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Quantifying Entanglement in a 68-Billion-Dimensional Quantum State Space

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Quantifying Entanglement in a 68-Billion-Dimensional Quantum State Space ARTICLE https://doi.org/10.1038/s41467-019-10810-z OPEN Quantifying entanglement in a 68-billion- dimensional quantum state space James Schneeloch 1,5, Christopher C. Tison1,2,3, Michael L. Fanto1,4, Paul M. Alsing1 & Gregory A. Howland 1,4,5 Entanglement is the powerful and enigmatic resource central to quantum information pro- cessing, which promises capabilities in computing, simulation, secure communication, and 1234567890():,; 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 modestly-sized systems. Here we demonstrate a method for rigorously quantifying high-dimensional 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 entanglement-of- formation of 7.11 ± .04 ebits shared by two spatially-entangled photons. With a Hilbert space exceeding 68 billion dimensions, we need 20-million-times fewer measurements than the uncompressed approach and 1018-times fewer measurements than tomography. Our tech- nique offers a universal method for quantifying entanglement in any large quantum system shared by two parties. 1 Air Force Research Laboratory, Information Directorate, Rome, NY 13441, USA. 2 Department of Physics, Florida Atlantic University, Boca Raton, FL 33431, USA. 3 Quanterion Solutions Incorporated, Utica, NY 13502, USA. 4 Rochester Institute of Technology, Rochester, NY 14623, USA. 5These authors contributed equally: James Schneeloch, Gregory A. Howland. Correspondence and requests for materials should be addressed to G.A.H. (email: [email protected]) NATURE COMMUNICATIONS | (2019) 10:2785 | https://doi.org/10.1038/s41467-019-10810-z | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-10810-z chieving a quantum advantage for information processing error analysis is straightforward and few computational resources requires scaling quantum systems to sizes that can pro- are needed. A fi vide signi cant quantum resources, including entangle- ment. Large quantum systems are now realized across many Results – platforms, including atomic simulators beyond 50 qubits1 3, Entropic witnesses of high-dimensional entanglement. Entan- nascent superconducting and trapped-ion-based quantum glement is revealed when subsystems of a quantum state are computers4,5, integrated-photonic circuits6–10, and photon pairs – specially correlated. A common situation divides a system entangled in high-dimensional variables11 16. between two parties, Alice and Bob, who make local measure- As quantum-information-based technologies mature, it will ments on their portion. Given two mutually unbiased, continuous b become useful to separate the physical layer providing quantum observables bx and k, they can measure discrete joint probability resources (e.g., trapped ions, photons) from the logical layer that distributions P(X , X ) and P(K , K ) by discretizing to pixel sizes utilizes those resources. For example, many imperfect qubits may a b a b ΔX and ΔK. Here bold notation indicates that X and K may form one logical qubit17,18 or thousands of atoms may coherently 19,20 (though need not) represent multidimensional coordinates. For act as a single-photon quantum memory . As with classical example, X and K might represent cartesian position and communication and computing, protocols and algorithms will be momentum that can be decomposed into horizontal and vertical implemented in the logical layer with minimal concern for the components such that X = (X, Y) and K = (K(x), K(y)). underlying platform. Because real-world systems are varied and A recent, quantitative entanglement witness40 uses these imperfect, the quantum resources they provide must be char- distributions to certify an amount of entanglement: acterized before use17. Certifying an amount of entanglement in a large quantum system is an essential but daunting task. While entanglement 2π d log À HðX jX ÞÀHðK jK ÞE ; ð1Þ witnesses21,22 and Bell tests23 can reveal entanglement’s presence, 2 Δ Δ a b a b f X K quantification generally requires a full estimation of the quantum state24. Beyond moderately sized states, the number of parameters to physically measure (i.e., the number of the measurements) where, for example, H(A|B) is the conditional Shannon entropy P E becomes overwhelming, making this approach unviable for cur- for (A, B). f is the entanglement of formation, a measure rent and future large-scale quantum technologies. describing the average number of Bell pairs required to synthesize Any practical method for quantitative entanglement certifica- the state. Equation (1) does not require full-state estimation but b tion must require only limited data. Two ideas can dramatically depends on an informed choice of bx and k. Still, in large systems, reduce the needed measurement resources. First is the develop- measuring these joint distributions remains oppressive. For P ment of quantitative entanglement witnesses, which bound the example, if Xa has 100 possible outcomes, determining (Xa, 25–28 2 amount of entanglement without full state estimation .Ina Xb) takes 100 joint measurements. Describing quantum recent landmark experiment, 4.1 entangled bits (ebits) of high- uncertainty with information-theoretic quantities is increasingly dimensional biphoton entanglement was certified using partial popular41,42. Entropies naturally link physical and logical layers state estimation29. One ebit describes the amount of entangle- and have useful mathematical properties. In particular, many ment in a maximally entangled, two-qubit state24. approximations to the joint distributions can only increase E Second, prior knowledge can be exploited to economize sam- conditional entropy. Because Eq. (1) bounds f from below, any pling. Certain features, or structure, are expected in specific sys- such substitution is valid. tems. In highly entangled quantum systems, for example, some observables should be highly correlated, the density matrix will be Improving an entropic entanglement witnesses for use with low rank, or the state may be nearly pure. Such assumptions can limited data. We use two entropic shortcuts to improve the be paired with numerical optimization to recover signals sampled entanglement witness. First, if the system is highly entangled, and below the Nyquist limit. One popular technique is compressed b b 30 x and k are well chosen, the joint distributions will be highly sensing , which has massively disrupted conventional thinking correlated; a measurement outcome for X should correlate to few about sampling. Applied to quantum systems, compressed sen- a outcomes for Xb. The distributions are therefore highly com- sing reduced measurement resources significantly for tasks, P 31–37 38,39 pressible. Consider replacing arbitrary groups of elements in including tomography and witnessing entanglement . (X , X ) with their average values to form a multilevel, com- Computational recovery techniques have substantial down- a b pressed estimate P~ðX ; X Þ. By multilevel, we mean that the new, sides. Because they are estimation techniques, conclusions drawn a b estimated distribution will appear as if it was sampled with from their results are contingent on the veracity of the initial varying resolution—fine detail in some regions and coarse detail assumptions. They are therefore unsuitable for closing loopholes in others. Because coarse graining cannot decrease conditional or verifying security. Numerical solvers are often proven correct P~ð ; Þ P~ð ; Þ under limited noise models and require hand-tuned parameters, entropy, Eq. (1) remains valid for Xa Xb and Ka Kb potentially adding artifacts and complicating error analysis. (see Supplemental Material: Proof arbitrary coarse-graining can- not decrease conditional entropy). Finally, the computational resources needed become prohibitive P~ð ; Þ P~ð ; Þ fi in very large systems. The largest quantum systems characterized Good estimates for Xa Xb and Ka Kb can be ef ciently measured by sampling at high resolution in correlated regions using these approaches remain considerably smaller than state-of- P the-art. and low resolution elsewhere. Note that the original ( ) and fi estimate (P~)) are full correlation matrices with N elements, but Here we provide an approach to entanglement quanti cation ~ that overcomes these downsides. First, we improve an entropic, only M N values measured to specify P. The witness is valid quantitative entanglement witness to operate on arbitrarily for arbitrary downsampling; it works best when the approximate downsampled data. Then we develop an adaptive, multilevel and actual distributions are most similar but can never E sampling procedure to rapidly obtain compressed distributions overestimate f or allow false positives. suitable for the witness. Crucially, our sampling assumptions are Second, if the observables are multi-dimensional such that they independent of the entanglement certification, so our method can can be decomposed into d marginal, component observables (e.g., guarantee security. Because we avoid numerical optimization, horizontal and vertical components) bx ¼ð^xð1Þ; x^ð2Þ; :::; ^xðdÞÞ 2 NATURE COMMUNICATIONS | (2019) 10:2785 | https://doi.org/10.1038/s41467-019-10810-z | www.nature.com/naturecommunications
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