qcond

Qcond is a term used in discussions of quantum information and quantum decision theory to denote a quantum conditioning operation or the conditioned quantum state of a system. In this usage, qcond(ρ, E) represents the post-conditioned state of a system described by the density operator ρ after conditioning on an event described by a positive operator E, with 0 ≤ E ≤ I.

In the standard projective-measurement case, a conditioning on a projector P leads to the conditioned state ρ conditioned on P given by ρ_P = P ρ P / Tr(P ρ). More generally, for a measurement described by a POVM {E_i} with measurement operators {K_i} such that E_i = K_i† K_i, the conditioned state upon obtaining outcome i is ρ_i = (K_i ρ K_i†) / Tr(K_i ρ K_i†), and the probability of that outcome is p_i = Tr(K_i ρ K_i†). The term qcond is used to emphasize the non-linear update of the state due to normalization after conditioning on the event.

Properties of qcond include dependence on the conditioning event, nonlinearity with respect to the prior state, and preservation of positivity with unit trace after normalization. It is applied in contexts such as quantum Bayesian inference, quantum state discrimination, and quantum control, where conditioning on partial information or measurement outcomes guides subsequent processing or decision-making.

In practice, qcond is often discussed alongside concepts like post-selection and conditioned quantum states. Related topics include quantum state update rules, POVMs, and the distinctions between conditional and unconditional quantum evolutions.