Temporal Quantum Correlations and Leggett-Garg Inequalities in Multilevel Systems

We show that the quantum bound for temporal correlations in a Leggett-Garg test, analogous to the Tsirelson bound for spatial correlations in a Bell test, strongly depends on the number of levels $N$ that can be accessed by the measurement apparatus via projective measurements. We provide exact bounds for small $N$ that exceed the known bound for the Leggett-Garg inequality, and we show that in the limit $N\to \infty$ the Leggett-Garg inequality can be violated up to its algebraic maximum.

Figure 1

The Leggett-Garg quantity $K_3$ for a spin of length $j=(N-1)/2$ precessing in a magnetic field with measurement times $\mathrm{\Omega }{t}_1=\pi$, $t_2-t_1=t_3-t_2=\tau$. The measurement is made with $M=N$ projectors (von Neumann scheme) in the $z$ direction. Inset: $K_3$ as a function of measurement time $\tau$ for various values of $N$. For $N=2$ the maximum is the familiar qubit or Lüders bound $K_3^{\max }=\frac{3}{2}$ (solid line). For $N=3$, however, the maximum value is 1.7565, and this increases with increasing $N$. Main panel: The black circles show the maximum value $K_3^{\max }$ as a function of system size $N=2j+1$ for the spin precession model with measurement times as above. The blue diamonds show the value of $K_3$ with $\tau$ fixed $\mathrm{\Omega }\tau =\pi /2$, and the solid line shows the asymptotic behavior $K_3^{\max }\sim 3-\sqrt{2/\pi j}$. In the limit $N\to \infty$, $K_3^{\max }$ tends to the algebraic bound of 3.

Figure 2

A plot of the data in Table 1. The maximum values for each $M$ (from SDP) are shown as straight lines.