CPT tests with the antihydrogen molecular ion

High precision radio-frequency, microwave and infrared spectroscopic measurements of the antihydrogen molecular ion $\bar{H}_{2}^{-}$ ($\bar{p}\bar{p}e^{+}$) compared with its normal matter counterpart provide direct tests of the CPT theorem. The sensitivity to a difference between the positron/antiproton and electron/proton mass ratios, and to a difference between the positron-antiproton and electron-proton hyperfine interactions, can exceed that obtained by comparing antihydrogen with hydrogen by several orders of magnitude. Practical schemes are outlined for measurements on a single $\bar{H}_{2}^{-}$ ion in a cryogenic Penning trap, that use non-destructive state identification by measuring the cyclotron frequency and bound-positron spin-flip frequency; and also for creating an $\bar{H}_{2}^{-}$ ion and initializing its quantum state.