Leggett–Garg Inequalities

Leggett–Garg inequalities test whether correlations at different times can be explained by definite values whose measurement can, in principle, leave their subsequent evolution unchanged. A violation rejects the combination of those premises and the experimental assumptions used to access the correlations. It does not reject every theory with a definite underlying state: invasive classical measurements can violate the same bound.

Helpful background. Born Rule relates sequential measurement models to outcome probabilities.

Three times and one joint distribution

Take a dichotomic variable $Q_i=Q(t_i)\in\{-1,+1\}$ at $t_1<t_2<t_3$. Macrorealism per se supplies a definite value at each time. Noninvasive measurability, together with an appropriate absence of influence from future choices and consistent preparation, permits the pairwise experiments to sample marginals of one joint distribution $p(Q_1,Q_2,Q_3)$.

For each fixed triple define

$$k=Q_1Q_2+Q_2Q_3-Q_1Q_3.$$

If $Q_1=Q_3$, then $k=2Q_1Q_2-1$, which is $1$ or $-3$. If $Q_1=-Q_3$, the first two terms cancel and $k=1$. Averaging therefore gives

$$-3\le K=C_{12}+C_{23}-C_{13}\le1, \qquad C_{ij}=\langle Q_iQ_j\rangle.$$

The averaging is elementary; the physically important step is identifying all three measured correlations with the same joint distribution. Separate experimental ensembles do not automatically justify that identification.

Coherent precession gives a violation

Consider a qubit with $H=\hbar\Omega\sigma_x/2$, measure $Q=\sigma_z$, and use ideal projective measurements in each two-time experiment. After a first outcome $q_i$, the state is its $\sigma_z$ eigenstate. A rotation through angle $\Omega(t_j-t_i)$ gives conditional mean

$$\mathbb E[Q_j\mid q_i]=q_i\cos[\Omega(t_j-t_i)].$$

Consequently $C_{ij}=\cos[\Omega(t_j-t_i)]$, independent of the first-outcome distribution. For equal intervals $\tau$,

$$K=2\cos(\Omega\tau)-\cos(2\Omega\tau).$$

At $\Omega\tau=\pi/3$, this is $3/2$. Equivalently, writing $c=\cos(\Omega\tau)$ gives $K=1+2c-2c^2$, whose maximum is $3/2$ at $c=1/2$.

There is no inconsistency with probability theory. The $C_{13}$ experiment omits the intermediate measurement. Inserting it changes the quantum dynamics and generally produces another correlation. Quantum theory does not identify these distinct interventions with passive readings of a common pre-existing trajectory.

The experimental inference needs a disturbance argument

A temporal sequence allows ordinary causal disturbance from earlier to later operations. This differs from a Bell experiment arranged so that distant setting choices and outcomes are spacelike separated. Apparatus calibration, control experiments and a declared measurement model are needed to bound the temporal disturbance a classical explanation can exploit. Ideal negative-result protocols attempt to acquire information without interaction in the relevant macrorealist description, but the inference remains conditional on how that claim is established.

See Leggett–Garg Tests for experimental reasoning and Tests of Macrorealism for the broader model class. A microscopic qubit illustrates the inequality; it does not by itself establish coherence between macroscopically distinct alternatives. That requires a separate account of the physical states and measurement resolution.

Exercise: detecting an inserted measurement

For $\Omega\tau=\pi/3$, compare $C_{13}$ when the middle projective measurement is omitted and when its outcome is ignored after it is performed.

Solution

Without the middle measurement, $C_{13}=\cos(2\Omega\tau)=-1/2$. With a measurement, each interval is a transition between $Q$ eigenstates, so conditional expectations compose and $C_{13}=\cos^2(\Omega\tau)=1/4$. Ignoring an outcome does not undo the measurement’s dynamical effect. Using the second value in the three-measurement experiment gives $K=1/2+1/2-1/4=3/4$, inside the classical bound.

References

• Emary, Clive, Neill Lambert, and Franco Nori, “Leggett–Garg inequalities,” Reports on Progress in Physics 77, 016001 (2014), doi:10.1088/0034-4885/77/1/016001 — temporal protocols and disturbance qualifications.
• Leggett, A. J., and Anupam Garg, “Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks?,” Physical Review Letters 54, 857–860 (1985), doi:10.1103/PhysRevLett.54.857 — original premises and test.