The Leggett-Garg Inequality
A temporal analogue of Bell inequalities, testing whether a single system's behavior over time admits a classical realist description.
Realism in time
Where Bell inequalities probe correlations between separated systems, the Leggett-Garg inequality probes correlations of one system across time. It follows from two classical assumptions: macrorealism, that a system is always in a definite state, and noninvasive measurability, that one can in principle measure it without disturbance. From these, temporal correlation functions must satisfy a bound that quantum systems can violate.
The inequality
Measure a two-valued observable Q (values plus or minus one) at three times and form K = C12 + C23 - C13, where Cij is the correlation between measurements at times i and j. Any macrorealistic, noninvasively measurable system obeys K <= 1. A coherently oscillating quantum system, such as a qubit undergoing Rabi evolution, can exceed this, reaching up to 1.5 in the ideal two-level case.
- Assumption: definite state at all times (macrorealism)
- Assumption: measurement need not disturb (noninvasiveness)
- Bound: K = C12 + C23 - C13 <= 1
- Quantum coherent oscillation: K up to 1.5
The measurement loophole
The noninvasiveness assumption is the hard part experimentally. A skeptic can attribute any violation to measurement disturbance rather than genuine quantum coherence. Careful experiments address this with ideal negative-result measurements, which condition only on runs where the detector did not interact, and with weak measurement, which limits disturbance by design. Violations have been observed in superconducting qubits, spins, and photonic systems.
Relevance
The Leggett-Garg framework tests whether quantum coherence persists at larger scales and over time, sharpening the boundary between quantum and classical behavior. In quantum computing it serves as a coherence witness: a violation certifies that a qubit genuinely superposes over the measured interval rather than merely being uncertain. It complements spatial tests like CHSH by interrogating the temporal dimension of quantum behavior.