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Witnessing Wigner Negativity

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Ulysse Chabaud1,2, Pierre-Emmanuel Emeriau3, and Frédéric Grosshans3

1Institute for Quantum Information and Matter, Caltech
2Université de Paris, IRIF, CNRS, France
3Sorbonne Université, CNRS, LIP6, F-75005 Paris, France

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Abstract

Negativity of the Wigner function is arguably one of the most striking non-classical features of quantum states. Beyond its fundamental relevance, it is also a necessary resource for quantum speedup with continuous variables. As quantum technologies emerge, the need to identify and characterize the resources which provide an advantage over existing classical technologies becomes more pressing. Here we derive witnesses for Wigner negativity of single mode and multimode quantum states, based on fidelities with Fock states, which can be reliably measured using standard detection setups. They possess a threshold expectation value indicating whether the measured state has a negative Wigner function. Moreover, the amount of violation provides an operational quantification of Wigner negativity. We phrase the problem of finding the threshold values for our witnesses as an infinite-dimensional linear optimisation. By relaxing and restricting the corresponding linear programs, we derive two hierarchies of semidefinite programs, which provide numerical sequences of increasingly tighter upper and lower bounds for the threshold values. We further show that both sequences converge to the threshold value. Moreover, our witnesses form a complete family – each Wigner negative state is detected by at least one witness – thus providing a reliable method for experimentally witnessing Wigner negativity of quantum states from few measurements. From a foundational perspective, our findings provide insights on the set of positive Wigner functions which still lacks a proper characterisation.

Continuous-variable quantum information uses information encoded in the continuous degrees of freedom of quantum systems and is a promising candidate for quantum computing. In continuous variables, quantum states may be represented equivalently in phase space via their Wigner function.

The negativity of the Wigner function is a sign of non-classicality and a necessary resource for any quantum computational speedup. Detecting this negativity for experimental quantum states is therefore crucial for the development of continuous-variable quantum technologies. However, this detection can be very difficult as it usually relies on quantum state tomography, which requires an exponential number of samples compared to the system size.

In this work, we propose an alternative, more efficient, approach which introduces specific observables equipped with threshold values, such that if the expectation value of an observable with an unknown quantum state exceeds its threshold value then the state is certified to exhibit Wigner negativity.

Our results pave the way for the characterisation of non-classical quantum states, with direct applications in quantum optics experiments. The mathematical aspects of our work also motivate further study of the set of quantum states with positive Wigner function.

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Cited by

[1] Mattia Walschaers, “Non-Gaussian Quantum States and Where to Find Them”, arXiv:2104.12596.

[2] Benjamin Morris, Lukas J. Fiderer, Ben Lang, and Daniel Goldwater, “Witnessing Bell violations through probabilistic negativity”, arXiv:2105.01685.

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Quantum

Mathematicians Prove 2D Version of Quantum Gravity Really Works

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Alexander Polyakov, a theoretical physicist now at Princeton University, caught a glimpse of the future of quantum theory in 1981. A range of mysteries, from the wiggling of strings to the binding of quarks into protons, demanded a new mathematical tool whose silhouette he could just make out.

“There are methods and formulae in science which serve as master keys to many apparently different problems,” he wrote in the introduction to a now famous four-page letter in Physics Letters B. “At the present time we have to develop an art of handling sums over random surfaces.”

Polyakov’s proposal proved powerful. In his paper he sketched out a formula that roughly described how to calculate averages of a wildly chaotic type of surface, the “Liouville field.” His work brought physicists into a new mathematical arena, one essential for unlocking the behavior of theoretical objects called strings and building a simplified model of quantum gravity.

Years of toil would lead Polyakov to breakthrough solutions for other theories in physics, but he never fully understood the mathematics behind the Liouville field.

Over the last seven years, however, a group of mathematicians has done what many researchers thought impossible. In a trilogy of landmark publications, they have recast Polyakov’s formula using fully rigorous mathematical language and proved that the Liouville field flawlessly models the phenomena Polyakov thought it would.

“It took us 40 years in math to make sense of four pages,” said Vincent Vargas, a mathematician at the French National Center for Scientific Research and co-author of the research with Rémi Rhodes of Aix-Marseille University, Antti Kupiainen of the University of Helsinki, François David of the French National Center for Scientific Research, and Colin Guillarmou of Paris-Saclay University.

The three papers forge a bridge between the pristine world of mathematics and the messy reality of physics — and they do so by breaking new ground in the mathematical field of probability theory. The work also touches on philosophical questions regarding the objects that take center stage in the leading theories of fundamental physics: quantum fields.

“This is a masterpiece in mathematical physics,” said Xin Sun, a mathematician at the University of Pennsylvania.

Infinite Fields

In physics today, the main actors in the most successful theories are fields — objects that fill space, taking on different values from place to place.

In classical physics, for example, a single field tells you everything about how a force pushes objects around. Take Earth’s magnetic field: The twitches of a compass needle reveal the field’s influence (its strength and direction) at every point on the planet.

Fields are central to quantum physics, too. However, the situation here is more complicated due to the deep randomness of quantum theory. From the quantum perspective, Earth doesn’t generate one magnetic field, but rather an infinite number of different ones. Some look almost like the field we observe in classical physics, but others are wildly different.

But physicists still want to make predictions — predictions that ideally match, in this case, what a mountaineer reads on a compass. Assimilating the infinite forms of a quantum field into a single prediction is the formidable task of a “quantum field theory,” or QFT. This is a model of how one or more quantum fields, each with their infinite variations, act and interact.

Driven by immense experimental support, QFTs have become the basic language of particle physics. The Standard Model is one such QFT, depicting fundamental particles like electrons as fuzzy bumps that emerge from an infinitude of electron fields. It has passed every experimental test to date (although various groups may be on the verge of finding the first holes).

Physicists play with many different QFTs. Some, like the Standard Model, aspire to model real particles moving through the four dimensions of our universe (three spatial dimensions plus one dimension of time). Others describe exotic particles in strange universes, from two-dimensional flatlands to six-dimensional uber-worlds. Their connection to reality is remote, but physicists study them in the hopes of gaining insights they can carry back into our own world.

Polyakov’s Liouville field theory is one such example.

Gravity’s Field

The Liouville field, which is based on an equation from complex analysis developed in the 1800s by the French mathematician Joseph Liouville, describes a completely random two-dimensional surface — that is, a surface, like Earth’s crust, but one in which the height of every point is chosen randomly. Such a planet would erupt with mountain ranges of infinitely tall peaks, each assigned by rolling a die with infinite faces.

Such an object might not seem like an informative model for physics, but randomness is not devoid of patterns. The bell curve, for example, tells you how likely you are to randomly pass a seven-foot basketball player on the street. Similarly, bulbous clouds and crinkly coastlines follow random patterns, but it’s nevertheless possible to discern consistent relationships between their large-scale and small-scale features.

Liouville theory can be used to identify patterns in the endless landscape of all possible random, jagged surfaces. Polyakov realized this chaotic topography was essential for modeling strings, which trace out surfaces as they move. The theory has also been applied to describe quantum gravity in a two-dimensional world. Einstein defined gravity as space-time’s curvature, but translating his description into the language of quantum field theory creates an infinite number of space-times — much as the Earth produces an infinite collection of magnetic fields. Liouville theory packages all those surfaces together into one object. It gives physicists the tools to measure the curvature —and hence, gravitation — at every location on a random 2D surface.

“Quantum gravity basically means random geometry, because quantum means random and gravity means geometry,” said Sun.

Polyakov’s first step in exploring the world of random surfaces was to write down an expression defining the odds of finding a particular spiky planet, much as the bell curve defines the odds of meeting someone of a particular height. But his formula did not lead to useful numerical predictions.

To solve a quantum field theory is to be able to use the field to predict observations. In practice, this means calculating a field’s “correlation functions,” which capture the field’s behavior by describing the extent to which a measurement of the field at one point relates, or correlates, to a measurement at another point. Calculating correlation functions in the photon field, for instance, can give you the textbook laws of quantum electromagnetism.

Polyakov was after something more abstract: the essence of random surfaces, similar to the statistical relationships that make a cloud a cloud or a coastline a coastline. He needed the correlations between the haphazard heights of the Liouville field. Over the decades he tried two different ways of calculating them. He started with a technique called the Feynman path integral and ended up developing a workaround known as the bootstrap. Both methods came up short in different ways, until the mathematicians behind the new work united them in a more precise formulation.

Add ’Em Up

You might imagine that accounting for the infinitely many forms a quantum field can take is next to impossible. And you would be right. In the 1940s Richard Feynman, a quantum physics pioneer, developed one prescription for dealing with this bewildering situation, but the method proved severely limited.

Take, again, Earth’s magnetic field. Your goal is to use quantum field theory to predict what you’ll observe when you take a compass reading at a particular location. To do this, Feynman proposed summing all the field’s forms together. He argued that your reading will represent some average of all the field’s possible forms. The procedure for adding up these infinite field configurations with the proper weighting is known as the Feynman path integral.

It’s an elegant idea that yields concrete answers only for select quantum fields. No known mathematical procedure can meaningfully average an infinite number of objects covering an infinite expanse of space in general. The path integral is more of a physics philosophy than an exact mathematical recipe. Mathematicians question its very existence as a valid operation and are bothered by the way physicists rely on it.

“I’m disturbed as a mathematician by something which is not defined,” said Eveliina Peltola, a mathematician at the University of Bonn in Germany.

Physicists can harness Feynman’s path integral to calculate exact correlation functions for only the most boring of fields — free fields, which do not interact with other fields or even with themselves. Otherwise, they have to fudge it, pretending the fields are free and adding in mild interactions, or “perturbations.” This procedure, known as perturbation theory, gets them correlation functions for most of the fields in the Standard Model, because nature’s forces happen to be quite feeble.

But it didn’t work for Polyakov. Although he initially speculated that the Liouville field might be amenable to the standard hack of adding mild perturbations, he found that it interacted with itself too strongly. Compared to a free field, the Liouville field seemed mathematically inscrutable, and its correlation functions appeared unattainable.

Up by the Bootstraps

Polyakov soon began looking for a workaround. In 1984, he teamed up with Alexander Belavin and Alexander Zamolodchikov to develop a technique called the bootstrap — a mathematical ladder that gradually leads to a field’s correlation functions.

To start climbing the ladder, you need a function which expresses the correlations between measurements at a mere three points in the field. This “three-point correlation function,” plus some additional information about the energies a particle of the field can take, forms the bottom rung of the bootstrap ladder.

From there you climb one point at a time: Use the three-point function to construct the four-point function, use the four-point function to construct the five-point function, and so on. But the procedure generates conflicting results if you start with the wrong three-point correlation function in the first rung.

Polyakov, Belavin and Zamolodchikov used the bootstrap to successfully solve a variety of simple QFT theories, but just as with the Feynman path integral, they couldn’t make it work for the Liouville field.

Then in the 1990s two pairs of physicists — Harald Dorn and Hans-Jörg Otto, and Zamolodchikov and his brother Alexei — managed to hit on the three-point correlation function that made it possible to scale the ladder, completely solving the Liouville field (and its simple description of quantum gravity). Their result, known by their initials as the DOZZ formula, let physicists make any prediction involving the Liouville field. But even the authors knew they had arrived at it partially by chance, not through sound mathematics.

“They were these kind of geniuses who guessed formulas,” said Vargas.

Educated guesses are useful in physics, but they don’t satisfy mathematicians, who afterward wanted to know where the DOZZ formula came from. The equation that solved the Liouville field should have come from some description of the field itself, even if no one had the faintest idea how to get it.

“It looked to me like science fiction,” said Kupiainen. “This is never going to be proven by anybody.”

Taming Wild Surfaces

In the early 2010s, Vargas and Kupiainen joined forces with the probability theorist Rémi Rhodes and the physicist François David. Their goal was to tie up the mathematical loose ends of the Liouville field — to formalize the Feynman path integral that Polyakov had abandoned and, just maybe, demystify the DOZZ formula.

As they began, they realized that a French mathematician named Jean-Pierre Kahane had discovered, decades earlier, what would turn out to be the key to Polyakov’s master theory.

“In some sense it’s completely crazy that Liouville was not defined before us,” Vargas said. “All the ingredients were there.”

The insight led to three milestone papers in mathematical physics completed between 2014 and 2020.

They first polished off the path integral, which had failed Polyakov because the Liouville field interacts strongly with itself, making it incompatible with Feynman’s perturbative tools. So instead, the mathematicians used Kahane’s ideas to recast the wild Liouville field as a somewhat milder random object known as the Gaussian free field. The peaks in the Gaussian free field don’t fluctuate to the same random extremes as the peaks in the Liouville field, making it possible for the mathematicians to calculate averages and other statistical measures in sensible ways.

“Somehow it’s all just using the Gaussian free field,” Peltola said. “From that they can construct everything in the theory.”

In 2014, they unveiled their result: a new and improved version of the path integral Polyakov had written down in 1981, but fully defined in terms of the trusted Gaussian free field. It’s a rare instance in which Feynman’s path integral philosophy has found a solid mathematical execution.

“Path integrals can exist, do exist,” said Jörg Teschner, a physicist at the German Electron Synchrotron.

With a rigorously defined path integral in hand, the researchers then tried to see if they could use it to get answers from the Liouville field and to derive its correlation functions. The target was the mythical DOZZ formula — but the gulf between it and the path integral seemed vast.

“We’d write in our papers, just for propaganda reasons, that we want to understand the DOZZ formula,” said Kupiainen.

The team spent years prodding their probabilistic path integral, confirming that it truly had all the features needed to make the bootstrap work. As they did so, they built on earlier work by Teschner. Eventually, Vargas, Kupiainen and Rhodes succeeded with a paper posted in 2017 and another in October 2020, with Colin Guillarmou. They derived DOZZ and other correlation functions from the path integral and showed that these formulas perfectly matched the equations physicists had reached using the bootstrap.

“Now we’re done,” Vargas said. “Both objects are the same.”

The work explains the origins of the DOZZ formula and connects the bootstrap procedure —which mathematicians had considered sketchy — with verified mathematical objects. Altogether, it resolves the final mysteries of the Liouville field.

“It’s somehow the end of an era,” said Peltola. “But I hope it’s also the beginning of some new, interesting things.”

New Hope for QFTs

Vargas and his collaborators now have a unicorn on their hands, a strongly interacting QFT perfectly described in a nonperturbative way by a brief mathematical formula that also makes numerical predictions.

Now the literal million-dollar question is: How far can these probabilistic methods go? Can they generate tidy formulas for all QFTs? Vargas is quick to dash such hopes, insisting that their tools are specific to the two-dimensional environment of Liouville theory. In higher dimensions, even free fields are too irregular, so he doubts the group’s methods will ever be able to handle the quantum behavior of gravitational fields in our universe.

But the fresh minting of Polyakov’s “master key” will open other doors. Its effects are already being felt in probability theory, where mathematicians can now wield previously dodgy physics formulas with impunity. Emboldened by the Liouville work, Sun and his collaborators have already imported equations from physics to solve two problems regarding random curves.

Physicists await tangible benefits too, further down the road. The rigorous construction of the Liouville field could inspire mathematicians to try their hand at proving features of other seemingly intractable QFTs — not just toy theories of gravity but descriptions of real particles and forces that bear directly on the deepest physical secrets of reality.

“[Mathematicians] will do things that we can’t even imagine,” said Davide Gaiotto, a theoretical physicist at the Perimeter Institute.

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Source: https://www.quantamagazine.org/mathematicians-prove-2d-version-of-quantum-gravity-really-works-20210617/

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Warm-starting quantum optimization

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Daniel J. Egger1, Jakub Mareček2, and Stefan Woerner1

1IBM Quantum, IBM Research – Zurich, Säumerstrasse 4, 8803 Rüschlikon, Switzerland
2Czech Technical University, Karlovo nam. 13, Prague 2, the Czech Republic

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Abstract

There is an increasing interest in quantum algorithms for problems of integer programming and combinatorial optimization. Classical solvers for such problems employ relaxations, which replace binary variables with continuous ones, for instance in the form of higher-dimensional matrix-valued problems (semidefinite programming). Under the Unique Games Conjecture, these relaxations often provide the best performance ratios available classically in polynomial time. Here, we discuss how to warm-start quantum optimization with an initial state corresponding to the solution of a relaxation of a combinatorial optimization problem and how to analyze properties of the associated quantum algorithms. In particular, this allows the quantum algorithm to inherit the performance guarantees of the classical algorithm. We illustrate this in the context of portfolio optimization, where our results indicate that warm-starting the Quantum Approximate Optimization Algorithm (QAOA) is particularly beneficial at low depth. Likewise, Recursive QAOA for MAXCUT problems shows a systematic increase in the size of the obtained cut for fully connected graphs with random weights, when Goemans-Williamson randomized rounding is utilized in a warm start. It is straightforward to apply the same ideas to other randomized-rounding schemes and optimization problems.

Many optimization problems in binary decision variables are hard to solve. In this work, we demonstrate how to leverage decades of research in classical optimization algorithms to warm-start quantum optimization algorithms. This allows the quantum algorithm to inherit the performance guarantees from the classical algorithm used in the warm-start.

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Experimental localisation of quantum entanglement through monitored classical mediator

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Soham Pal1, Priya Batra1, Tanjung Krisnanda2, Tomasz Paterek2,3,4, and T. S. Mahesh1

1Department of Physics, Indian Institute of Science Education and Research, Pune 411008, India
2School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
3MajuLab, International Joint Research Unit UMI 3654, CNRS, Université Côte d’Azur, Sorbonne Université, National University of Singapore, Nanyang Technological University, Singapore
4Institute of Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, 80-308 Gdańsk, Poland

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Abstract

Quantum entanglement is a form of correlation between quantum particles that cannot be increased via local operations and classical communication. It has therefore been proposed that an increment of quantum entanglement between probes that are interacting solely via a mediator implies non-classicality of the mediator. Indeed, under certain assumptions regarding the initial state, entanglement gain between the probes indicates quantum coherence in the mediator. Going beyond such assumptions, there exist other initial states which produce entanglement between the probes via only local interactions with a classical mediator. In this process the initial entanglement between any probe and the rest of the system “flows through” the classical mediator and gets localised between the probes. Here we theoretically characterise maximal entanglement gain via classical mediator and experimentally demonstrate, using liquid-state NMR spectroscopy, the optimal growth of quantum correlations between two nuclear spin qubits interacting through a mediator qubit in a classical state. We additionally monitor, i.e., dephase, the mediator in order to emphasise its classical character. Our results indicate the necessity of verifying features of the initial state if entanglement gain between the probes is used as a figure of merit for witnessing non-classical mediator. Such methods were proposed to have exemplary applications in quantum optomechanics, quantum biology and quantum gravity.

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[2] Laszlo Gyongyosi, “Quantum State Optimization and Computational Pathway Evaluation for Gate-Model Quantum Computers”, Scientific Reports 10, 4543 (2020).

[3] Laszlo Gyongyosi and Sandor Imre, “Entanglement accessibility measures for the quantum Internet”, Quantum Information Processing 19 4, 115 (2020).

[4] Laszlo Gyongyosi, “Unsupervised Quantum Gate Control for Gate-Model Quantum Computers”, Scientific Reports 10, 10701 (2020).

[5] Richard Howl, Vlatko Vedral, Devang Naik, Marios Christodoulou, Carlo Rovelli, and Aditya Iyer, “Non-Gaussianity as a signature of a quantum theory of gravity”, arXiv:2004.01189.

[6] Laszlo Gyongyosi and Sandor Imre, “Routing space exploration for scalable routing in the quantum Internet”, Scientific Reports 10, 11874 (2020).

[7] Laszlo Gyongyosi and Sandor Imre, “Circuit Depth Reduction for Gate-Model Quantum Computers”, Scientific Reports 10, 11229 (2020).

[8] Tanjung Krisnanda, “Distribution of quantum entanglement: Principles and applications”, arXiv:2003.08657.

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[11] Laszlo Gyongyosi, “Decoherence dynamics estimation for superconducting gate-model quantum computers”, Quantum Information Processing 19 10, 369 (2020).

[12] Laszlo Gyongyosi and Sandor Imre, “Entanglement concentration service for the quantum Internet”, Quantum Information Processing 19 8, 221 (2020).

[13] B. Sharmila, “Signatures of nonclassical effects in tomograms”, arXiv:2009.09798.

[14] B. Sharmila, V. R. Krithika, Soham Pal, T. S. Mahesh, S. Lakshmibala, and V. Balakrishnan, “Tomographic entanglement indicators from NMR experiments”, arXiv:2105.08555.

[15] Laszlo Gyongyosi and Sandor Imre, “Scalable distributed gate-model quantum computers”, Scientific Reports 11, 5172 (2021).

[16] Laszlo Gyongyosi and Sandor Imre, “Resource prioritization and balancing for the quantum internet”, Scientific Reports 10, 22390 (2020).

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Quantum

Optimal Multi-port-based Teleportation Schemes

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on

Marek Mozrzymas1, Michał Studziński2, and Piotr Kopszak1

1Institute for Theoretical Physics, University of Wrocław, 50-204 Wrocław, Poland
2Institute of Theoretical Physics and Astrophysics, National Quantum Information Centre, University of Gdańsk, 80-952 Gdańsk, Poland

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Abstract

In this paper, we introduce optimal versions of a multi-port based teleportation scheme allowing to send a large amount of quantum information. We fully characterise probabilistic and deterministic case by presenting expressions for the average probability of success and entanglement fidelity. In the probabilistic case, the final expression depends only on global parameters describing the problem, such as the number of ports $N$, the number of teleported systems $k$, and local dimension $d$. It allows us to show square improvement in the number of ports with respect to the non-optimal case. We also show that the number of teleported systems can grow when the number $N$ of ports increases as $o(N)$ still giving high efficiency. In the deterministic case, we connect entanglement fidelity with the maximal eigenvalue of a generalised teleportation matrix. In both cases the optimal set of measurements and the optimal state shared between sender and receiver is presented. All the results are obtained by formulating and solving primal and dual SDP problems, which due to existing symmetries can be solved analytically. We use extensively tools from representation theory and formulate new results that could be of the separate interest for the potential readers.

► BibTeX data

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Cited by

[1] Piotr Kopszak, Marek Mozrzymas, Michał Studziński, and Michał Horodecki, “Multiport based teleportation — transmission of a large amount of quantum information”, arXiv:2008.00856.

The above citations are from SAO/NASA ADS (last updated successfully 2021-06-17 13:16:24). The list may be incomplete as not all publishers provide suitable and complete citation data.

Could not fetch Crossref cited-by data during last attempt 2021-06-17 13:16:22: Could not fetch cited-by data for 10.22331/q-2021-06-17-477 from Crossref. This is normal if the DOI was registered recently.

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Source: https://quantum-journal.org/papers/q-2021-06-17-477/

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