Shadows and Mirrors: Reconstructing Quantum States of Atom Motion

Imagine that a pair of coins are tossed in a black box. The box reports only one of the following three results at random: (1) the outcome of the first coin (heads or tails), (2) the outcome of the second coin (heads or tails), or (3) whether the outcomes of the two coins matched or were different. Our task is to construct a joint probability distribution of the four possible outcomes of the coins (HH, TT, HT, TH) based on many observations of the black box outputs. Now suppose that after many trials, the black box reports that each coin comes up heads two‐thirds of the time when measured individually, yet the coins never match when they are compared. (Clearly the results of the coin tosses have been correlated—perhaps a joker in the black box flips the coins and then changes the outcomes appropriately). We seek a distribution that both reflects this correlation and obeys the marginal distributions of each coin as two‐thirds heads, one‐third tails (see the three tables on page 23). The only way to satisfy both requirements is to force the joint probability $\mathrm{P(}\mathrm{TT})$ of getting two tails to be negative! Mathematically, this is because $\mathrm{P(}\mathrm{HH}\mathrm{)+P(}\mathrm{TT})$ is observed to be zero, yet we expect $\mathrm{P(}\mathrm{HH})$ to be greater than $\mathrm{P(}\mathrm{TT}\mathrm{),}$ because the individual coins are weighted toward heads.