Figure 4: Quantum circuits which realize examples of different classes of causal relations between two qubits.

Circuits for four combinations of cause-effect and common-cause mechanisms: (a) PROBC, (b) PHYSC, (c) PROBQ and (d) COH⊂PHYSQ. In all circuits, C and E are initially prepared in the maximally entangled state and F is discarded at the end. The examples differ only in the choice of the gate , as described in the text. Panels with two circuits represent an equal probabilistic mixture of both scenarios. Black squares represent the trace operation, grey squares represent complete-dephasing operations, grey triangles represent preparations of the completely mixed state, grey circles represent the classical XNOR gate, which sets b=−ed for binary variables b,e,d taking the values {±1} and the two-qubit unitary U_BF|DE is the partial swap given by equation (2). Below each example, we specify the causal map obtained from the state ρ_CE and the gate via . Lowercase letters c,b,d represent classical binary variables, P(cb|d) represents a conditional probability distribution over these, δ_x,y denotes the Kronecker delta function and u(x) denotes the uniform distribution over x.