Leggett–Garg Tests

A Leggett–Garg experiment tests whether correlations at different times admit a common description with definite values and suitably noninvasive measurements. The inequality itself is derived in Leggett–Garg Inequalities. Here the key issue is experimental: temporal measurement can disturb the later dynamics, so a violation needs a precise statement of what disturbance has been excluded.

Pairwise temporal correlations

For a dichotomic quantity $Q=\pm1$ at $t_1<t_2<t_3$, measure

$$C_{ij}=\sum_{q_i,q_j=\pm1}q_iq_j\,p(q_i,q_j\mid ij), \qquad K=C_{12}+C_{23}-C_{13}.$$

Under a common joint distribution of definite $Q_1,Q_2,Q_3$, with the same statistics applicable to the different pairwise protocols, $K\le1$. The label $ij$ matters: the three experiments have different intervening measurements. Their identification with marginals of one common distribution requires the noninvasiveness assumption or a justified replacement.

A driven qubit with $H=\hbar\Omega X/2$ and ideal projective $Z$ measurements gives, for the appropriate identical preparations,

$$C_{ij}=\cos[\Omega(t_j-t_i)].$$

At equal spacing $\Omega(t_2-t_1)=\pi/3$, one obtains $K=2\cos(\pi/3)-\cos(2\pi/3)=3/2$. This is a quantum temporal-correlation prediction, not a proof that the measuring apparatus was noninvasive.

Addressing the disturbance objection

An ideal negative-result protocol couples a detector to only one alternative. A null result is then treated as noninteracting under a macrorealist account in which the system occupied the other alternative. The physical implementation must control hidden coupling, preparation changes, and selection effects. A null result in quantum theory can still update the state.

Other approaches calibrate invasiveness, use weak measurements with an explicit response model, or compare later marginal probabilities with and without an earlier intervention. These strategies answer related but not identical questions. Weak coupling alone does not prove zero disturbance, and a correction based on quantum theory should not be advertised as a model-independent rejection of every classical explanation.

The Emary–Lambert–Nori review develops these distinctions and the “clumsiness loophole”: an invasive classical device may mimic a temporal-inequality violation.

Estimation and controls

For $N_{ij}$ independent binary-product trials, the sample correlation has approximate standard error

$$\sigma_{ij}\simeq \sqrt{\frac{1-\widehat C_{ij}^{\,2}}{N_{ij}}}.$$

If data sets are independent, their variances add in $K$. Drift, shared calibration, serial dependence, and postselection can violate that assumption. Interleave protocols, keep the acceptance rule fixed, and record failures as specified by the witness rather than discarding inconvenient outcomes.

A violation establishes incompatibility with the complete tested hypothesis and auxiliary controls. It does not isolate “realism” while leaving measurement disturbance unconstrained. A microscopic implementation also does not, by itself, establish a macroscopically distinct superposition; see Tests of Macrorealism.

Exercise

Suppose $C_{12}=C_{23}=0.5$ and $C_{13}=-0.5$, each estimated from $N$ independent trials. Estimate the statistical standard error of $K$.

Solution

Each variance is approximately $0.75/N$. Therefore $\sigma_K=\sqrt{2.25/N}=1.5/\sqrt N$. For $N=900$ per protocol, the statistical error is $0.05$. This does not include a bound on invasive disturbance, which is a separate inferential requirement.

References

• Emary, C., Lambert, N., and Nori, F., “Leggett–Garg inequalities,” Reports on Progress in Physics 77, 016001 (2014), arXiv:1304.5133 — experimental protocols, disturbance, and interpretation.