# Why do quantum operations need to be reversible?

Why do quantum operations need to be reversible?

Is it because of the physical implementation of the qubits or operators neesd to be unitary so that the resultant states fit in the Bloch sphere?

"Why" is quite a nebulous concept that, ultimately, comes back to your understanding of quantum mechanics. For me, the way that I set up QM is with a set of postulates. The first postulate is that quantum states are represented by vectors in a Hilbert space of length 1.

A second postulate is essentially that operations are linear. (This is not how I would phrase it if I were teaching the whole subject, but has the essential content for the argument.)

If you then assert that an operation must map valid quantum states to valid quantum states (what else makes sense?) then you can prove that the only valid linear operation is a unitary. Unitaries are reversible.

It's not really about the Bloch Sphere directly because the Bloch Sphere only really works for qubits, and this is a more general result. There is, however, a visualisation of the above discussion that, in the case of qubits, maps to the Bloch Sphere.

(I have measurement as an extra postulate. This contains non-linear, and also irreversible operations. Although some might assert that these are indeed unitary, and hence reversible, in a larger space. In the context of quantum computing, it is worth pointing out that entire computations can be done just with irreversible measurements, e.g. measurement-based computation, entirely doing away with unitary operations!)

Not all quantum operations are reversible. Consider for example the quantum channel $$\Phi$$ defined as $$\Phi(\rho)=\operatorname{tr}(\rho)\sigma$$. This is a channel that gives as output the same state regardless of the input. This is not a reversible operation, as there is no information in the output that allows you to infer what the input was (you can imagine this describing the act of completely ignoring whatever state you were given and always produce as output a specific state $$\sigma$$).

The operations that are reversible are the ones described by unitary operators (or in the density matrix formalism, by maps of the form $$\rho\mapsto U\rho U^\dagger$$ with $$U$$ unitary). Any operation mapping pure states to pure states is going to be unitary. These are the operations describing the evolution of a quantum state without loss of information.

On a more general level, I would say that requiring the time evolution in a physical theory to be reversible is akin to assuming that the state at a given time is fully determined by the state of the system at previous times, and that different initial conditions lead to different final states.