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In the ordered computational basis $|0⟩,|1⟩$, the Pauli X gate, Pauli Y gate, Pauli Z gate, Hadamard gate and T gate areIn the ordered two-qubit computational basis $|00⟩,|01⟩,|10⟩,|11⟩$, with the first qubit controlling the second, the controlled-NOT gate isHere the T gate uses the conventional representative with first diagonal entry one; another common representative differs only by a global phase.

Use deterministic two-qubit entanglement dilution, retaining the specified complex probability amplitudes. Alice implements a quantum instrument on $A$ with Kraus operatorsThe Kraus representation completeness condition holds even for complex $\alpha ,\beta$:To realize this using the permitted quantum gates and a basic measurement, Alice initializes an ancilla qubit $C$ to $|0⟩$. Conditional on her computational-basis state being $j$, she applies to $C$ the unitary matrixEach has orthonormal columns. Therefore $W=|0⟩⟨0{|}_A\otimes V_0+|1⟩⟨1{|}_A\otimes V_1$ is a unitary gate, withA computational-basis measurement of $C$ implements the desired quantum instrument. Its two unnormalized branches on $AB$ areEach has probability $1/2$. Alice sends her outcome $m$ as one classical bit. For $m=0$, both parties keep their qubits unchanged. For $m=1$, both apply a Pauli X gate, givingThus both outcomes produce the exact target using one classical bit and local operations. The construction includes $\alpha =0$ or $\beta =0$ and arbitrary relative quantum phases; it requires no quantum channel between the parties.

A matching in a graph is a subset $M\subseteq E$ whose edges have pairwise disjoint endpoints. It is a perfect matching when every vertex is incident to exactly one edge in $M$.