Why are only unitary operations allowed in quantum information theory? In the quantum information theory, any operations for a quantum state have to be unitary operations. Why is this restriction needed? Can't we make a non-unitary operation to a state? I know that a unitary operation is a reversible operation, but I don't even know why it is necessary.
 A: 
Why is this restriction needed?

Here's a pretty simple way to convince yourself this is true:
Applying any physical action (up to measurement) on a state essentially means exposing it to a specific Hamiltonian $H$ and using that Hamiltonian to evolve that state in time (the only way to evolve a state in time nontrivially is through the Hamiltonian) through the Schrodinger equation
$$H(t)|\psi(t)\rangle = i\partial_t|\psi(t)\rangle.$$
The solution to this equation is given by
$$|\psi(t)\rangle=U(t,0)|\psi(0)\rangle,$$
where $U(t,0)$ is an operator that satisfies
$$H(t)U(t,0)=i\partial_tU(t,0).$$
The solution of this equation is well known, and is given by the so-called Dyson series
$$U(t,0)=\text{T}\exp\left(-i\int_{0}^{t}H(t')\mathrm{d}t'\right),$$
where the $\text{T}\exp$ represents the time-ordered exponential (see wikipedia for the full discussion).
The important thing to note is that the Hamiltonian $H$ is a Hermitian operator (by the axioms of quantum mechanics), and consequently $U(t,0)$ is a Unitary operator. Since any physical operation on a state really corresponds to altering the Hamiltonian and using that to evolve the state, we see that any operation applicable to quantum information must be unitary.

Can't we make a non-unitary operation to a state?

The answer is technically no but realistically yes. As I just demonstrated, any operation on a state that evolves it in time must be unitary. However, that is only if the Hamiltonian is Hermitian. So what would it mean to have a non-Hermitian Hamiltonian?
Well, consider dividing our system (Hilbert space) into two subsystems, the internal system $\mathcal{H}_{i}$ and the environment $\mathcal{H}_{e}$. That is, $\mathcal{H}=\mathcal{H}_{i}\otimes\mathcal{H}_{e}$. Then we know that $H$ is Hermitian when acting on the entire system $\mathcal{H}$, but we aren't guaranteed that $H$ is Hermitian when acting on just the internal system $\mathcal{H}_i$. In particular, if there is some interaction between the internal system and the environment, states in the internal system can "bleed" into the environment. This effectively leads to a non-Hermitian form of the internal Hamiltonian, with eigenvalues
$$E_n-i\Gamma_n,$$
leading to the time evolution
$$|n(t)\rangle=e^{-iE_nt}e^{-\Gamma t}|n(0)\rangle.$$
That is, the eigenstates in the internal system decay into the environment over time. This effectively leads to a non-unitary time evolution in the internal system. A great reference for this type of "non-Hermitian quantum mechanics" is this book by Nimrod Moiseyev.
This effect on non-isolated systems is actually very important in quantum information, when you want to actually build a quantum computer. Isolating the environment to maintain unitarity becomes a huge challenge, that is still being solved.
I hope this helped! If anything was unclear, just leave a comment and I'll try to explain it in more detail.
