Ambiguous measurements, signaling, and violations of Leggett-Garg inequalities

Ambiguous measurements do not reveal complete information about the system under test. Their quantum-mechanical counterparts are semiweak (or in the limit, weak) measurements and here we discuss their role in tests of the Leggett-Garg inequalities. We show that, while ambiguous measurements allow one to forgo the usual noninvasive measurability assumption, to derive a Leggett-Garg inequality that may be violated, we are forced to introduce another assumption that equates the invasive influence of ambiguous and unambiguous detectors. Based on this assumption, we derive signaling conditions that should be fulfilled for the plausibility of the Leggett-Garg test. We then propose an experiment on a three-level system with a direct quantum-optics realization that satisfies all signaling constraints and violates a Leggett-Garg inequality.

Figure 1

Sketch of a three-level system realized as optical channels $A,B$, and $C$ with nontrivial time evolution generated by the blocks labeled $U$. We initialize by injecting a photon into channel $C$. Three configurations are shown. (a) No measurement at $t_2$. (b) Unambiguous measurement. With blocking elements in channels $B$ and $C$, detection of the photon at $t_3$ means that we can infer that the photon was in channel $A$ at time $t_2$. (c) Ambiguous measurement. With only channel $C$ blocked, detection of the photon at $t_3$ means that, from a macrorealistic point of view, the photon was in either channel $A$ or $B$ at time $t_2$.

Figure 2

The signaling quantifier ${\delta }_{\mathrm{A}}^{\mathrm{QM}}\left(B\right)$ as a function of parameters $\theta$ and $\varphi$ for ambiguous measurements of a three-level system. Parameter $\chi$ was chosen to set ${\delta }_{\mathrm{A}}^{\mathrm{QM}}\left(A\right)=0$. The black lines indicate parameters for which ${\delta }_{\mathrm{A}}^{\mathrm{QM}}\left(B\right)=-{\delta }_{\mathrm{A}}^{\mathrm{QM}}\left(C\right)=0$ and both NSIT and ESIT are obeyed.

Figure 3

The ambiguously measured LG correlator $K_{\mathrm{A}}^{\mathrm{QM}}$ for our three-level system as a function of angles $\theta$ and $\varphi$. Red and orange colors correspond to $K_{\mathrm{A}}^{\mathrm{QM}}>1$; blue corresponds to $K_{\mathrm{A}}^{\mathrm{QM}}<1$. The black lines are the no-signaling lines from Fig. 2 along which ${\delta }_{\mathrm{A}}^{\mathrm{QM}}\left(n_3\right)=0;\forall n_3$. In the top-left quadrant we see the no-signaling line intersect a $K>1$ region, such that we have NSIT, ESIT, and a violation of the LGI.

Figure 4

Two cuts through Fig. 3 for the inverted-measurement case plus corresponding results for the weak measurements of Sec. 6. (a) The correlator $K_{\mathrm{A}}^{\mathrm{QM}}$ as a function of $\theta$ along the no-signaling line that goes through the orange region in Fig. 3. Since ${\mathrm{\Delta }}_{\mathrm{A}}=0$ along this line, the LGI reverts to $K_{\mathrm{A}}\le 1$ and violations of Eq. (15) occur within the indicated red region. Black line, inverted measurement; blue line, weak measurement. (b) The correlator $K_{\mathrm{A}}^{\mathrm{QM}}$ along the straight-line cut in Fig. 3 from $\varphi =0$ to $\pi$ with $\theta =0.831\pi$. The red region shows the right-hand side of Eq. (15) and, since generally we have signaling here, this quantity is greater than 1. Indeed, only near the maximum of $K_{\mathrm{A}}^{\mathrm{QM}}$ does $1+{\mathrm{\Delta }}_{\mathrm{A}}^{\mathrm{QM}}$ drop significantly such that we obtain a violation of Eq. (15). N.B.: The maximum of the $K_{\mathrm{A}}^{\mathrm{QM}}$ curve here is slightly displaced from the no-signaling point and thus has a value slightly higher than the no-signaling maximum (1.482 vs 1.464). (c) The same as panel (b) but for the weakly measured case with $\theta =0.856\pi$. Again the maximum is slightly offset from the no-signaling maximum (1.173 vs 1.147).