Biased Non-Causal Game

The standard formulation of quantum theory assumes that events are ordered is a background global causal structure. Recently in Ref.[$\href{http://www.nature.com/ncomms/journal/v3/n10/full/ncomms2076.html}{Nat. Commun. {\bf3}, 1092 (2012)}$], the authors have developed a new formalism, namely, the \emph{process matrix} formalism, which is locally in agreement with quantum physics but assumes no global causal order. They have further shown that there exist \emph{non-causal} correlations originating from \emph{inseparable} process matrices that violate a \emph{causal inequality} (CI) derived under the assumption that events are ordered with respect to some global causal relation. This CI can be understood as a guessing game, where two separate parties, say Alice and Bob, generate random bits (say input bit) in their respective local laboratories. Bob generates another random bit (say decision bit) which determines their goal: whether Alice has to guess Bob's bit or vice-verse. Here we study this causal game but with biased bits and derive a biased causal inequality (BCI). We then study the possibility of violation of this BCI by inseparable process matrices. Interestingly, we show that there exist \emph{inseparable} qubit process matrices that can be used to violate the BCI for an arbitrary bias in the decision bit. In such scenario, we also derive the maximal violation of the BCI under local operations involving traceless binary observables. However, for biased input bits, we find that there is a threshold bias beyond which no valid qubit process matrix can be used to violate the causal inequality under \emph{measurement-repreparation} type operation.