Abstract
Entanglement is the powerful and enigmatic resource central to quantum information processing, which promises capabilities in computing, simulation, secure communication, and metrology beyond what is possible for classical devices. Exactly quantifying the entanglement of an unknown system requires completely determining its quantum state, a task which demands an intractable number of measurements even for modestlysized systems. Here we demonstrate a method for rigorously quantifying highdimensional entanglement from extremely limited data. We improve an entropic, quantitative entanglement witness to operate directly on compressed experimental data acquired via an adaptive, multilevel sampling procedure. Only 6,456 measurements are needed to certify an entanglementofformation of 7.11 ± .04 ebits shared by two spatiallyentangled photons. With a Hilbert space exceeding 68 billion dimensions, we need 20milliontimes fewer measurements than the uncompressed approach and 10^{18}times fewer measurements than tomography. Our technique offers a universal method for quantifying entanglement in any large quantum system shared by two parties.
Introduction
Achieving a quantum advantage for information processing requires scaling quantum systems to sizes that can provide significant quantum resources, including entanglement. Large quantum systems are now realized across many platforms, including atomic simulators beyond 50 qubits^{1,2,3}, nascent superconducting and trappedionbased quantum computers^{4,5}, integratedphotonic circuits^{6,7,8,9,10}, and photon pairs entangled in highdimensional variables^{11,12,13,14,15,16}.
As quantuminformationbased technologies mature, it will become useful to separate the physical layer providing quantum resources (e.g., trapped ions, photons) from the logical layer that utilizes those resources. For example, many imperfect qubits may form one logical qubit^{17,18} or thousands of atoms may coherently act as a singlephoton quantum memory^{19,20}. As with classical communication and computing, protocols and algorithms will be implemented in the logical layer with minimal concern for the underlying platform. Because realworld systems are varied and imperfect, the quantum resources they provide must be characterized before use^{17}.
Certifying an amount of entanglement in a large quantum system is an essential but daunting task. While entanglement witnesses^{21,22} and Bell tests^{23} can reveal entanglement’s presence, quantification generally requires a full estimation of the quantum state^{24}. Beyond moderately sized states, the number of parameters to physically measure (i.e., the number of the measurements) becomes overwhelming, making this approach unviable for current and future largescale quantum technologies.
Any practical method for quantitative entanglement certification must require only limited data. Two ideas can dramatically reduce the needed measurement resources. First is the development of quantitative entanglement witnesses, which bound the amount of entanglement without full state estimation^{25,26,27,28}. In a recent landmark experiment, 4.1 entangled bits (ebits) of highdimensional biphoton entanglement was certified using partial state estimation^{29}. One ebit describes the amount of entanglement in a maximally entangled, twoqubit state^{24}.
Second, prior knowledge can be exploited to economize sampling. Certain features, or structure, are expected in specific systems. In highly entangled quantum systems, for example, some observables should be highly correlated, the density matrix will be low rank, or the state may be nearly pure. Such assumptions can be paired with numerical optimization to recover signals sampled below the Nyquist limit. One popular technique is compressed sensing^{30}, which has massively disrupted conventional thinking about sampling. Applied to quantum systems, compressed sensing reduced measurement resources significantly for tasks, including tomography^{31,32,33,34,35,36,37} and witnessing entanglement^{38,39}.
Computational recovery techniques have substantial downsides. Because they are estimation techniques, conclusions drawn from their results are contingent on the veracity of the initial assumptions. They are therefore unsuitable for closing loopholes or verifying security. Numerical solvers are often proven correct under limited noise models and require handtuned parameters, potentially adding artifacts and complicating error analysis. Finally, the computational resources needed become prohibitive in very large systems. The largest quantum systems characterized using these approaches remain considerably smaller than stateoftheart.
Here we provide an approach to entanglement quantification that overcomes these downsides. First, we improve an entropic, quantitative entanglement witness to operate on arbitrarily downsampled data. Then we develop an adaptive, multilevel sampling procedure to rapidly obtain compressed distributions suitable for the witness. Crucially, our sampling assumptions are independent of the entanglement certification, so our method can guarantee security. Because we avoid numerical optimization, error analysis is straightforward and few computational resources are needed.
Results
Entropic witnesses of highdimensional entanglement
Entanglement is revealed when subsystems of a quantum state are specially correlated. A common situation divides a system between two parties, Alice and Bob, who make local measurements on their portion. Given two mutually unbiased, continuous observables \(\widehat {\mathbf{x}}\) and \(\widehat {\mathbf{k}}\), they can measure discrete joint probability distributions P(X_{a}, X_{b}) and P(K_{a}, K_{b}) by discretizing to pixel sizes Δ_{X} and Δ_{K}. Here bold notation indicates that X and K may (though need not) represent multidimensional coordinates. For example, X and K might represent cartesian position and momentum that can be decomposed into horizontal and vertical components such that X = (X, Y) and K = (K^{(x)}, K^{(y)}).
A recent, quantitative entanglement witness^{40} uses these distributions to certify an amount of entanglement:
where, for example, H(AB) is the conditional Shannon entropy for P(A, B). E_{f} is the entanglement of formation, a measure describing the average number of Bell pairs required to synthesize the state. Equation (1) does not require fullstate estimation but depends on an informed choice of \(\widehat {\mathbf{x}}\) and \(\widehat {\mathbf{k}}\). Still, in large systems, measuring these joint distributions remains oppressive. For example, if X_{a} has 100 possible outcomes, determining P(X_{a}, X_{b}) takes 100^{2} joint measurements. Describing quantum uncertainty with informationtheoretic quantities is increasingly popular^{41,42}. Entropies naturally link physical and logical layers and have useful mathematical properties. In particular, many approximations to the joint distributions can only increase conditional entropy. Because Eq. (1) bounds E_{f} from below, any such substitution is valid.
Improving an entropic entanglement witnesses for use with limited data
We use two entropic shortcuts to improve the entanglement witness. First, if the system is highly entangled, and \(\widehat {\mathbf{x}}\) and \(\widehat {\mathbf{k}}\) are well chosen, the joint distributions will be highly correlated; a measurement outcome for X_{a} should correlate to few outcomes for X_{b}. The distributions are therefore highly compressible. Consider replacing arbitrary groups of elements in P(X_{a}, X_{b}) with their average values to form a multilevel, compressed estimate \(\tilde P({\mathbf{X}}_{\mathrm{a}},{\mathbf{X}}_{\mathrm{b}})\). By multilevel, we mean that the new, estimated distribution will appear as if it was sampled with varying resolution—fine detail in some regions and coarse detail in others. Because coarse graining cannot decrease conditional entropy, Eq. (1) remains valid for \(\tilde P({\mathbf{X}}_{\mathrm{a}},{\mathbf{X}}_{\mathrm{b}})\) and \(\tilde P({\mathbf{K}}_{\mathrm{a}},{\mathbf{K}}_{\mathrm{b}})\) (see Supplemental Material: Proof arbitrary coarsegraining cannot decrease conditional entropy).
Good estimates for \(\tilde P({\mathbf{X}}_{\mathrm{a}},{\mathbf{X}}_{\mathrm{b}})\) and \(\tilde P({\mathbf{K}}_{\mathrm{a}},{\mathbf{K}}_{\mathrm{b}})\) can be efficiently measured by sampling at high resolution in correlated regions and low resolution elsewhere. Note that the original (P) and estimate (\(\tilde P\))) are full correlation matrices with N elements, but only \(M \ll N\) values measured to specify \(\tilde P\). The witness is valid for arbitrary downsampling; it works best when the approximate and actual distributions are most similar but can never overestimate E_{f} or allow false positives.
Second, if the observables are multidimensional such that they can be decomposed into d marginal, component observables (e.g., horizontal and vertical components) \(\widehat {\mathbf{x}} = (\hat x^{(1)},\hat x^{(2)},...,\hat x^{(d)})\) (similar for \(\widehat {\mathbf{k}}\)), the conditional entropies have the property
with equality when P(X_{a}, X_{b}) is separable. If we expect nearly separable jointdistributions, the reduced, marginal jointdistributions \(P(X_{\mathrm{a}}^{(i)},X_{\mathrm{b}}^{(i)})\) can be separately measured but still capture nearly all of the correlations present. For example, in a twodimensional cartesian scenario, we might separately measure horizontal correlations P(X_{a}, X_{b}), \(P(K_{\mathrm{a}}^{(x)},K_{\mathrm{b}}^{(x)})\) and vertical correlations P(Y_{a}, Y_{b}), \(P(K_{\mathrm{a}}^{(y)},K_{\mathrm{b}}^{(y)})\). For dcomponent observables, this is a dth power reduction in the number of measurements. Like the first shortcut, this approximation also cannot overestimate E_{f}.
Combining both improvements, our new quantitative entanglement witness is
Proofofconcept experimental setup
As a test experimental system, we use photon pairs entangled in their transversespatial degrees of freedom^{43,44}, where the transverse plane is perpendicular to the optic axis. Our test bed, given in Fig. 1a, creates photon pairs via spontaneous parametric downconversion (see “Methods”). Generated photons are positively correlated in transverseposition and anticorrelated in transversemomentum. This state closely approximates the original form of the Einstein–Podolsky–Rosen paradox. Because position \(\widehat {\mathbf{x}} = (\hat x,\hat y)\) and momentum \(\widehat {\mathbf{k}} = (\hat k^{(x)},\hat k^{(y)})\) (where \(\widehat {\mathbf{k}} = \widehat {\mathbf{p}}/\hbar\)) observables are continuous, this state is very high dimensional.
After creation, the twin photons are separated at a beam splitter and enter identical measurement apparatuses, where a basis selection system allows for interrogating position or momentum. A digital micromirror device (DMD)—an array of individually addressable micromirrors—is placed in the output plane. By placing patterns on the signal and idler DMDs and using coincidence detection, rectangular regions of the position or momentum jointdistributions are sampled at arbitrary resolution.
Adaptive, multilevel data acquisition
We measure jointdistributions \(\tilde P(X_{\mathrm{a}},X_{\mathrm{b}})\), \(\tilde P(Y_{\mathrm{a}},Y_{\mathrm{b}})\), \(\tilde P(K_{\mathrm{a}}^{(x)},K_{\mathrm{b}}^{(x)})\), and \(\tilde P(K_a^{(y)},K_{\mathrm{b}}^{(y)})\). Finding compressed distributions requires a multilevel partitioning of the jointspace that is not known a priori. Our adaptive approach is inspired by quadtree image compression^{45}. An example is shown in Fig. 1b–g. First, all DMD mirrors are directed toward the detector to obtain a total coincidence rate R_{T}. Then the jointspace is divided into four quadrants (c), which are independently sampled. If the count rate in the ith quadrant exceeds a threshold αR_{T} (0 ≤ α ≤ 1), the region is recursively split and the process is repeated. The algorithm rapidly identifies important regions of the jointspace for highresolution sampling.
We set the maximum resolution of our system to 512 × 512 pixelsperphoton for a 512^{4}dimensional jointspace. The recovered jointdistributions in position and momentum are given in Fig. 2a–d. Figure 2e, f show \(\tilde P(X_{\mathrm{a}},X_{\mathrm{b}})\) with the partitioning overlaid. These display the expected strong position and momentum correlations. A histogram showing the number of partitions at various scales is given in Fig. 2g; most partitions are either 1 × 1 or 2 × 2 pixels in size. Only 6456 partitions are needed to accurately cover the 512^{4}dimensional space—an astonishing 20millionfold improvement versus using the unimproved witness. Over 10^{21} measurements are needed to perform full, unbiased tomography.
The entanglement witness (Eq. (3)) applied to the data in Fig. 2 is shown in Fig. 3. For short acquisition times, there is a systematic bias toward estimating a large E_{f}. This occurs because many of the poorly correlated regions have not yet accumulated any detection events, resulting in a systematic bias toward low conditional entropies. Statistical error is low in this region because the highly correlated regions have high count rates and rapidly reach statistical significance. With additional measurement time, the initial bias diminishes and statistical error decreases. To our knowledge, 7.11 ± .04 ebits is the largest quantity of entanglement experimentally certified in a quantum system. More than 14 maximally pairwiseentangled logical qubits are needed to describe an equal amount of entanglement. We do not require advanced postprocessing such as numerical optimization, estimation, or noise reduction; however, we do postselect on coincident detection events and optionally subtract accidental coincidences (see “Methods”). Our witness does not explicitly require any postprocessing and is suitable for use in adversarial scenarios given a pristine experimental system.
The performance of our technique as a function of maximum discretization resolution is shown in Fig. 4. Figure 4a shows the approximate distribution partition number as a function of discretization dimension and the improvement factor over naive sampling. Figure 4b shows the certified E_{f}, with and without accidental subtraction, along with the ideal E_{f} for our source under a doubleGaussian approximation^{44}. Because our pump laser is not Gaussian (Fig. 1a), the actual E_{f} is slightly less but difficult to simulate. Error bars enclosing two standard deviations are scarcely visible. For low resolution, <1000 measurements witness entanglement. Progressively refining to higher effective resolution allows more entanglement to be certified until the maximum is reached.
Discussion
We have shown an efficient method for performing informationbased entanglement certification in a very large quantum system. An alternative, important metric for quantifying entanglement in highdimensional systems is the entanglement dimensionality, or Schmidt rank, which describes the number of modes over which the entanglement is distributed^{22,46,47,48}. In contrast, entanglement measures quantify entanglement as a resource of entangled bits without regard for their distribution. Efficiently certifying the entanglement dimensionality faces many of the same problems as certifying a number ebits, such as the intractability of full tomography and the desire to avoid side effects from prior assumptions. Recently, Bavaresco et al. used measurements in only two bases to efficiently certify over nine entangled dimensions between orbitalangularmomentum entangled photon pairs without special assumptions about the underlying state^{49}.
The number of entangled dimensions and the number of entangled bits are complementary but distinct characterizations of entanglement^{50}. If a density matrix cannot be decomposed into pure states with Schmidt rank <d, then the state is at least ddimensionally entangled. However, a ddimensional entangled state may possess an arbitrarily small amount of entanglement. Consider a system with a large Schmidt rank but where one coefficient of the Schmidt decomposition is much larger than the others. This system will have a large entanglement dimensionality but require few entangled bits to synthesize. In this way, a given entanglement dimensionality D provides an upper bound on the entanglement of formation E_{f} such that \(0 < E_{\mathrm{f}} \le \log _2D\). In contrast, a given E_{f} provides a lower bound to the entanglement dimensionality \(D \ge 2^{E_{\mathrm{f}}}\), describing the situation where all D dimensions are maximally entangled. Our quantitative witness therefore also certifies entanglement dimensionality but may dramatically underestimate when the target system is not nearmaximally entangled (e.g., with additive noise or nonuniform marginals). In our case, we certify 2^{7.11} ≥ 138 maximally entangled dimensions with background subtraction and 2^{3.43} ≥ 10 maximally entangled dimensions without background subtraction. To our knowledge, 10 entangled dimensions is the largest certified entanglement dimensionality without assumptions about the state.
Our approach shows a path forward for certifying quantum resources in large quantum systems, where we exploit prior knowledge without conventional downsides. We show the power of an informationtheoretic approach to characterizing quantum systems, and how compression can be leveraged without computational signal recovery. Though the method presented here is limited to Einstein–Podolsky–Rosentype systems where entanglement is shared by two parties, we expect that similar techniques for manybody systems utilizing higherorder correlations will soon follow.
Methods
Experimental apparatus
The 810nm, spatially entangled photon pairs are produced via spontaneous parametric downconversion (SPDC)^{44}. The pump laser is a 405nm diode laser (CrystaLaser DL405025SO) attenuated to 7.9 mW with a 356 μm (x) × 334 μm (y) beam waist. A spectral cleanup filter (Semrock Versachrome TBP01400/16) removes unwanted the 810nm light. The pump laser is not spatially filtered. The nonlinear crystal is a 3mmlong BiBO crystal oriented for typeI, degenerate, collinear SPDC. The crystal is held at 32.3 °C in an oven for longterm stability. A lowpass interference filter (Semrock LP442) removes remaining pump light, followed by a telescope relay system (f_{1} = 50 mm, f_{2} = 100 mm) that magnifies the SPDC field ≈2×. A halfwaveplate and polarizing beamsplitter allow switching between imaging (\(\widehat {\mathbf{x}}\)) and Fouriertransforming (\(\widehat {\mathbf{k}}\)) beampaths; a beam block is placed in the unused path.
The DMDs (TI Lightcrafter 4500) are computer controlled via a digital video port (HDMI). A 512 × 1024 physicalpixel area was used for data given in this manuscript. Because the DMD has twice the vertical pixel density, this corresponds to a square area. The 10mm effective focal length, aspheric lenses (Thorlabs AC080010) couple light into 100 micron core multimode fibers connected to photoncounting detector modules (Excelitas SPCMAQ4C10). The 810/10 nm bandpass filters (Thorlabs FBS81010) are placed before the fiber coupling. A timecorrelated singlephoton counting module (PicoQuant HydraHarp400) produces histograms of photonpair relative arrival times. We postselect on coincident detections within a 1ns coincidence window centered on the histogram peak. With all DMD mirrors pointed toward the detectors, there are approximately 26,400 total coincidences/s.
Data collection
The apparatus must be adjusted to separately measure the four reduced, jointprobability distributions P(X_{a}, X_{b}), P(Y_{a}, Y_{b}), \(P(K_{\mathrm{a}}^{(x)},K_{\mathrm{b}}^{(x)})\), and \(P(K_{\mathrm{a}}^{(y)},K_{\mathrm{b}}^{(y)})\). For example, to access the horizontal, jointposition distribution P(X_{a}, X_{b}), we adjust the halfwaveplates to direct light down the imaging beampaths so the DMDs lie in an image plane of the nonlinear crystal. To access a particular, rectangular element of the distribution, local, onedimensional “tophat” patterns are placed on signal (a) and idler (b) DMDs that only vary horizontally. In the regions where light should be directed to the detectors, all vertical pixels are used. The local images’ outerproduct defines the rectangular region of the jointspace P(X_{a}, X_{b}) that is being sampled.
To instead access the vertical, jointposition distribution P(Y_{a}, Y_{b}), local DMD patterns are used that only vary vertically. The jointmomentum distributions are similarly sampled, with the halfwaveplates instead adjusted to send light down the Fouriertransforming optical path so that the DMDs sit in the farfield of the nonlinear crystal.
Adaptive sampling algorithm
For each configuration, experimental data are stored in nodes in a quadtree decomposition of P whose levels describe increasingly fine detail. The ith node corresponds to a square area of \(\tilde P\) at location \((x_{\mathrm{a}}^i,x_{\mathrm{b}}^i)\) with span \(w_{\mathrm{a}}^i = w_{\mathrm{b}}^i = w\). Nodes are sampled by placing the corresponding, onedimensional local patterns on the DMDs and generating a coincidence histogram during acquisition time T_{a} = 0.5 s. Coincidences C_{i} are counted within a 1ns coincidence window centered on the coincidence peak; accidental coincidences A_{i} are counted in a 1ns window displaced 2 ns from the coincidence window. Coincidence and accidental values are appended to a list each time the node is sampled. The estimated count rate \(R_i = \left\langle {C_i} \right\rangle /\epsilon _iT_{\mathrm{a}}\), where \(\epsilon _i\) is a calibrated, relative fiber coupling efficiency. Optionally, A_{i} can be subtracted from C_{i} for accidental removal. Uncertainty is computed by assuming Poissonian counting statistics for C_{i} and A_{i} and applying standard, algebraic propagation of error through the calculation of the entanglement quantity (Eq. (3)).
The data collection algorithm consists of a partitioning phase followed by an iterative phase. During partitioning, the algorithm repeatedly iterates through a scanlist of leaves of the tree. Node i is considered stable when sgn(αR_{T} − R_{i}) is known to at least β standard deviations of certainty, where splitting threshold α (0 ≤ α ≤ 1) and stability criterion β are userchosen heuristics. Stable nodes are no longer measured. If a node is stable and R_{i} ≥ αR_{T}, the node is split into four equalsized subquadrants, which are initially unstable and added to the scanlist. Optionally, a maximum resolution (maximum tree depth) may be set.
The transition to the iterative phase occurs when the percentage of unstable leaves is <Γ, a userchosen parameter. At this point, stability is ignored and all leaf nodes are scanned repeatedly and guaranteed to have the same total acquisition time. Various final stopping criteria can be used; we chose a fixed total run time. Note that heuristic parameters α, β, and γ may be changed during operation if desired. For the data shown in this manuscript, α = 0.002, β = 2, and Γ = 0.15 with a 30h runtime.
The probability distribution \(\tilde P\) is computed by uniformly distributing the estimated count rate (with or without accidental subtraction) from each leaf node across its constituent elements in \(\tilde P\), followed by normalization.
Data availability
The data supporting the results presented in this manuscript is available from the corresponding author G.A.H. upon request.
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Acknowledgements
We gratefully acknowledge support from the OSD ARAP QSEP program and Air Force Office of Scientific Research LRIR 14RI02COR. J.S. acknowledges support from the National Research Council Research Associate Program. Any opinions, findings, and conclusions or recommendations expressed in this article are those of the authors and do not necessarily reflect the views of AFRL.
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G.A.H. and J.S. conceived of the idea and contributed equally. J.S. derived the entanglement witness and led the theoretical analysis. G.A.H. and C.C.T. developed the data collection algorithm. G.A.H. performed the experiment with help from M.L.F. and analyzed the data with help from C.C.T. and J.S. P.M.A. participated in useful scientific discussions. G.A.H. wrote the manuscript with contributions from all authors.
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Schneeloch, J., Tison, C.C., Fanto, M.L. et al. Quantifying entanglement in a 68billiondimensional quantum state space. Nat Commun 10, 2785 (2019). https://doi.org/10.1038/s4146701910810z
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