Quantum Entanglement

Last session you wrote joint wavefunctions for identical particles as sums of products. Today the particles can be told apart, because they sit in different laboratories. Take two qubits, where a qubit is any quantum system with two perfectly distinguishable states labelled 0 and 1: spin up or down, or a photon polarised horizontally (H) or vertically (V). The pair has four basis states, |00⟩, |01⟩, |10⟩ and |11⟩, the first digit for Alice's particle and the second for Bob's. A product state, also called separable, is one that can be written as a definite state of Alice's particle times a definite state of Bob's.

Now consider |Φ+⟩ = (1/√2)(|00⟩ + |11⟩), read 'phi-plus'. Suppose Alice's particle were in a|0⟩ + b|1⟩ and Bob's in c|0⟩ + d|1⟩. Multiplying out gives a·c|00⟩ + a·d|01⟩ + b·c|10⟩ + b·d|11⟩. To match |Φ+⟩, a·d and b·c must vanish while a·c and b·d both equal 1/√2. But if a·d is zero, then a or d is zero, which destroys a·c or b·d. No assignment works, and that is the definition: a state of two systems is entangled when it cannot be written as a product of states of the individual systems. The quick test: α|00⟩ + β|01⟩ + γ|10⟩ + δ|11⟩ is a product exactly when αδ − βγ = 0; for |Φ+⟩ it is one half.

Pause on what this means. The pair is in a pure state, the most complete description quantum mechanics allows, yet neither particle has a state of its own. Erwin Schrödinger, who coined the word entanglement in 1935, called it 'not one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought'. Know everything about two dice and you know each die; in an entangled pair, the information lives in the relationship.

Make the qubits photons, 0 for horizontal and 1 for vertical, and give Alice and Bob each a polarisation analyser, a polarising beam splitter that sends every photon to one of two detectors: 'pass' if it is found along the analyser's axis, 'blocked' if found perpendicular to it. With both analysers horizontal, the Born rule (Chapter 9) gives one half for both pass, one half for both blocked and zero for a mismatch. They agree on every pair, yet neither can predict a single result: each sees a sequence as random as fair coin tosses.

Now rotate both analysers to the same new angle, say 30 degrees. The agreement does not suffer, because |Φ+⟩ has the same form in every basis of linear polarisation: it equals (1/√2)(|θ⟩|θ⟩ + |θ + 90°⟩|θ + 90°⟩) for any angle θ, an equal superposition of 'both along θ' and 'both perpendicular to θ'. Matched analysers at any angle give perfect correlation, while each local outcome stays perfectly random. The correlation is not stored in either photon. It belongs to the pair.

Ordinary correlations are different. In his 1981 essay 'Bertlmann's socks and the nature of reality', John Bell described his colleague Reinhold Bertlmann, who always wore socks of different colours: see one pink sock and you know the other is not pink, because the colours were fixed when he dressed. The quantum version of the socks is a mixture, a source emitting |HH⟩ or |VV⟩ at random. It agrees perfectly at matched horizontal settings, but at matched 45-degree settings each photon passes with probability one half independently of its partner, so pairs agree only half the time, while |Φ+⟩ still agrees every time. Superposition is not ignorance (Chapter 15).

On 15 May 1935 Physical Review published a four-page paper by Albert Einstein and his younger associates Boris Podolsky and Nathan Rosen of the Institute for Advanced Study in Princeton: 'Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?' EPR demanded that a complete theory contain a counterpart for every element of physical reality, and offered a test for one: if, without in any way disturbing a system, we can predict with certainty the value of a physical quantity, then there exists an element of physical reality corresponding to it.

Their thought experiment used two particles that interact and fly apart with perfectly correlated positions and perfectly anticorrelated momenta, which is allowed because the relative position x₁ − x₂ and the total momentum p₁ + p₂ are compatible observables: in their commutator, particle 1's iħ cancels particle 2's. Measure particle 1's position and you can predict particle 2's with certainty; measure its momentum instead and you can predict particle 2's. If nothing done here can disturb a particle far away, an assumption called locality, then particle 2 must have a definite position and a definite momentum together. No quantum state assigns both (Chapter 8), so, EPR concluded, the quantum description is incomplete.

The natural completion is a hidden-variable theory: extra properties, fixed when the pair is created and carried by each particle, that settle in advance what every measurement will give. It is local if each outcome depends only on that particle's hidden variables and its own detector's setting, never on a distant one, exactly like Bertlmann's socks. Niels Bohr replied that October, under the same title, that the EPR criterion of reality is ambiguous. In 1951 David Bohm recast the argument with two spin-one-half particles of total spin zero, the two-outcome form that later experiments, usually with photon polarisation, put to the test. Einstein's famous phrase came in a 1947 letter to Max Born: physics, he insisted, should be free of 'spooky actions at a distance'.

To handle mismatched settings, score each result +1 (pass, or spin up) or −1 (blocked, or spin down), and define the correlation E as the average product of Alice's and Bob's scores: the probability that they agree minus the probability that they disagree. E = 1 is perfect agreement, E = −1 perfect disagreement, and E = 0 means no more relation than two independent coins.

For photons in |Φ+⟩, put Alice's analyser at angle α and Bob's at β. The amplitude for both to pass is cos(α − β) divided by √2, so both pass with probability one half times cos²(α − β), and both are blocked equally often: coincidences follow the cosine squared of the relative angle θ = α − β. Agreement has probability cos²θ, and the correlation is the cosine of twice the angle, E = cos 2θ: 1 at 0 degrees, 0.71 at 22.5, 0.5 at 30, zero at 45 and −1 at 90. It is Malus's law of 1809, the rule that a polariser passes a fraction cos²θ of polarised light, reborn as a law about pairs.

Bohm's spin version adds one twist. In the singlet |Ψ−⟩ = (1/√2)(|↑↓⟩ − |↓↑⟩), Stern–Gerlach magnets along the same axis always give opposite results. Tilt Bob's magnet by θ, and Chapter 17's rule for spin at an angle, cos²(θ/2), gives the probability of opposite results, so the correlation is minus the cosine of the angle, E = −cos θ. Spins use half-angles because two spin states are mutually exclusive when their axes are 180 degrees apart, while two polarisation states are mutually exclusive at only 90 degrees. A polariser turned by θ behaves like a magnet turned by 2θ.

Could hidden variables mimic these curves? In the simplest local model, drawn beside the quantum curve in the lab, each pair carries a shared, randomly chosen hidden polarisation angle, and each analyser outputs 'pass' if its axis lies within 45 degrees of it. Each side alone sees fifty-fifty, and the model matches quantum mechanics at 0, 45 and 90 degrees, but in between its correlation falls along a straight line, E = 1 − 4θ/π with θ in radians: 0.50 at 22.5 degrees, where quantum mechanics predicts 0.71 and experiments agree. In 1964 John Bell proved that no local model can match the quantum predictions for every combination of settings.

Here is the scheme every first-year student pitches to me. Alice and Bob share entangled pairs a light-year apart, and Alice encodes a message by switching her analyser between two angles. With her analyser at angle θ to Bob's, his photon passes either together with hers, with probability one half times cos²θ, or while hers is blocked, with probability one half times sin²θ. Since cos²θ + sin²θ = 1, the total is exactly one half whatever θ is, and still one half if Alice measures nothing at all. This is no-signalling, and the no-communication theorem guarantees it for every entangled state and any operation Alice performs on her own particle.

The correlations are real, but they live only in the comparison. Each list alone looks like noise, and the cosine law emerges only when the lists are lined up pair by pair, which needs an ordinary channel at or below the speed of light. Nor can Alice choose her outcomes: she chooses the question, never the answer. If she could force her photon to pass, she would force Bob's, and relativity would be in ruins. Nature's randomness is the bodyguard of causality.

Does Alice's measurement at least push Bob's photon into line? Textbook quantum mechanics updates the joint state once Alice records 'pass' at angle α, describing Bob's photon as polarised along α. Whether that update is a physical change or a revision of information depends on interpretation (Chapter 22): pilot-wave theory posits a nonlocal influence that can never be harnessed to signal, many-worlds denies collapse altogether, and information-based views call it bookkeeping. All agree that Bob's own statistics never change, so no energy is delivered and no message crosses. And when the measurements are too far apart for light to link them, relativity lets observers disagree about which came first, so 'Alice pushed Bob' and 'Bob pushed Alice' fit the data equally well, and no experiment can single out a pusher.

Entangled pairs arise when two systems are born together under a shared conservation law. Early sources were atomic cascades, in which an excited calcium atom emits two photons in quick succession with correlated polarisations, measured by Carl Kocher and Eugene Commins at Berkeley in 1967. Since the late 1980s the workhorse has been spontaneous parametric down-conversion, or SPDC, in a nonlinear crystal such as beta-barium borate, whose response to light is not strictly proportional to the field, letting light waves mix. Occasionally a pump photon vanishes and two lower-energy photons, the signal and the idler, are born at the same instant.

Spontaneous: no input beam triggers the split; it is seeded by vacuum fluctuations. Parametric: the crystal ends exactly as it began, keeping no energy. Down-conversion: each daughter has a lower frequency than its parent. Energy conservation makes the reciprocals of the wavelengths add, 1/λ_pump = 1/λ_signal + 1/λ_idler, so a violet pump photon of wavelength 405 nm and energy 3.06 eV can become two photons of wavelength 810 nm and energy 1.53 eV each. Momentum conservation, called phase matching, sends the pair out on opposite sides of the pump beam. The process is feeble: even optimised waveguide sources convert only a few pump photons per million, and bulk crystals far fewer.

Polarisation entanglement needs one more trick. In type-II down-conversion, one daughter emerges horizontally polarised and the other vertically, along two cones of light tilted away from the pump. Where the cones intersect, a photon could be the horizontal one from the first cone or the vertical one from the second, and nothing records which. As in the double-slit experiment of Chapter 7, indistinguishable alternatives add as amplitudes, so the pair emerges in a superposition of 'H here, V there' and 'V here, H there', which birefringent plates can turn into any Bell state, including |Φ+⟩. Paul Kwiat, Anton Zeilinger and colleagues reported this bright source in December 1995.

Since the 1990s entanglement has been treated as a resource, counted in ebits, one ebit being one maximally entangled pair. Teleportation, proposed in 1993 by Charles Bennett and five colleagues, transfers an unknown quantum state from Alice to Bob: Alice makes a joint measurement on the particle carrying the state and her half of a shared pair, and sends Bob two ordinary bits; Bob applies one of four corrections, and his particle takes on the original state while Alice's measurement erases it from her particle, so no copy is made. Until the bits arrive, Bob's particle is pure noise. Zeilinger's group in Innsbruck demonstrated teleportation with photons in 1997.

In 1991 Artur Ekert proposed entanglement-based cryptography: Alice and Bob measure shared pairs at random settings, turn the matched results into a shared secret key, and use the rest to test the correlations, which an eavesdropper cannot touch without degrading them detectably. In 2017 the satellite Micius distributed entangled photon pairs to two ground stations 1,203 kilometres apart. In computing, the Hadamard and CNOT gates of Chapter 23 turn |00⟩ into |Φ+⟩ in two steps, and large-scale entanglement is an essential ingredient of quantum algorithms, though by itself it guarantees no speed-up.

Two properties make the resource precious. Entanglement is monogamous: a particle maximally entangled with one partner cannot be entangled with any other, which is why an eavesdropper cannot share the correlations. And it is fragile: stray interactions with the environment spread it away, the decoherence of Chapter 22, so protecting entanglement is the central engineering challenge of quantum technology.