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Tom Siegfried
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Tensor networks get entangled with quantum gravity

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First of two parts

One of the first steps toward becoming a scientist is discovering the difference between speed and velocity.

To nonscientists, it’s usually a meaningless distinction. Fast is fast, slow is slow. But speed, technically, refers only to rate of motion. Velocity encompasses both speed and direction. In science, you usually want to know more than just how fast something is going; you also want to know where it is going. Hence the need to know direction, and to analyze velocity, not just speed. Numbers like velocity that express both a magnitude and a direction are known as vectors.

Vectors are great for describing the motion of a particle. But now suppose you need to analyze something more complicated, where multiple magnitudes and directions are involved. Perhaps you’re an engineer calculating stresses and strains in an elastic material. Or a neuroscientist tracing the changing forces on water flow near nerve cells. Or a physicist attempting to describe gravity in the cosmos. For all that, you need tensors. And they might even help you unify gravitational theory with quantum physics.

Tensors accommodate multiple numerical values (a vector is actually a simple special case of a tensor). While the ideas behind tensors stretch back to Gauss, they were first fully described in the 1890s by the Italian mathematician Gregorio Ricci-Curbastro, with the help of his student Tullio Levi-Civita. (Tensors were given their name in 1898 by Woldemar Voigt, a German crystallographer, who was studying stresses and strains in nonrigid bodies.)

Ricci (as he is commonly known) was influenced by the German mathematician Bernhard Riemann in developing advanced calculus with applications to complicated geometrical problems. In particular, this approach proved valuable in studying coordinate systems. Tensors help make sense of the relationships in the system that stay the same when you change the coordinates. That turned out to be just the thing Einstein needed in his theory of gravity, general relativity. His friend Marcel Grossmann explained tensors to him and they became the essential feature of general relativity’s mathematics.

And now, in a recent development, some physicists think tensors of a sort could help solve the longstanding problem of unifying general relativity with quantum mechanics. It’s part of a popular new line of research using tensors to quantify quantum entanglement, which some physicists believe has something to do with gravity.

Entanglement is that spooky connection between separated particles that disturbed Einstein so much. Somehow a measurement of one of a pair of particles affects what you’ll find when you measure its distant partner, or so it seems. But this “entanglement” is a clear-cut consequence of quantum physics for particles that share a common origin or interaction. It leads to some weird phenomena, but it’s all very sensible mathematically, as described by the “quantum state.” Entangled particles belong to a single quantum state.

A quantum state determines the mathematical expression (called the wave function) that can be used to predict the outcome of measurements of a particle — whether the direction that it spins is pointing up or down, for instance. When describing multiple particles — such as those in materials exhibiting quantum properties such as superconductivity — quantum states can get very complicated. Coping with them is made easier by analyzing the network of entanglement among those many particles. And patterns of such network connections can be described using tensors.

“Tensor networks are representations of quantum many-body states of matter based on their local entanglement structure,” physicist Román Orús writes in a recent paper posted at arXiv.org. “In a way, we could say that one uses entanglement to build up the many-body wave function.”

Put another way, Orús says, the entire wave function can be thought of as built from smaller tensor subnetworks, kind of like Legos. Entanglement is the glue holding the Legos together.

“Tensor network methods represent quantum states in terms of networks of interconnected tensors, which in turn capture the relevant entanglement properties of a system,” Orús writes in another recent paper, to be published in Annals of Physics.

While the basic idea of tensor networks goes back decades, they became more widely used to study certain quantum systems in the 1990s. In the last few years, ideas from quantum information theory have spawned an explosion of new methods using tensor networks to aid various calculations. Instead of struggling with complicated equations, physicists can analyze systems using tensor network diagrams, similar to the way Feynman diagrams are used in other aspects of quantum physics.

“This is a new language for condensed matter physics (and in fact, for all quantum physics) that makes everything much more visual and which brings new intuitions, ideas and results,” Orús writes.

Most recently, tensor networks have illuminated the notion that quantum entanglement is related to gravity. In Einstein’s general relativity, gravity is the effect of the geometry of spacetime. Analyses suggest that the geometry in which a quantum state exists is determined by the entanglement tensor network.

“By pushing this idea to the limit,” Orús notes, “a number of works have proposed that geometry and curvature (and hence gravity) could emerge naturally from the pattern of entanglement present in quantum states.”

If so, tensor networks could be the key to unlocking the mystery of quantum gravity. And in fact, another clue to quantum gravity, known as the holographic principle, seems naturally linked to a particular type of tensor network. That’s a connection worth exploring further.

Follow me on Twitter: @tom_siegfried

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