Tests of Macrorealism

Tests of macrorealism examine a package of assumptions about a specified coarse observable: it has a definite value, suitable measurements can reveal it without changing subsequent evolution, and future choices do not alter earlier facts. Each assumption needs an operational role. Rejecting their conjunction is more precise than claiming that an experiment disproves reality.

Definite values do not imply noninvasiveness

Let a classical two-state system carry $Q=\pm1$. A measurement can reveal $Q$ and also flip it. That invasive apparatus is fully compatible with definite values while changing later correlations. Therefore a temporal-inequality violation cannot be interpreted without controls on disturbance or an explicitly stated operational replacement.

The canonical Leggett–Garg inequality gives one consequence of a common joint distribution. Leggett–Garg Tests explains how its pairwise correlations are measured. Other witnesses examine whether inserting a measurement changes a later marginal distribution.

No-signaling in time

For an optional intermediate measurement with outcome $a$, define

$$\Delta(b)= \sum_a p(a,b\mid\text{measurement inserted}) -p(b\mid\text{measurement omitted}).$$

The condition $\Delta(b)=0$ is often called no-signaling in time. Unlike spatial no-signaling, it does not follow generally from quantum theory: a physical intervention can affect the future through an ordinary causal channel.

As a simple example, prepare $\lvert+x\rangle$ and immediately measure $X$ at the end. Without an intermediate measurement, $p(+x)=1$. Inserting an unread $Z$ measurement produces $I/2$, so $p(+x)=1/2$ and $\Delta(+x)=-1/2$. This is a clear disturbance witness. It does not rule out an invasive classical model of the apparatus.

Passing one such marginal test is also insufficient to certify every hidden variable is undisturbed. Disturbances may cancel in the observed distribution or appear only for another preparation. Appropriate combinations of temporal consistency conditions and assumptions are needed for a stronger macrorealist inference.

What is macroscopic in the test?

The tested quantity might be a current direction, a position region, a collective spin, or an effective two-level degree of freedom. Report the branch separation, resolution, participating degrees of freedom, and coherence time. A two-outcome signal can arise in a microscopic device; the number of detector clicks says nothing by itself about macroscopic distinctness.

Coarse graining can make interference inaccessible without modifying microscopic quantum dynamics. A negative result may therefore reflect insufficient resolution or environmental decoherence, rather than a fundamental boundary between quantum and classical matter.

Conversely, a successful witness should identify which package is excluded. The strongest interpretation requires independently credible controls on preparation, invasiveness, detector response, and sample selection. It should not promote a bound on one coarse observable into a conclusion about every property of the object.

Exercise

A classical measurement flips $Q$ with probability $f$. Before measurement $p(Q=+1)=p$. With no subsequent evolution, compute the change in the final $+1$ probability. Can an invasive measurement pass this one test?

Solution

After measurement $p'=(1-f)p+f(1-p)$, so $p'-p=f(1-2p)$. At $p=1/2$ the marginal is unchanged for every $f$, despite real disturbance on individual runs. Testing several preparations is therefore more informative than observing one unchanged marginal.

References

• Emary, C., Lambert, N., and Nori, F., “Leggett–Garg inequalities,” Reports on Progress in Physics 77, 016001 (2014), doi:10.1088/0034-4885/77/1/016001 — macrorealism, noninvasiveness, and experimental qualifications.
• Nimmrichter, S., and Hornberger, K., “Macroscopicity of mechanical quantum superposition states,” Physical Review Letters 110, 160403 (2013), arXiv:1205.3447 — a different, modification-based measure of experimental macroscopicity.