Open Quantum Systems and Decoherence
Every system treated so far in this part has been closed: a state vector, a Hamiltonian, and unitary evolution for all time. No system in a laboratory is closed. A trapped ion radiates, a superconducting circuit shares its substrate with two-level defects, a molecule in flight scatters background gas and emits thermal photons, and an apparatus is by construction coupled to whatever will record its reading. This chapter is the systematic theory of what remains when the rest of the world is traced away: the density operator [Landau:1927] [vonNeumann:1927], the completely positive maps that carry it forward [Stinespring:1955] [Kraus:1971] [Choi:1975], the master equations that generate those maps in continuous time [Lindblad:1976] [Gorini:1976], and the loss of interference — decoherence — that they describe [Zeh:1970] [Zurek:1981].
Its place in the book is fixed by what it explains. The entanglement of Entanglement and Bell Tests is here turned from a resource into a nuisance: coupling to an environment entangles a system with degrees of freedom nobody monitors, and the interference terms that make quantum mechanics quantum are not destroyed but exported, becoming inaccessible to any local measurement. That process is fast, and it is quantitative — the measured decay of a mesoscopic superposition in a microwave cavity [Brune:1996] and of interference fringes of hot molecules [Hackermueller:2004] are the two cleanest instances, and this chapter states what will be derived and what was measured in each. Decoherence explains why the classical world of Hamiltonian Mechanics looks classical, and this treatise records plainly what it does not do: it does not select an outcome, and the residual question belongs to Interpretations (Evidence-Anchored). The standard monographs are [Breuer:2002] [Schlosshauer:2007].
Open Quantum Systems and Decoherence: all derivations of this chapter are pending.
States of an open system
The density operator
[Reserved: the density operator introduced independently by Landau, from the damping problem in wave mechanics [Landau:1927], and by von Neumann in his probabilistic reconstruction of quantum mechanics [vonNeumann:1927] [vonNeumann:1932]; its defining properties \(\rho=\rho^{\dagger}\), \(\rho\geq0\), \(\tr\rho=1\); expectation values as \(\avg{A}=\tr(\rho A)\); the von Neumann equation \(\ii\hbar\,\dd\rho/\dd t=\comm{H}{\rho}\) as the state-operator form of the Schrödinger evolution of The Postulates of Quantum Mechanics; pure states as the projectors \(\rho^{2}=\rho\) and the purity \(\tr\rho^{2}\) as the scalar that measures departure from them.]
Reduced states and the partial trace
[Reserved: the partial trace as the unique map reproducing the statistics of every observable of the subsystem; the Schmidt decomposition of a bipartite pure state [Schmidt:1907] and the equality of the two reduced spectra; purification, so that every mixed state is the restriction of a pure state on a larger space, which is the state-space face of Stinespring's dilation theorem [Stinespring:1955]; the bipartite machinery of Entanglement and Bell Tests reused here with the roles reversed.]
Proper and improper mixtures
[Reserved: the distinction, due in its sharp form to d'Espagnat [dEspagnat:1976], between a statistical mixture prepared by a classical randomizer and the reduced state of one half of an entangled pair; the two are described by the same operator and are indistinguishable by any measurement on the subsystem alone, and they are not the same physical situation — the second can be undone by recohering the environment, as the spin echo of Section 87.7.2 demonstrates. Why this is the point at which decoherence stops short of solving the measurement problem (Section 87.4.6).]
Entropy and mixedness
[Reserved: the von Neumann entropy \(S=-\tr(\rho\ln\rho)\) [vonNeumann:1932], its vanishing exactly on pure states, its invariance under unitary evolution — hence the impossibility of a closed system's entropy growing, which is why an environment is mandatory; subadditivity and the strong subadditivity inequality of Lieb and Ruskai [Lieb:1973]; entanglement entropy of a bipartition as the reduced-state entropy; the relation to the thermodynamic entropy of Statistical Mechanics.]
Quantum channels
Complete positivity
[Reserved: why positivity of a map is not enough — the transpose is positive and its extension to half of an entangled pair is not, so a physical map must be completely positive; Stinespring's dilation theorem for completely positive maps on operator algebras [Stinespring:1955]; Choi's finite-dimensional criterion, that a map is completely positive if and only if its Choi matrix is positive semidefinite [Choi:1975]; trace preservation, and the CPTP maps as exactly the reduced dynamics obtainable from a unitary evolution on system plus environment in a product initial state.]
The Kraus representation
[Reserved: Kraus's operator-sum form \(\rho\mapsto\sum_{k}K_{k}\rho K_{k}^{\dagger}\) with \(\sum_{k}K_{k}^{\dagger}K_{k}=\identity\) [Kraus:1971]; the non-uniqueness of the operator set up to an isometry, and what is therefore physical about it; the reading of the \(K_{k}\) as measurement outcomes on the environment, which connects the formalism to the generalized measurements of The Postulates of Quantum Mechanics; the number of Kraus operators bounded by the square of the dimension.]
The standard channels
[Reserved: the qubit channels that every later section uses — dephasing, which destroys coherences in a fixed basis while leaving populations untouched; amplitude damping, the channel of spontaneous emission (Quantum Optics and the Photon); the depolarizing channel; and the two relaxation times \(T_{1}\) and \(T_{2}\) they define, with the inequality \(T_{2}\leq2T_{1}\) that separates energy relaxation from pure dephasing [Nielsen:2010].]
Process tomography
[Reserved: reconstruction of an unknown channel from the states it produces on a tomographically complete input set, the prescription of Chuang and Nielsen [Chuang:1997]; its first full application to a two-qubit gate [OBrien:2004]; the scaling of the required number of settings, and randomized benchmarking as the practical substitute; a measured process matrix is what makes “decoherence” a number rather than a word.]
Master equations
Projection-operator derivations
[Reserved: the exact Nakajima–Zwanzig equation obtained by projecting the total von Neumann equation onto the relevant part of the state [Nakajima:1958] [Zwanzig:1960], with its memory kernel and inhomogeneous initial term; Redfield's earlier relaxation theory for magnetic resonance [Redfield:1957]; the Born and Markov approximations, weak coupling and short environment correlation time, stated as inequalities on time scales rather than as slogans; the secular (rotating-wave) approximation, without which the resulting equation can produce negative probabilities.]
The Lindblad–Gorini–Kossakowski–Sudarshan generator
[Reserved: the theorem fixing the most general generator of a completely positive trace-preserving semigroup, proved for bounded generators by Lindblad [Lindblad:1976] and, in \(N\)-level systems, by Gorini, Kossakowski and Sudarshan [Gorini:1976]: a Hamiltonian commutator plus a dissipator built from jump operators \(L_{k}\) with non-negative rates. Davies's derivation of the semigroup as the weak coupling limit [Davies:1974], which is what ties the abstract theorem to a physical bath; the detailed-balance condition that makes the Gibbs state stationary; why no Markovian generator can be exact.]
Quantum Brownian motion
[Reserved: the influence functional of Feynman and Vernon [Feynman:1963], integrating out a linear bath exactly; the Caldeira–Leggett model of a particle bilinearly coupled to a continuum of oscillators, its Ohmic spectral density and its high-temperature master equation [Caldeira:1983], together with the demonstration that dissipation suppresses macroscopic tunnelling [Caldeira:1981]; the position-diffusion term whose coefficient sets the decoherence rate of a spatial superposition; the spin–boson problem and its phase diagram [Leggett:1987]; the classical limit returning the Langevin equation of Experiment: Brownian Motion and Avogadro's Number and the transport coefficients of Nonequilibrium Thermodynamics and Transport.]
Unravellings and quantum trajectories
[Reserved: the Monte Carlo wave-function method, in which the Lindblad evolution is recovered as the average over stochastic pure state trajectories punctuated by jumps [Dalibard:1992], and the diffusive alternative of quantum-state diffusion [Gisin:1992]; that the unravelling is not unique but is fixed by what the environment is actually measured with; the observation of individual quantum jumps in a single trapped ion by electron shelving [Nagourney:1986], which turned an ensemble formalism into a record of single events.]
The fluorescence of a single trapped ion driven hard on a strong transition, and coupled weakly to a long-lived third level, is not a smoothly decaying signal. It is a random telegraph: the ion is either fully bright or completely dark, it switches between the two abruptly on the time scale of the detector, and the intervals it spends in each are distributed exponentially [Nagourney:1986]. Averaged over many such records the signal reproduces the smooth exponential relaxation of ensemble theory exactly. The measurement therefore establishes something an ensemble experiment cannot: that the exponential decay law of an ensemble carries no information about whether the individual system changes gradually or in a single event, and that here it does the latter.
Derivation. Let the ion have a bright configuration \(B\), cycling on the strong transition and scattering photons at a rate the detector resolves, and a dark configuration \(D\), shelved in the metastable level and scattering none; let \(\Gamma_{\uparrow}\) be the rate of the transition \(B\to D\) and \(\Gamma_{\downarrow}\) that of \(D\to B\). Markovian dynamics — the environment forgetting between events, which is the approximation stated as an inequality on time scales in Section 87.3.1 — means the switching probability per unit time is constant, so the probability of still being bright after a time \(t\) obeys \(\dd P/\dd t=-\Gamma_{\uparrow}P\) and
an exponential distribution of bright intervals with mean \(1/\Gamma_{\uparrow}\), and likewise for the dark ones with \(\Gamma_{\downarrow}\). Every individual record is therefore a two-valued telegraph with exponentially distributed dwell times, which is what is seen. The ensemble probability \(p(t)\) of being dark obeys the rate equation
whose solution relaxes smoothly and exponentially to \(\Gamma_{\uparrow}/(\Gamma_{\uparrow}+\Gamma_{\downarrow})\). The smooth ensemble curve and the discontinuous single record are thus not in conflict — the first is the mean of the second — and no measurement of the first could have decided between them. That is the general content of an unravelling: a stochastic pure-state trajectory whose average is the Lindblad evolution of Section 87.3.2. It is also why the unravelling is not unique. Which trajectories are realized is fixed by what the environment is monitored with, and here it is fixed by the fact that the experiment counts photons.
∎Beyond Markov
[Reserved: structured environments in which the memory kernel cannot be collapsed — solid-state qubits with \(1/f\) noise, photonic band gaps, strongly coupled molecular aggregates; measures of non-Markovianity based on the backflow of distinguishability [Breuer:2009] and on divisibility of the dynamical map [Rivas:2010], and the fact that the two do not agree; why non-Markovian dynamics is not merely a correction but changes what the coherence time means.]
Decoherence
Environment-induced superselection
[Reserved: Zeh's argument that macroscopic systems are never isolated and that this, not any modification of the theory, is what makes their superpositions unobservable [Zeh:1970]; Zurek's pointer-basis analysis [Zurek:1981] and environment-induced superselection [Zurek:1982], in which the interaction with the environment singles out a preferred basis and forbids superpositions across it; the review that consolidated the programme [Zurek:2003]; the honest statement that einselection derives a superselection rule from ordinary unitary dynamics and postulates nothing new.]
Pointer states
[Reserved: the pointer states as those least perturbed by the system–environment coupling — exact eigenstates of the interaction in the ideal case, and predictability-sieve selected states in general [Zurek:1981] [Zurek:2003]; why position is the pointer observable for a particle scattering photons and air molecules while energy eigenstates are pointer states for a weakly damped oscillator; coherent states as the pointer states of the harmonic oscillator, the classical limit already met in Elementary Quantum Systems and Hamilton–Jacobi Theory and the Optical–Mechanical Analogy.]
Decoherence rates
[Reserved: the scattering formula of Joos and Zeh [Joos:1985] for the decay of spatial coherence, linear in the scattering rate and quadratic in the separation for small separations; the resulting estimates, which give decoherence times shorter than relaxation times by enormous factors — a dust grain in air loses spatial coherence over micrometre separations in far less than a nanosecond, while a single atom in ultrahigh vacuum stays coherent for seconds; the collisional, thermal-photon and gravitational channels compared; why the numbers, not the mechanism, are what make classicality inevitable.]
Quantum Darwinism
[Reserved: Zurek's account of objectivity as redundancy — the pointer observable is the one whose record is imprinted many times over in disjoint fragments of the environment, so that many observers can learn it independently without disturbing it [Zurek:2009]; the redundancy plateau in the partial information plot as the signature; the first laboratory signatures of the plateau in photonic cluster states [Ciampini:2018]; the claim's status as a consequence of unitary dynamics plus a factorized environment, not a new postulate.]
Thermalization of an isolated system
[Reserved: the complementary route to classical statistics, in which the “bath” is the rest of the same closed system — Deutsch's random-matrix argument [Deutsch:1991] and Srednicki's eigenstate thermalization hypothesis [Srednicki:1994], under which each individual energy eigenstate already encodes thermal expectation values; its direct test in an isolated ultracold Bose gas, where the growth of entanglement entropy was measured while the global state stayed pure [Kaufman:2016]; the link forward to the statistical mechanics of Statistical Mechanics and back to the scattering description of Scattering Theory.]
A small, completely isolated quantum system — an ultracold gas of a few atoms in an optical lattice, prepared in a pure state and shielded from every bath — relaxes so that each local observable takes the value a thermal ensemble would assign it, while the global state remains pure and its evolution remains unitary. What grows during the relaxation is the entanglement entropy of a subregion, measured by interfering two identical copies of the system; it rises and saturates at the value the thermodynamic entropy of that subregion would have, while the entropy of the whole stays zero throughout [Kaufman:2016]. Thermalization therefore needs no external environment: a system large enough to contain its own complement is its own bath, and the environment of Section 87.2 may be internal.
Eigenstate thermalization: that the expectation of a local observable in an individual many-body energy eigenstate is a smooth function of the energy density alone, so that any superposition confined to a narrow energy window relaxes to the microcanonical value of every local observable; and that the entanglement entropy of a small subregion then grows to the thermodynamic entropy that subregion would carry at the corresponding temperature
What decoherence does not settle
[Reserved: the precise statement of the limit — decoherence converts an improper mixture in a preferred basis into a state operationally indistinguishable from a classical ensemble, and does not select which member is realized; the circularity objection, that the interference terms are still present in the global state; why this leaves the measurement problem exactly where Interpretations (Evidence-Anchored) takes it up, and why no experiment in this treatise distinguishes the readings. The section is to say this plainly rather than allow the formalism to suggest more than it proves.]
Decoherence observed
Cavity quantum electrodynamics
[Reserved: the Paris experiment in which a mesoscopic superposition of two coherent states of a microwave field, separated by a few photons in phase space, was created with a circular Rydberg atom and its decoherence followed in real time with a second atom [Brune:1996]; the observed scaling of the decoherence rate with the square of the separation, the first direct measurement of the rate rather than of its consequences; later reconstruction of the full Wigner function of such a state and of its decay [Deleglise:2008]; the apparatus belongs with Quantum Optics and the Photon.]
A superposition of two coherent states of a microwave cavity field, separated by a few photons in phase space, loses its interference much faster than the field loses its energy, and the ratio of the two rates is the square of the phase-space separation. In the Paris experiment a circular Rydberg atom prepared such a superposition and a second atom, sent after a controlled delay, read out the coherence that survived [Brune:1996]; the coherence decayed exponentially in the delay, with a time constant shorter than the cavity's own energy damping time by the measured factor, and the dependence on the separation was measured by varying it. The observation is a direct one — the rate itself, read against a delay — and not classicality inferred after the fact from an outcome that looks classical.
Derivation. Grant the generator of Section 87.3.2 for a cavity losing photons to a zero-temperature environment at rate \(\kappa\), with the single jump operator \(L=\sqrt{\kappa}\,a\):
Coherent states are eigenstates of the annihilation operator, \(a\ket{\alpha}=\alpha\ket{\alpha}\) and \(\bra{\beta}a^{\dagger}=\beta^{*}\bra{\beta}\), so on the coherence \(\rho=\ket{\alpha}\bra{\beta}\) every term of Equation (87.3) acts by multiplication:
The imaginary part of the bracket only turns the phase. Its real part is \(\operatorname{Re}(\alpha\beta^{*})-\tfrac{1}{2}\abs{\alpha}^{2} -\tfrac{1}{2}\abs{\beta}^{2}=-\tfrac{1}{2}\abs{\alpha-\beta}^{2}\), so the modulus of the coherence decays at the rate
whereas the mean photon number obeys \(\dd\langle n\rangle/\dd t=-\kappa\langle n\rangle\) and so relaxes at the rate \(\kappa\). The ratio is
the square of the phase-space separation. At a separation of ten photon amplitudes the interference is gone fifty times faster than the energy; for a separation of macroscopic size, where \(\abs{\alpha-\beta}^{2}\) counts the quanta by which the two branches differ, the factor is astronomical. This is the quantitative content of the claim that large objects look classical: nothing in Equation (87.3) changes with size except one number in an exponent, and no new physics is invoked at any size.
∎Engineered reservoirs in ion traps
[Reserved: superpositions of motional states of a single trapped ion exposed to a deliberately applied, characterized noise, so that the environment is a control parameter rather than a nuisance [Myatt:2000] [Turchette:2000]; the measured decay of coherence against the amplitude of the applied noise and against the size of the superposition, confirming the predicted scaling over an order of magnitude; why an engineered reservoir is the cleanest possible test of the Lindblad form of Section 87.3.2.]
Matter-wave interferometry
[Reserved: interference of large molecules [Arndt:1999] as the platform (theory in Matter Waves); controlled collisional decoherence, in which fringe contrast decays exponentially with background gas pressure exactly as predicted [Hornberger:2003]; thermal decoherence, in which the molecules are heated until they emit their own blackbody photons and destroy their own interference [Hackermueller:2004] — decoherence with no external environment at all, the environment being the internal vibrational degrees of freedom; both are quantitative confirmations of the Joos–Zeh formula [Joos:1985].]
A beam of large molecules crossing a grating interferometer produces fringes whose contrast can be read directly. Admitting background gas reduces that contrast exponentially in the gas pressure, with no shift of the fringe positions and no loss of molecules beyond ordinary attenuation [Hornberger:2003]. Heating the molecules until they emit their own thermal photons reduces it in the same way, the environment in that case being the molecule's own internal vibrational degrees of freedom, with nothing external involved at all [Hackermueller:2004]. In both cases the contrast returns when the disturbance is removed, and in neither does anything about the molecule's own dynamics change. What is measured is therefore the export of coherence into a record kept somewhere else, not a modification of the Schrödinger equation of The Postulates of Quantum Mechanics.
Derivation. Let the interferometer put the molecule in a superposition of two paths separated by \(\Delta x\), and let a single environment particle scatter from it. Write the environment state after the scattering as \(\ket{\chi_{1}}\) or \(\ket{\chi_{2}}\) according to which path the molecule took. Linearity of the joint evolution gives
and tracing out the environment multiplies the molecule's off-diagonal element by \(\langle\chi_{1}|\chi_{2}\rangle\). Nothing has been done to the molecule — its two path amplitudes are untouched, and its momentum is barely changed — yet the fringe contrast is now \(\abs{\langle\chi_{1}|\chi_{2}\rangle}\). When the scattered particle's wavelength is short compared with \(\Delta x\) the two environment states are essentially orthogonal, that overlap is essentially zero, and a single event suffices. The visibility after a time \(t\) is then simply the probability that no such event has occurred,
with \(n\) the number density of the gas, proportional to its pressure at fixed temperature, \(\sigma\) the collision cross section of Scattering Theory and \(v_{\text{rel}}\) the mean relative speed. Hence the observed exponential in pressure, with a decay constant that is an ordinary collision rate and contains no new parameter. Two features deserve marking. This is not attenuation: the molecule is deflected by a negligible angle and still arrives, but arrives without its fringe. And the total state in Equation (87.7) stays pure throughout, so the coherence has been relocated rather than destroyed — which is the distinction of Section 87.1.3, and the reason the refocusing of Section 87.7.2 can sometimes retrieve it.
∎Solid-state qubits
[Reserved: coherent oscillations in a single-Cooper-pair box [Nakamura:1999] and the coherence times of superconducting and spin qubits since — from nanoseconds to hundreds of microseconds, an improvement of five orders of magnitude achieved by removing environments one at a time; \(1/f\) charge and flux noise and two-level defects as the dominant channels; why this is the branch of physics in which decoherence theory is used as an engineering tool.]
The quantum Zeno effect
The Misra–Sudarshan theorem
[Reserved: the theorem that for a state with a Hamiltonian of bounded second moment the short-time survival probability is quadratic in the elapsed time, \(1-(\Delta E\,t/\hbar)^{2}\), so that \(N\) ideal projective measurements in a fixed interval leave a survival probability tending to one as \(N\to\infty\) — a continuously observed unstable system never decays [Misra:1977]; the conditions under which the quadratic regime is destroyed, and the anti-Zeno regime in which measurement accelerates decay; the relation to the projection postulate of The Postulates of Quantum Mechanics.]
Observations
[Reserved: the trapped-ion demonstration of Itano and collaborators, in which the induced transition between two hyperfine levels was progressively suppressed as the number of interrogating optical pulses grew [Itano:1990]; the objection that the result follows from ordinary unitary dynamics of the three-level system and requires no collapse postulate [Ballentine:1991], which this treatise states as part of the result rather than as a footnote; observation of both the Zeno and the anti-Zeno regime in the tunnelling decay of atoms from an accelerating optical lattice [Fischer:2001], where the crossover between them was measured.]
A driven transition between two levels of a trapped ion, which transfers the whole population in a fixed time when left alone, is progressively suppressed as the number of interrogating pulses applied during that time is increased: the population surviving in the initial level rises monotonically towards unity with the number of pulses [Itano:1990]. The effect is real and reproducible. Its interpretation is not settled by the observation, and this treatise says so as part of the result rather than in a footnote: the same numbers follow from the ordinary unitary dynamics of the three-level system with its driving and probe fields, so the experiment does not by itself demonstrate a collapse of the state, and what it does and does not decide belongs with Interpretations (Evidence-Anchored).
Derivation. Let \(\ket{\psi}\) be the initial state and \(H\) the generator of its evolution. Expanding \(\ket{\psi(t)}=\ee^{-\ii Ht/\hbar}\ket{\psi}\) to second order in \(t\) and taking the overlap with \(\ket{\psi}\), the terms linear in \(t\) cancel between the amplitude and its conjugate and there remains
The initial decay is quadratic, not linear, and that is the whole of the effect: a genuine rate would give \(P(t)=1-\lambda t\), for which the argument below does nothing at all. Divide a fixed interval \(T\) into \(N\) equal parts and project onto \(\ket{\psi}\) after each. If each projection resets the state, the survival probability is the product of \(N\) independent factors,
so a continuously watched system never leaves its initial state. The hypotheses carry the result and must be stated with it. The second moment \(\Delta E\) must be finite, which fails for a Hamiltonian unbounded in the relevant sense and is exactly the case of a decay that is exponential at all times; and the interval \(T/N\) must still lie inside the quadratic regime of Equation (87.9), which for a decay into a broad continuum is extremely short. When the second condition fails and the probe instead broadens the level into the continuum, the same analysis gives an increase of the decay rate — the anti-Zeno regime, which is measured in the same systems.
∎Fighting decoherence
Decoherence-free subspaces
[Reserved: symmetry of the system–environment coupling leaving a subspace on which the dissipator acts trivially [Lidar:1998]; the singlet of two qubits under collective dephasing as the minimal example; its realization as a memory in two trapped ions, with a measured coherence time an order of magnitude longer than that of a bare qubit [Kielpinski:2001].]
Dynamical decoupling
[Reserved: Hahn's spin echo [Hahn:1950], in which a refocusing pulse recovers coherence lost to static inhomogeneity — the direct demonstration that such loss is not irreversible, and the sharpest illustration of Section 87.1.3; the generalization to pulse sequences that average an arbitrary slow environment to zero [Viola:1999]; the filter-function picture relating a sequence to the noise spectrum it suppresses, which turns a qubit into a spectrometer of its own environment.]
An ensemble of spins precessing in a slightly inhomogeneous static field loses its transverse magnetization in a time \(T_{2}^{*}\) fixed by the spread of the field. A single \(\pi\) pulse applied at time \(\tau\) makes the magnetization reappear at time \(2\tau\), essentially undiminished [Hahn:1950]. The echo can be repeated many times, and the envelope of successive echoes decays on a much longer time \(T_{2}\). The disappearance of the signal is therefore not by itself irreversible — only the residual envelope is — and the two time scales measure two different things: the inhomogeneity of the field, and the rate at which that inhomogeneity itself fluctuates.
Derivation. Give each spin of the ensemble a detuning \(\delta\) from the mean Larmor frequency, constant in time but differing from spin to spin with some distribution \(p(\delta)\). In the frame rotating at the mean frequency its transverse component acquires the phase \(\phi(t)=\delta t\), so the ensemble-averaged transverse magnetization at time \(\tau\) is
the Fourier transform of the detuning distribution, which decays on the time \(T_{2}^{*}\) set by the reciprocal of its width. No individual spin has lost anything at that point; their phases have merely fanned out. A \(\pi\) pulse about an axis in the transverse plane conjugates the accumulated phase, \(\phi\mapsto-\phi\). Evolving for a further time \(t\) gives the total phase \(-\delta\tau+\delta t\), and
simultaneously and whatever \(p(\delta)\) may be, so the whole ensemble rephases and Equation (87.11) returns to its initial value. The only property of \(\delta\) used is that it is the same before and after the pulse. Whatever part of \(\delta\) fluctuates on the time scale \(\tau\) accumulates different phases in the two halves of the interval, and for that part the cancellation is incomplete — which is precisely the separation between \(T_{2}^{*}\) and \(T_{2}\), and the reason a longer \(\tau\) recovers less. The echo is the sharpest available demonstration of Section 87.1.3: a state operationally indistinguishable from a classical ensemble can nevertheless be brought back, so looking like a mixture is not the same as being one.
∎Quantum error correction
[Reserved: Shor's nine-qubit code [Shor:1995] and Steane's seven-qubit code [Steane:1996], which encode one logical qubit so that the syndrome of an error can be measured without measuring the state; the discretization of errors, whereby correcting a finite set of Pauli errors corrects a continuum; the threshold theorem as a statement about rates rather than about principles; the experimental demonstration that a logical qubit's error rate falls as the code distance grows [Acharya:2023], which is the point at which the theory of this chapter became a construction rather than a limit.]