  We develop a unitary dependence theory to characterize the behaviors of
quantum circuits and states in terms of how quantum gates manipulate qubits and
determine their measurement probabilities. A qubit has dependence on a 1-qubit
unitary gate if its measurement probabilities depend on the parameters of the
gate. A 1-qubit unitary creates such a dependence onto the target qubit, and a
CNOT gate copies all the dependences from the control qubit to the target
qubit. The complete dependence picture of the output state details the
connections qubits may have when being manipulated or measured. Compared to the
conventional entanglement description of quantum circuits and states, the
dependence picture offers more practical information, easier generalization to
many-qubit systems, and better robustness upon partitioning of the system. The
unitary dependence theory is a useful tool for understanding quantum circuits
and states that is based on the practical ideas of how manipulations and
measurements are affected by elementary gates.
