Superposition & Measurement

Today we apply the grammar of chapter 14 to the simplest system there is, a qubit, short for quantum bit: anything with exactly two perfectly distinguishable states, labelled |0⟩ and |1⟩, such as a photon polarized horizontally or vertically, or an electron whose spin along a chosen axis is up or down. Because the Schrödinger equation is linear, as chapter 10 showed, any weighted sum of allowed states is also allowed. That sum is a superposition, |ψ⟩ = α|0⟩ + β|1⟩, and the weights α and β are complex numbers called probability amplitudes.

The amplitudes carry the odds. By the Born rule of chapter 9, asking 'zero or one?' returns 0 with probability |α|² and 1 with probability |β|², so |α|² + |β|² = 1, the normalization condition; the state 0.6|0⟩ + 0.8|1⟩ qualifies, since 0.36 + 0.64 = 1. Amplitudes also carry a phase. An overall phase shared by both terms changes nothing measurable, but the relative phase between α and β is physical. A general qubit needs just two angles: the amplitude of |0⟩ is the cosine of θ/2, and the amplitude of |1⟩ is the sine of θ/2, turned in the complex plane by a phase angle φ. The angle θ sets the odds; φ decides how the state interferes.

Now the crucial subtlety. A superposition is not 'both 0 and 1 at once', a slogan that sounds profound and explains nothing. It is one perfectly definite state, and what makes it a superposition is its relation to a particular measurement. Take |+⟩ = (1/√2)(|0⟩ + |1⟩). Asked 'zero or one?', it answers at random. But it is an eigenstate of the Pauli matrix σₓ of chapter 14, whose eigenstates |+⟩ and |−⟩ = (1/√2)(|0⟩ − |1⟩) form the X basis, so asked 'plus or minus?', it answers plus every time. For a photon, |+⟩ is light polarized at 45 degrees, which a polarizer turned to 45 degrees passes with certainty. Superposition is a relationship between a state and a basis.

Here is the most tempting mistake in quantum mechanics. I flip a coin, slam a teacup over it, and ask you the odds. Fifty-fifty, you say, and you are right, but only about your knowledge: the coin is already heads or tails. A probabilistic blend of definite states like this is called a mixture, and its probabilities measure ignorance. Is |+⟩ just a quantum coin under a cup? Measure only in Z and you cannot tell, because both give 0 half the time. To catch the difference, ask a question that makes the possibilities interfere.

Ask the X question. The amplitude for plus is the sum α + β divided by √2; for |+⟩, with α = β = 1/√2, it is exactly 1, so plus comes up every time, while the amplitude for minus, α − β divided by √2, cancels to zero. That cancellation is destructive interference, and only amplitudes, which can be negative or complex, can do it. Each hidden coin is really |0⟩ or really |1⟩, and each of those gives plus with even odds, so any blend gives plus exactly half the time. Probabilities add; amplitudes add and then get squared. That rule from the double slit of chapter 7 is the whole difference between superposition and ignorance.

Experiment settles it. In 1986 Philippe Grangier, Gérard Roger and Alain Aspect at the Institut d'Optique in Orsay sent single photons to a beam splitter and showed that each was registered on one side or the other, with joint detections on both sides far rarer than any classical wave allows. Then, with the same source and detectors, they sent the photons through a Mach-Zehnder interferometer, which splits and then recombines the two possible paths, and recorded fringes with a visibility above 98 percent. Had every photon secretly taken one arm, each exit would have received half the photons whatever the arm lengths. Instead, at the right setting, nearly every photon left by one exit, steered by a relative phase no hidden-coin story contains.

Now make measurement precise. In chapter 14, each observable became a Hermitian operator whose eigenvalues are the possible results and whose eigenstates |k⟩ form a basis. A projective measurement, the textbook ideal, obeys three rules. First, the outcome is always an eigenvalue. Second, the probability of result k is the squared magnitude of ⟨k|ψ⟩, the overlap between |k⟩ and the state: |α|² for 0 and |β|² for 1 in a Z measurement. Third, immediately afterwards the system is in the matching eigenstate |k⟩. This third rule is the projection postulate, better known as collapse: α|0⟩ + β|1⟩ jumps to whichever of |0⟩ or |1⟩ was found.

Max Born proposed the probability rule in 1926; in 1927 Werner Heisenberg wrote that every determination of position reduces the wave packet; Paul Dirac's textbook of 1930 said a measurement makes the system jump into an eigenstate of the quantity measured. In 1932 John von Neumann made the jump an axiom, distinguishing two utterly different kinds of change: process 1, the abrupt, unpredictable change caused by measurement, and process 2, smooth, deterministic Schrödinger evolution. Keep that pair in mind. It returns to haunt us.

This randomness is not that of a sloppy instrument. Prepare a million qubits in exactly the state 0.6|0⟩ + 0.8|1⟩ and measure each perfectly: nothing in the theory says which will give 1, yet very nearly 64 percent will. Single outcomes are random; the statistics are exact. Whether collapse is a physical process, an update of information, or never happens at all is interpretation-dependent, the quarrel of chapter 22. But every physicist uses the rule the same way: after a result, predict the future from the eigenstate found.

The aftermath rule has a testable consequence. Measure Z and get 1, and the state is now |1⟩; measure Z again at once and, since the overlap of |1⟩ with itself is 1, you get 1 with certainty. This repeatability is what makes a measurement a measurement: a reading that changed every time you glanced at it would record nothing. It also motivated the collapse rule, since the second answer is certain even though the first was not.

Now ask sideways. Prepare |0⟩ and measure X: plus and minus come up fifty-fifty, because |0⟩ = (1/√2)(|+⟩ + |−⟩). Suppose you get plus. Measure Z again and you find 0 or 1 at random: the X measurement erased the Z information. Z and X are incompatible observables. Their matrices do not commute, as you found in chapter 14, so no state is an eigenstate of both, and a sharp answer to one question forces a maximally uncertain answer to the other. In chapter 17, silver atoms will do exactly this in chains of Stern-Gerlach magnets.

You can watch the same logic with three polarizing filters. Two crossed at 90 degrees, vertical then horizontal, block essentially all light. Slide a third, at 45 degrees, between them, and one-eighth of the original unpolarized light reappears. Wave optics gets the intensities from Malus's law; the quantum account follows each indivisible photon. The diagonal filter measures it in the diagonal basis: it passes with probability equal to the cosine squared of 45°, one half, emerging diagonally polarized with its vertical history wiped out, and gets another fifty-fifty chance at the last filter. If filters merely read fixed labels, adding one could only remove light. Here it lets more through, because measurement changes the state.

Repeatability has a startling corollary. Suppose a resonant radio wave steadily rotates a qubit from |0⟩ toward |1⟩, which it would reach after a time T. At short times the amplitude of |1⟩ grows in proportion to the elapsed time, so the probability grows only as the time squared. Check early and you almost certainly find 0, and collapse resets the state to |0⟩, restarting the slow creep. With N checks spread over T, each finds a flip with probability of only about (π/2N)², so the total risk is roughly π²/(4N), shrinking toward zero as N grows. In 1977 Baidyanath Misra and George Sudarshan named this the quantum Zeno effect, after Zeno of Elea, whose paradox argued that a flying arrow, motionless at every instant, never moves.

It was tested in 1990 at the National Institute of Standards and Technology in Boulder by Wayne Itano, Daniel Heinzen, John Bollinger and David Wineland. They held about 5,000 beryllium ions in a Penning trap and applied a radio-frequency pulse, lasting 256 ms, that would carry every ion between two sublevels of its ground state. During it they fired 1 to 64 brief ultraviolet laser pulses: an ion in one sublevel scattered light, an ion in the other stayed dark, so each pulse was a Z measurement. With one pulse, essentially every ion made the transition; with two, half did; with 64, fewer than 8 percent did, in agreement with the calculations. Wineland later shared the 2012 Nobel Prize in Physics with Serge Haroche.

One detail matters enormously. Each ion in the scattering sublevel threw off about 72 photons per pulse, but the collecting optics could catch only about 2 in every 10,000 of them, far fewer than one per ion. The authors stressed that what counts is the number of photons scattered, not detected: the measurement was complete once photons flew off carrying the information, and no eye, screen or mind was needed. Nor does interruption always slow things down. In 2001 Mark Raizen's group at the University of Texas at Austin saw both the Zeno effect and the anti-Zeno effect, in which frequent measurement speeds decay up, in sodium atoms tunnelling out of a trap made of laser light.

Now the confession. Quantum mechanics, as usually taught, runs on two rules for change. Rule A is the Schrödinger equation: deterministic, continuous, reversible and linear, so a superposition going in comes out as the same superposition of the separately evolved parts. Such evolution is called unitary, because it preserves the length of every state and every overlap between states. Rule B is collapse: random, abrupt, irreversible and not linear at all, since it discards every branch but one. Rule B is invoked for measurements, but a measuring device is made of atoms obeying Rule A like everything else. Which rule applies, and when?

Apply Rule A to the measurement itself. A perfect detector that shows '0' for |0⟩ and '1' for |1⟩ produces |0⟩|A₀⟩ and |1⟩|A₁⟩, where |A₀⟩ and |A₁⟩ are pointer readings. Linearity then forces a qubit that starts as α|0⟩ + β|1⟩ to end as α|0⟩|A₀⟩ + β|1⟩|A₁⟩: not one reading but a superposition of two, with the pointer entangled with the qubit, a word chapter 20 makes precise. Yet every notebook records one result, with frequencies |α|² and |β|². In 1995 the philosopher Tim Maudlin put the clash crisply: the wavefunction is a complete description; it always evolves by the linear Schrödinger equation; measurements have single, definite outcomes. All three cannot be true. Collapse theories modify the equation, pilot-wave theory adds particle positions, and many-worlds denies the single outcome.

Because the chain from qubit to detector to eye to brain can be extended indefinitely, some physicists once wondered whether it ends only at a conscious mind; Eugene Wigner floated the idea in 1961 and later abandoned it, and no experiment supports it. The mainstream view is plain: a measurement is a physical interaction that records information in an apparatus or its environment, and, as the beryllium ions showed, quantum systems respond to that record whether or not anyone reads it. Interpretations disagree about whether collapse is real, but none of the mainstream ones lets a mind cause it. The genuine puzzle is Rule A against Rule B.

In 1935 Erwin Schrödinger imagined a cat sealed in a box with a radioactive atom, a Geiger counter and a vial of poison: Rule A, applied without limit, would leave the cat in a superposition of alive and dead. He meant it as a criticism of taking the formalism too literally. So why do we never see dust or cats in superpositions? The leading answer is decoherence, developed by Dieter Zeh from 1970 and by Wojciech Zurek, Erich Joos and others in the 1980s. A large object is never alone: air molecules strike it and light bounces off it, and every encounter leaves a faint record of where it is, just as the detector recorded the qubit.

Once the surroundings hold a record, the object is entangled with them, and for any measurement on the object alone, the interference terms are multiplied by the overlap between the environment's two records. When the records differ, that overlap plunges toward zero and the object behaves exactly like a mixture. The speed is staggering. For a dust grain a hundredth of a millimetre across, superposed across its own width, standard estimates give a decoherence time of about 10⁻³¹ seconds in air, 10⁻¹⁴ seconds in the best laboratory vacuum, and one second in deep space, lit only by the cosmic microwave background. A warm, breathing cat never stands a chance.

Decoherence has been switched on and off in the laboratory. In 1996 Michel Brune, Serge Haroche and colleagues in Paris watched a superposition of two microwave fields of a few photons, held in a superconducting cavity, lose its coherence, faster the more different its components were. In 2003 Klaus Hornberger, Markus Arndt, Anton Zeilinger and colleagues in Vienna leaked gas into an interferometer for C₇₀ fullerene molecules: the higher the pressure, the more collisions recorded the molecules' paths, and the fainter the fringes, as predicted.

Decoherence explains why interference vanishes for big things, why the everyday world looks definite in position rather than in some exotic basis, and why records are stable. What it does not do, by itself, is pick one outcome: the total state of object plus environment still holds both branches, which merely can no longer interfere. Whether that ends the story is where the interpretations of chapter 22 part company.