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Matrix multiplication does not allow us to have a quantum logic gate which takes input 2 qubits and gives in the output 1 qubit.

But why is not that useful? How are we supposed to build the quantum bool algebra if we do not have such gates?

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Indeed, it is useful to consider these types of situations. Sometimes we might measure a qubit and throw it away, leaving us one less qubit than before in our system. The issue you bring up is, what is the right mathematical description of this process?

It can't be matrix multiplication as you've described. Let's consider a mapping from a two-qubit to single-qubit space. We might try to accomplish this with a non-square matrix of dimensions $2 \times 4$ acting on our statevector (of dimension four). However, it is a fact of linear algebra that any such matrix must send at least one vector to zero. This means such a transformation could never represent a quantum process, since a vector of zeros has no interpretation in terms of probabilities.

So what is the right way to think about these kinds of processes? The answer is not in terms of unitary gates, but rather so-called "quantum channels" which act on "density matrices". These topics are addressed in Nielsen and Chuang, and also in the IBM Qiskit Textbook if you're interested. Though depending on your interests, you might be satisfied just knowing their existence.

More to your concern about a complete "quantum boolean algebra": unitary gates and measurements are truly enough. The fact that quantum gates have the same input lines as output is no limitation, which can be understood from a number of standpoints such as the theory of reversible computing.

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