam-splitter (PBS). Each of the photons is made incident on a phase-only spatial light modulator (SLM, Hamamatsu X10468-02) that is placed in the Fourier plane of the crystal using a lens (See Fig. A.2. For the reflected photon to be manipulated by the SLM, we use a half-wave-plate (HWP) to rotate its polarisation from vertical to horizontal. Computer-generated holograms displayed on each SLM allow us to select particular spatial modes of the incident light and convert them into a Gaussian mode, which effectively couples (using a 10X objective) into a single-mode fiber (SMF) that carries these filtered photons to a single photon avalanche detector (SPAD). This effectively allows us to perform projective measurements of any complex spatial mode. Time coincident events between the two detectors are registered by a coincidence counting logic (CC).
The accuracy of the projective measurement performed by the combination of an SLM and SMF is ensured through the use of an intensity flattening telescope (IFT) [29]. With this technique, we afocally decrease the size of the mode that is propagating from the SLM to the objective lens, thus removing the Gaussian component introduced by the use of an SMF and recovering the orthogonality between spatial modes of a given basis. This is especially important when measuring coherent superpositions of spatial modes or pixel MUBs. In our experiment, the back-propagated mode is magnified by a factor of 3 (see Fig. A.3), which allows us to increase the bandwidth of the measured modes  = 950µm) propagates through a telescope system composed of two lenses with f1 = 250 mm and f2 = 50 mm. After the telescope, the pump beam waist is located at a distance of 50 mm away from the second lens, with a size of wp = 188µm. We place the crystal such that this final beam waist is at its longitudinal center. while minimizing losses that would hinder our efficiency. The quality of our measurements also relies on the design of our bases. We choose to work with the Pixel basis, a discrete position-momentum basis composed of circular macro-pixels arranged inside a circle. The mutually unbiased bases to the pixel basis are a coherent superposition of the pixel states with a specific phase relationship between them [41]. As described in the main text, in order to obtain the highest fidelities to a maximally entangled state, we try and equalise the probabilities of measuring each pixel mode by adapting the size of the macro-pixels, the spacing between them, and the size of the circle they are in, according to the characteristics of the jointtransverse-momentum-amplitude (JTMA) of the state we are producing. Doing so, we improve the visibility of the correlations in mutually unbiased pixel bases, which increases the fidelity of our state to the maximally entangled state. Consider a bipartite quantum system with Hilbert space H AB = H A ⊗ H B , with equal local dimensions dim(H A ) = dim(H B ) = d, and an a priori unknown state ρ of this system. For the certification of the Schmidt number w.r.t. the bipartition into subsystems A and B, we want to use only a few measurement settings (local product bases) to give a lower bound on the fidelity F (ρ, Φ) to the target state Φ ⟩ = ∑ n λ n nn ⟩ AB , given by: The entanglement dimensionality can be deduced from the fidelity taking into account that for any state ρ of Schmidt number r ≤ d, the fidelity of Eq. (A.1) is bounded by: where the sum runs over the i m with m ∈ {0, ..., d − 1} such that λ im ≥ λ i ′ m ∀m ≥ m ′ . Hence, any state with F (ρ, Φ) > B r (Φ) must have an entanglement dimensionality of at least r + 1.
Our goal is to obtain a (lower bound on the) fidelity that is as large as possible for the target state whose Schmidt rank (the number of non-vanishing coefficients λ n ) is as close as possible to the local dimension d. Here, we want to focus on the case where the target state is the maximally entangled state Φ + ⟩, i.e., where λ n = 1 √ d ∀ n. The method for certifying high-dimensional entanglement (in particular, the Schmidt number) described in Ref. [22] works in the following way. First, one designates a standard basis { mn ⟩} m,n=0,...,d−1 , measures locally w.r.t. this basis, obtaining estimates for the matrix elements {⟨ mn ρ mn ⟩} m,n from the coincidence counts {N ij } i,j in the chosen basis via Then, one measures in M of d possible mutually unbiased bases { ĩ kj * k ⟩} i,j , where the label k ∈ {0, 1, . . . , d − 1} labels the chosen basis, and the asterisk denotes complex conjugation w.r.t. the standard basis. The local basis vectors are constructed according to the prescription by Wootters and Fields [41], that is, where ω = exp( 2π i d ) are the complex d-th roots of unity, and we hence refer to these bases as Wootters-Fields (WF) bases. When d is an odd prime, which we will assume from now on, the set of all bases { j k ⟩} j together with the standard basis { m ⟩} m forms a complete set of d + 1 mutually unbiased bases with the property that the overlaps between any two basis states of any two different bases from the set have the same magnitude. Measurements in any of these d bases then provide the matrix elements {⟨ĩ kj * k ρ ĩ kj * k ⟩} i,j for any of the M chosen values of k. For simplicity, let us now first concentrate on the case where one measures only in a single of these Wootters-Fields bases, i.e., the case M = 1. We use the corresponding matrix elements to bound the target state fidelity F (ρ, Φ) by splitting it in two contributions, The first term F 1 (ρ, Φ) can be calculated directly from the measurements in the standard basis, while the term F 2 (ρ, Φ) can be bounded from below by the quantityF 2 (ρ, Φ) ≤ F 2 (ρ, Φ), given bỹ where we have noted that ∑ d−1 m,n=0 ⟨ mn ρ mn ⟩ = 1 by construction due to Eq. (A.3), and the prefactorγ mm ′ nn ′ is given byγ With the fidelity bound for this particular target state at hand, one can then certify a Schmidt number of r whenever one finds a fidelity (bound) larger than r d , see [22]. To prove this, we focus on the information given by the elements of the density matrix that we obtain from measuring in the WF basis. In particular, the sum over the correlated WF matrix elements can be split into three terms, i.e., which are given by where we have introduced the quantity We can then bound the real part appearing in the summands of Σ 3 by their modulus, i.e., and use the Cauchy-Schwarz inequality and the spectral decomposition ρ = ∑ i p i ψ i ⟩⟨ ψ i such that Finally, one notes that for any single basis choice (labelled by k), one has which vanishes whenever (m−m ′ −n+n ′ ) mod (d) ≠ 0, and is equal to d otherwise, resulting inγ mm ′ nn ′ as in Eq. (A.7).
A.III. Changing the designation of 'standard basis' The fidelity bound we have derived is based on designating one of the measured bases as 'standard basis', while the other is a WF basis constructed w.r.t. the standard basis. However, the bases are mutually unbiased and because the target state is maximally entangled, we can also reverse their roles to obtain another fidelity bound that can differ in its value from the original bound, depending on the estimated matrix elements (i.e., depending on the experimental data). Moreover, if one has measured in several of the WF bases, any of them can be designated the 'standard basis' and one may combine it either with the original standard basis, or with any of the other WF bases for which one has taken data.

a. Exchanging standard basis and WF basis
To see which modifications this entails, let us first spell out the specific relationship between these bases. Suppose we designate the WF basis { ñ k ⟩} n labelled by k as and since c (k, st.) mnm ′ n ′ = c mnm ′ n ′ , the coefficientγ mm ′ nn ′ in Eq. (A.7) is unaffected. We can thus simply exchange the role of the standard basis and any one of the WF bases to obtain a fidelity bound of the form In this section, we discuss how we bound the entanglement of formation (E oF ) of our bipartite state from the measurement data for any two MUBs. Here, the E oF quantifies how many Bell states are required to convert to a single copy of our high-dimensional entangled state via local operations and classical communication (LOCC) [45].
Using the uncertainty relations reviewed in Ref. [46], the bound on the E oF is given by  Figure A.4 shows the E oF achieved in our experiment as a function of dimension, compared with the theoretical maximum of log 2 (d). As can be seen, the E oF increases until approximately d = 31, where it reaches a maximum value of 4.0 ± 0.1 ebits. As the dimension is increased further, experimental imperfections and the inherent dimensionality limits of the source bring this number back down.