Quantum entanglement

• 1. Introduction

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  Quantum entanglement produced a distance record in 2017, when the Micius satellite sent entangled photons to ground bases in Lijian, Yunnan and Delingha, Qinghai. Yin and colleagues reported that a two-photon pair survived a journey of 1,203 km and still violated a Bell inequality. Transmission efficiency improved by an order of magnitude over earlier fiber-optic experiments.

  When particles are entangled, no single particle has a state of its own. The group has one state, and it holds however far apart the particles sit. Physicists treat this as the main feature that quantum mechanics has and classical mechanics lacks.

  The idea troubled some of the greatest minds in physics. Why did Albert Einstein think it exposed a flaw in the theory? How did experimenters eventually settle the argument? And why does a phenomenon that cannot send a message faster than light now sit at the center of research into communication and computation?

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• 2. A Quarrel Over Reality

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  By 1931, Albert Einstein had formulated a thought experiment involving a box that emits a photon. He pointed out that the experimenter's choice of measurement on the box would change what could be predicted about the photon, even when the photon was very far away. The argument belonged to the long, collegial Bohr-Einstein debates with Niels Bohr over how to interpret quantum mechanics.

  In that same year, Hermann Weyl wrote in his textbook on group theory and quantum mechanics that systems built from several interacting parts show a kind of Gestalt. In his words, "the whole is greater than the sum of its parts". In 1932, Erwin Schrödinger derived the defining equations of entanglement but left them unpublished. In 1935, Grete Hermann studied the mathematics of an electron interacting with a photon and noticed the same phenomenon.

  Later in 1935, Einstein, Boris Podolsky and Nathan Rosen published the paper that gave the Einstein-Podolsky-Rosen paradox its name. They claimed that "the quantum-mechanical description of physical reality given by wave functions is not complete". Their thought experiment began with two systems that interact and then separate. Afterward, they argued, quantum mechanics could not describe either system individually.

  Einstein and others considered such behavior impossible, because it broke the principle of local realism in causality. Their conclusion was that the accepted formulation of quantum mechanics had to be incomplete.

  Soon after the paper appeared, Schrödinger wrote Einstein a letter in German. In it he used the word Verschränkung, which he himself translated as entanglement. A full paper followed, in which he wrote: "I would not call entanglement one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought."

  Schrödinger shared Einstein's unease, because entanglement seemed to break the speed limit on information that relativity implies. Einstein later mocked the theory for what he called "spukhafte Fernwirkung", or "spooky action at a distance". The paradox they described could be built from a single pair of particles whose spins add up to zero.

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• 3. Socks and Spins

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  Take a pair of entangled particles prepared so that their total spin is known to be zero. If one particle is found with clockwise spin along a chosen axis, the other, measured along the same axis, shows anticlockwise spin.

  A spin-zero particle that decays into two spin-1/2 particles is the standard example. With no orbital angular momentum, conservation of angular momentum keeps the total spin at zero after the decay. Whenever one particle is measured spin up on an axis, the other is found spin down on that axis. This arrangement is called the singlet state.

  John Bell explained why such perfect opposition is not strange by itself, using a colleague named Bertlmann who always wore socks of mismatched colors. Bell wrote that "Which colour he will have on a given foot on a given day is quite unpredictable". But on observing "that the first sock is pink you can be already sure that the second sock will not be pink." Hidden variables could do the same job. Particles might be made in pairs, one carrying "up" and the other "down", from the start.

  The truly remarkable features appear only when several distinct experiments are compared, such as spin measurements along different axes. Entanglement is also not the same thing as the correlation of everyday life or classical probability. It is better described as potential correlation, which a suitable experiment can turn into actual correlation. Classical probability cannot replicate the correlations an entangled state produces.

  Mathematically, an entangled state is one that cannot be factored into a product of states of its local pieces. Maximal knowledge about the whole system does not give maximal knowledge about its parts. Measurement ends the link, because the particles become entangled with their environment, and the state describing the particles alone is no longer entangled. The trouble begins when the two measurements are placed so far apart that no signal could connect them.

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• 4. The Timing Problem

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  David Bohm introduced a version of the paradox built on the singlet state. A source emits particles and sends them in opposite directions, and each pair is entangled. In the standard textbook account, a spin measurement on one particle collapses the wave function of the whole pair. Each outcome, up or down, has a probability of 50%.

  If both spins are measured along the same axis, they come out anti-correlated. The random result on one side seems to have been passed to the other, so that the second particle can make the "right choice" when its turn comes.

  Special relativity sharpens the puzzle. The distance and timing of the two measurements can be chosen so that the interval between them is spacelike. Any causal link between them would then have to travel faster than light, which relativity forbids. It is not even possible to say which measurement came first, since some inertial frames put one first and others put the other first. Different observers would disagree about cause and effect.

  Two observers named Alice and Bob show why no message can get through. Alice measures system A and obtains 0 or 1 with equal probability. If she gets 0, she knows with certainty that Bob will find 1, and the reverse. But the outcome is random, and she cannot choose which state the pair collapses into. She therefore cannot send Bob information by acting on her system.

  The general argument is known as the no-communication theorem. Entanglement can produce statistical correlations between distant events, but it cannot carry a signal faster than light. Einstein's way out was to suppose that each particle carries its instructions with it from the start.

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• 5. Bell's Verdict

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  In 1964, John S. Bell turned the quarrel into something a laboratory could settle. He showed that any theory obeying local realism must keep the strength of certain correlations below an upper limit, now called Bell's inequality. He also showed that quantum theory predicts violations of that limit for certain entangled systems.

  The hidden-variable idea held that values fixed at the moment of separation decide every measurement outcome. Nothing would need to travel between the particles. This picture breaks down when spins are measured along different axes. Over many pairs of such measurements, a local hidden-variable view would always satisfy Bell's inequality. Experiments have shown that it is not satisfied.

  The reason lies in the uncertainty principle. Spin measurements along different axes cannot have definite values at the same time, so they are incompatible. Classical physics allows any number of properties to be measured at once with arbitrary accuracy. Compatible measurements have been proven unable to show Bell-violating correlations, which makes entanglement fundamentally non-classical. Even when measurements are made in moving relativistic frames, where each happens first in its own frame, the results remain correlated.

  The experiments arrived slowly. In 1946, John Archibald Wheeler suggested studying the polarization of pairs of gamma-ray photons produced by electron-positron annihilation. Chien-Shiung Wu and I. Shaknov carried this out in 1949, showing that the entangled pairs imagined by EPR could be made in a laboratory. Despite Schrödinger's claim of its importance, little work on entanglement appeared for decades afterward.

  Stuart Freedman and John Clauser performed pioneering Bell tests in 1972, and Alain Aspect followed with experiments in 1982. The earliest tests generated their entangled particles using atomic cascades. To date, every Bell test has found local hidden variables inconsistent with how physical systems behave.

  The violation of Bell inequalities is often called quantum nonlocality, a label that draws objections. Critics argue that it wrongly suggests faster-than-light signals must explain the result. Even so, the term has become a widespread convention.

  Early tests left a gap, since it could not be ruled out that the result at one site was subtly sent to the other. Loophole-free Bell tests closed it by separating the locations so widely that light would have needed longer than the interval between measurements. In one case it would have needed 10,000 times longer.

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• 6. Entanglement at Work

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  Bell actively discouraged students from pursuing work like his, calling it too esoteric. Yet after a talk at Oxford, a student named Artur Ekert suggested that violating a Bell inequality could serve as a resource for communication. Ekert went on to publish a quantum key distribution protocol called E91 built on that idea.

  Physicists now describe entanglement as a resource, much as energy is a resource for mechanical operations. In the "distant labs" setting, two labs, A and B, can each perform any quantum operation locally and exchange only classical information. These operations, known as LOCC, cannot produce entanglement between the labs. With a supply of entangled states, though, a larger class of transformations becomes possible.

  Quantum teleportation was proposed in 1992 and realized experimentally in 1997. Alice performs a joint measurement on her unknown state and her half of an entangled pair, then tells Bob the results by telephone. Bob uses them to operate on his half and make it equal to her original state. Her measurement erases the state in her lab, so it is transferred rather than copied.

  Entanglement swapping is a variant that lets two parties who have never interacted share an entangled state. One source emits an entangled pair, A and B, and a second source emits another pair, C and D. Measuring B and C in the basis of Bell states leaves A and D entangled. Using entanglement as a resource has a cost, because states are consumed in the process.

  Beginning in the mid-1990s, Anton Zeilinger generated entanglement through parametric down-conversion to develop entanglement swapping and to demonstrate quantum cryptography with entangled photons. In 2022, the Nobel Prize in Physics went to Aspect, Clauser and Zeilinger "for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science".

  Superdense coding and teleportation are among the best-known applications. Most researchers believe entanglement is necessary for quantum computing, although some dispute this. Proving the security of quantum key distribution under standard assumptions does not require entanglement. The device-independent security of such systems does exploit entanglement between the communicating partners.

  In August 2014, Brazilian researcher Gabriela Barreto Lemos of the University of Vienna and her team took pictures of objects using photons that never interacted with them. Those photons were entangled with other photons that did. The idea has since been adapted to make infrared images using only standard cameras that are insensitive to infrared.

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• 7. Many Kinds of Entanglement

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  Four Bell states form the standard menu of two-qubit entanglement. All four are maximally entangled and together form an orthonormal basis for the two qubits. They show how quantum mechanics can violate Bell-type inequalities.

  Three qubits bring new variety, with the Greenberger-Horne-Zeilinger state and the W state as the two classic examples. Tracing out any one qubit turns the GHZ state into a separable state, while the W state stays entangled after the same step. Systems of three or more parts can show several qualitatively different kinds of entanglement. A single particle cannot be maximally entangled with more than one other particle at a time, a property called monogamy.

  Werner states, a family describing pairs of particles, show that entanglement alone is not enough to violate a Bell inequality. Bell noted this himself in his 1964 paper. For appropriate parameter choices, Werner states are entangled yet always admit a local hidden-variable model.

  Not every entangled state is equally valuable as a resource. In the simplest classification, two states belong to the same local unitary class if a local unitary operation converts one into the other, and there are infinitely many such classes, even for two pure qubits. In the coarser SLOCC classification, two pure qubits fall into just two classes, entangled and separable.

  A very coarse classification asks whether many copies of a state can be turned into at least one pure entangled state. Such states are called distillable. It came as a surprise that some entangled states are not distillable, and these are called bound entangled. Another scheme sorts entangled states into non-local, steerable, and neither, and all three sets are non-empty.

  The von Neumann entropy of a subsystem measures entanglement for bipartite pure states. A pure two-particle state has zero entropy as a whole, yet each half has entropy greater than zero. For mixed states, no single measure is standard, and many are NP-hard to compute as the system grows. For two-qubit and qubit-qutrit systems, the Peres-Horodecki criterion gives a complete test for entanglement.

  Spontaneous parametric down-conversion is one of the most common ways to make a pair of photons entangled in polarization. Others include photons from the decay cascade of a bi-exciton in a quantum dot, the Hong-Ou-Mandel effect, and Hardy's interferometer for an electron and a positron. Spin squeezed states and NOON states, which are entangled, are good candidates for enhancing precision measurements. Photons, electrons, molecules and even small diamonds have all been entangled in experiments.

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• 8. Frontiers

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  In 2023, the Large Hadron Collider measured entanglement at the highest energy yet, using the ATLAS detector on top-quark pairs. The work built on theory first proposed in 2021, and the effect was observed at more than five sigma. The top quark is the heaviest known particle, with a very short lifetime. It is the only quark that decays before hadronization, so its spin information passes almost intact to decay products the detector can catch. The effect was later confirmed in the CMS detector.

  In 2020, researchers reported entanglement between the motion of a millimetre-sized mechanical oscillator and a distant cloud of atoms acting as a spin system. Later work entangled two mechanical oscillators. Physicists at Brookhaven National Laboratory used high-energy electron-proton collisions to show entanglement within protons. They found that quarks and gluons are interdependent rather than isolated, with maximal entanglement.

  Quantum field theory offers its own hint. The Reeh-Schlieder theorem is sometimes read as saying that entanglement is omnipresent in the quantum vacuum.

  The problem of time in quantum gravity gives entanglement another role. Standard quantum theory treats time as an independent background, while general relativity treats it as a dynamical variable tied to matter. The Wheeler-DeWitt equation predicts a universe that is timeless or static. Don Page and William Wootters suggested that the universe still appears to evolve for observers inside it, because of energy entanglement between an evolving system and a clock system.

  Gravity raises a related question. In the anti-de Sitter space, a model space the source calls unrealistic, the AdS/CFT correspondence links a quantum gravitational system to a quantum field theory without gravity. Using it, Mark Van Raamsdonk suggested that spacetime itself arises from entangled quantum degrees of freedom living on the boundary of that space.

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