Leggett–Garg inequalities test whether a system can be described as possessing definite macroscopic values at all relevant times while those values are, in principle, measurable without disturbing future evolution.
The mathematics resembles a Bell inequality but the structure is temporal rather than spatial. Measurements occur on one system at different times. The central challenge is therefore different too: a measurement at an earlier time can physically disturb the later outcome. Any claimed violation must confront this invasiveness loophole explicitly.
Guide 29 owns Bell/CHSH nonlocality. Guide 54 owns continuous measurement. Guide 83 owns weak measurement. This guide owns macrorealism, two-time correlators, Leggett–Garg inequalities, no-signalling-in-time conditions, ideal-negative-result logic and quantum violations.
Assume definite values Q(t)=±1 and noninvasive measurability → derive temporal correlation bounds → compare with sequential quantum measurements → diagnose violation while auditing measurement disturbance.
1. Macrorealism per se
The first Leggett–Garg premise says that a sufficiently macroscopic system is at every relevant time in one of its macroscopically distinct states.
Represent the measured property by a dichotomic variable
Q(t)=±1.
Under macrorealism per se, Q has a definite value whether or not it is measured.
2. Noninvasive measurability
The second premise says that it is possible, in principle, to determine Q without affecting its subsequent dynamics.
This is the crucial assumption distinguishing a Leggett–Garg test from ordinary sequential quantum measurements.
A projective quantum measurement generally disturbs the state, so an observed violation cannot be interpreted without explaining why a macrorealist is not allowed to attribute it entirely to that disturbance.
3. Induction or arrow-of-time assumption
Standard discussions also assume that future measurement choices do not influence already existing past values.
This temporal causal assumption is usually implicit but should be separated from the noninvasiveness assumption.
4. Two-time correlators
For measurement times t₁,t₂,t₃, define
C_ij=⟨Q(t_i)Q(t_j)⟩.
Under macrorealism and noninvasive measurability, these pairwise averages can be treated as marginals of one joint probability distribution for the three definite values q₁,q₂,q₃∈{±1}.
5. Derive the three-time Leggett–Garg bound
For one deterministic assignment q₁,q₂,q₃ define
K_3=q₁q₂+q₂q₃−q₁q₃.
If q₁=q₃, then q₁q₃=1 and the first two terms sum to either +2 or −2, giving K₃=1 or −3.
If q₁=−q₃, then q₁q₃=−1 and q₁q₂+q₂q₃=q₂(q₁+q₃)=0, giving K₃=1.
Therefore every deterministic assignment obeys
−3≤K_3≤1.
Averaging over hidden-variable distributions preserves the upper bound:
C_{12}+C_{23}−C_{13}≤1.
6. Equal-time-spacing qubit model
Take observable Q=Z and Hamiltonian
H=(ℏΩ/2)X.
For ideal sequential projective measurements separated by time τ, the two-time correlation is
C(τ)=cos(Ωτ).
For equal spacing,
C_{12}=C_{23}=cosθ
and
C_{13}=cos2θ
with θ=Ωτ.
7. Quantum prediction
The quantum Leggett–Garg combination is
K_3(θ)=2cosθ−cos2θ.
Using cos2θ=2cos²θ−1, let x=cosθ:
K_3=1+2x−2x².
This parabola is maximised at x=1/2, giving
K_3^{max}=3/2.
The maximum occurs at θ=π/3 modulo the relevant symmetries.
8. Worked numerical violation
Set Ωτ=π/3.
C_{12}=C_{23}=cos(π/3)=1/2;C_{13}=cos(2π/3)=−1/2;K_3=1/2+1/2−(−1/2)=3/2.
This exceeds the macrorealistic upper bound of one.
9. Why sequential quantum measurements give cos(Ωτ)
Suppose a first projective Z measurement yields q=±1. Immediately afterward the state is the corresponding Z eigenstate.
Evolution under X rotation for time τ rotates its Bloch Z component to q cosθ.
The conditional expected later outcome is therefore
E[Q_j|q_i]=q_i cosθ.
Multiplying by q_i and averaging gives C_{ij}=cosθ.
10. The invasiveness loophole
A macrorealist can say: the earlier measurement kicked the system, and that kick caused the later correlations that violate the inequality.
Unlike a spacelike Bell test, one cannot simply separate the measurements so that no physical influence can travel between them.
Therefore a convincing Leggett–Garg experiment must justify or operationally test noninvasiveness.
11. Ideal negative-result measurement
Leggett and Garg proposed a macrorealist strategy: design a detector that interacts only if the system occupies one state. Keep only trials in which no detector interaction occurred.
A macrorealist who believes the system occupied a definite state can then regard a null result as revealing the alternative without physical interaction.
Quantum mechanically, the null result still updates the state. The point is to close the loophole using assumptions a macrorealist themselves should accept.
12. Clumsiness loophole
Real detectors can disturb the system even in nominally negative-result configurations through stray fields, imperfect selectivity or timing changes.
This is often called the clumsiness loophole.
Control experiments should bound ordinary disturbance large enough to explain the observed violation.
13. No signalling in time
A direct operational test of disturbance asks whether inserting an earlier measurement changes the marginal statistics of a later measurement.
For times t₁<t₂, a no-signalling-in-time condition is
P(Q_2=q)=Σ_{q_1}P(Q_1=q_1,Q_2=q)
where the left side comes from an experiment without the earlier measurement and the right from one with it.
Violation indicates that the earlier intervention changes later observed statistics.
14. NSIT and Leggett–Garg inequalities are not identical tests
Leggett–Garg inequalities constrain temporal correlations under macrorealist assumptions.
NSIT conditions test specific measurement disturbance at the level of marginals.
A set of appropriately chosen NSIT and arrow-of-time conditions can form necessary and sufficient conditions for macrorealism in specified sequential scenarios, whereas one LG inequality is only one facet of the macrorealist polytope.
15. Weak measurements as a disturbance-control strategy
Guide 83 explains weak measurements. In a Leggett–Garg context, weak coupling can reduce disturbance per trial while still allowing correlations to be estimated from large ensembles.
However, weak measurement is not literally noninvasive. Residual backaction and postselection must be modelled and bounded.
16. Continuous weak monitoring
A continuously monitored qubit generates a noisy measurement record correlated with Q(t).
Two-time correlators can be reconstructed from the record, but measurement backaction is simultaneously modifying the system.
A stochastic master equation provides the correct joint model of record and backaction. Treating the record as passive classical observation would be inconsistent.
17. Stationarity assumptions
Some experimental variants reduce the number of separate measurements by assuming stationarity, so a correlation depends only on time difference.
This can simplify protocols but introduces another assumption. Drift, relaxation or preparation dependence can violate stationarity and bias the inferred Leggett–Garg combination.
18. Decoherence suppresses violation
For a damped coherent oscillation, a simple phenomenological correlator can take the form
C(τ)=e^{-Γτ}cos(Ωτ).
Then
K_3=2e^{-Γτ}cosθ−e^{-2Γτ}cos2θ.
Increasing Γ generally reduces the accessible violation because temporal coherence is lost before the required rotations develop.
19. Strong measurement can also suppress coherent violation
Very frequent projective measurement can produce Zeno dynamics, inhibiting the coherent transition needed to build nontrivial temporal correlations.
Thus measurement plays two roles: it provides information and it changes dynamics.
A Leggett–Garg experiment sits precisely at this interface.
20. Higher-order inequalities
With more measurement times one can construct chains such as
K_n=C_{12}+C_{23}+…+C_{n−1,n}−C_{1n}
with macrorealist bounds depending on n and sign convention.
Longer temporal chains can reveal different aspects of macrorealistic correlation structure but also accumulate more sensitivity to drift and invasiveness.
21. Temporal CHSH-type inequalities
One can also construct temporal analogues of CHSH using two possible measurements at an earlier time and two at a later time.
The algebraic resemblance to Bell tests does not remove the temporal signalling issue: the first measurement can influence the second.
22. Macrorealism is not simply classical mechanics
A classical stochastic process with invasive measurements can violate an inequality derived under noninvasive measurability.
Therefore “violation means quantum” is too compressed unless measurement invasiveness has been controlled under the relevant macrorealist model.
23. Relation to contextuality
Temporal correlation inequalities can be embedded in broader contextuality frameworks when compatible measurement contexts and operational equivalences are specified.
Guide 56 owns contextuality generally. The Leggett–Garg setting remains distinguished by its explicit macrorealist and temporal interpretation.
24. Relation to histories
Sequential alternatives can be represented by class operators or quasiprobabilities over histories.
When interference between histories prevents a consistent positive joint probability distribution, temporal correlations can violate classical bounds.
This offers another mathematical language for the same obstruction: incompatible histories resist assignment of one noncontextual joint classical distribution.
25. Experimental systems
Leggett–Garg-style tests have been implemented in superconducting qubits, spins, photons, neutrino-related analyses and other two-level or multilevel systems.
Comparing them requires care because the measurement protocol, invasiveness controls and assumptions differ substantially.
The relevant achievement is not simply the numerical value of K but the degree to which the experiment closes alternative disturbance explanations under its stated assumptions.
26. Common misconceptions
“Leggett–Garg is Bell’s theorem in time.” The algebra is related, but temporal measurement disturbance creates a fundamentally different loophole structure.
“K₃>1 alone disproves every classical model.” It rules out the tested macrorealist package only when its assumptions, especially noninvasive measurability, are justified.
“Weak measurement is perfectly noninvasive.” It reduces disturbance per trial but does not eliminate it exactly.
“A flat late-time observable is macrorealistic.” Macrorealism concerns definite values and compatible temporal statistics, not merely absence of visible oscillation.
27. Worked synthesis problem
Let a coherently oscillating qubit have equal time separation τ with Ωτ=π/3.
C_{12}=1/2.C_{23}=1/2.C_{13}=−1/2.K_3=3/2.- The macrorealist upper bound is one.
The quantum calculation violates the inequality by 0.5. The experimental interpretation still requires an invasiveness analysis.
28. Practice set
- State macrorealism per se.
- State noninvasive measurability.
- Define C_ij.
- Derive the deterministic values of K₃.
- State the standard upper Leggett–Garg bound.
- For equal-spacing qubit oscillations, write K₃(θ).
- What is its maximum quantum value in the ideal model?
- What is the invasiveness loophole?
- What does NSIT test?
- Why is weak measurement not automatically loophole free?
Answers
- The tested macroscopic observable has a definite value at each relevant time independent of observation.
- It is possible in principle to determine that value without altering later dynamics.
⟨Q(t_i)Q(t_j)⟩.- For q_i=±1,
q₁q₂+q₂q₃−q₁q₃equals either 1 or −3. K₃≤1.2cosθ−cos2θ.- 3/2.
- The earlier measurement may disturb the system enough to explain the later correlations classically.
- Whether inserting an earlier measurement changes later marginal statistics.
- Finite weak coupling still produces some backaction and the disturbance must be bounded.
Sources and further study
[1] Anthony J. Leggett and Anupam Garg, Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks?, Physical Review Letters 54, 857 (1985). Foundational Leggett–Garg inequality.
[2] Clive Emary, Neill Lambert and Franco Nori, Leggett–Garg inequalities, Reports on Progress in Physics 77, 016001 (2014). Comprehensive review of theory and experiments.
[3] Johannes Kofler and Časlav Brukner, Conditions for quantum violation of macroscopic realism, Physical Review Letters 99, 180403 (2007).
[4] J. J. Halliwell, Leggett-Garg inequalities and no-signaling in time: A quasiprobability approach. Temporal probabilities, NSIT and histories.
[5] G. C. Knee and colleagues, Violation of a Leggett–Garg inequality with ideal non-invasive measurements, Nature Communications 3, 606 (2012). Experimental treatment of noninvasiveness assumptions.
Continue through Quantum Mathematics — Batch 21
Guide 81: Quantum Darwinism, Pointer States, Environment as Witness, Redundancy and Decoherence develops environmental objectivity. Guide 82: Quantum Discord, Classical–Quantum States, Conditional Entropy and Measurement Disturbance develops measurement-sensitive correlations. Guide 83: Weak Measurements, Weak Values, Postselection and Pointer Shifts develops low-disturbance measurement and its limits.