Exact operator bounds on Leggett-Garg violations

Abstract

Leggett-Garg (LG) inequalities test whether correlations measured at different times admit a positive joint probability distribution over histories. A violation requires negative values in any joint quasiprobability distribution reproducing those correlations, but such distributions can have negative values even without a violation. Spacetime density kernels extend the density-matrix description to multiple times and are used to study timelike entanglement. We establish a universal bound that relates LG violations to their negative eigenvalue weight. For LG tests involving any number of measurements, we determine the least compatible kernel negativity through an exact relation valid for any system dimension, initial state, dynamics, or measurement rule. This minimum is nonzero if and only if at least one LG inequality is violated. Three-time Lüders LG tests certify at most one eighth of negativity, whereas measurements resolving individual spin levels can approach one half. Including additional incompatible qubit observables gives a further exact minimum that can exceed the LG-only ceiling without increasing the LG violation.

Leggett–Garg (LG) violations require negative history weights, but temporal quasiprobabilities and spacetime density kernels can be negative without violating an LG inequality. This paper determines the minimum negative eigenvalue weight $\mathcal N(H)=(\|H\|_1-1)/2$ among Hermitian, unit-trace kernels reproducing specified LG pair correlations. For three times, the exact minimum is $\mathcal N_{\min}=[\max_\alpha K_\alpha-1]_+/4$, where the four LG combinations have classical bound one. A witness with spectrum $\{1,-3\}$ establishes the lower bound; a diagonal kernel on outcome registers attains it. For an $n$-time cycle, the result becomes $\mathcal N_{\min}^{(n)}=[\max_\sigma K_\sigma-(n-2)]_+/\Delta_n$, with $\Delta_n=2(n-2)$ for even $n$ and $2(n-1)$ for odd $n$. Explicit signed-history constructions establish attainment. These relations depend only on the correlations, not on dimension, preparation, dynamics, or measurement update. The known three-time Lüders bound $K\leq3/2$ therefore certifies at most $1/8$ negativity. Level-resolving spin measurements can exceed this and approach $1/2$ as spin increases. Adding two averages involving an incompatible qubit observable yields another exact minimum, proved by symmetry averaging and diagonalization; one example increases the certificate from $0.125$ to approximately $0.2286$ without changing the LG violation. The minimization fixes only the stated averages and does not require compatible kernels to arise from a particular physical process. Additional moments can force greater negativity. The general multitime result concerns cycle correlations, while the spin examples are protocol-family results, not dimension-wise global optima.