Suppose we construct a non-unitary quantum system α in hilbert space. It entails that this system would have no direct parallel in quantum circuitry as it is a requirement that all quantum gates are unitary and therefore the function constructed from them is also unitary. But does this mean that we cannot practically implement α?A potential parallel is found in the simulation of classical computations on a quantum computer, achieved by simulating an information-destroying NAND gate using an information-preserving Toffoli gate. If a non-unitary quantum system was such because it implemented a 'quantum NAND gate', it would be possible to redesign it as a unitary system using a Toffoli gate. It is not clear to me (and I cannot immediately find literature on this) if there are clearly defined bounds on what non-unitary operations can be simulated using unitary equivalents. Is there a clearly defined subset of non-unitary quantum systems where α is practically implementable? Is it, perhaps, universal?
When is a non-unitary quantum system only theoretical?
quantum computing;quantum information
I complete here Peters answer with a characterization of physical maps as CPTP maps. As you know, if the system is isolated, the only operations you can implement are the unitary operations. But, as you noticed, if you use an ancillary subsystem and throw a subsystem away, you can implement some other operations. The set of physical operations are the completely positive trace preserving maps. (Following most of the literature, Ill call them CPTP maps), and Stinesprings dilation theorem indeed ensures that any CPTP map can be written as the partial trace of a unitary acting on a larger Hilbert space.Since we are looking at non isolated subsystem, its quantum state has to be described by a density matrix $$. In full generality, a density matrix has to be positive ($0$) and having a unit trace ($\mathrm{Tr}=1$). Let suppose we have a generic physical map $\mathcal{E}:\mathcal{E}()$. To be physical $\mathcal{E}()$ needs to be a physical state for any input state $$, that is, we should have $:0$, $\mathcal{E}()0$ and $\mathrm{Tr}\mathcal{E}()=\mathrm{Tr}$. In other words, $\mathcal{E}$ has to be positive and trace preserving.However, it is not enough. For example, the transpose $\mathcal{T}$, is a positive trace preserving map, which is unphysical. The idea is the one behind the Peres-Hordecki criterion: suppose you have a channel performing $\mathcal{T}$ on the thystem $A$. You can add another system $B$ of the same dimension, where you perform the identity $\mathcal{I}$ i.e. wher you dont do anything. The global map on $AB$ is then $\mathcal{T}_A\mathcal{I}_B$, which is not positive any more: some linear algebra shows $\mathcal{T}_A\mathcal{I}_B(_{AB})$ has $d(d-1)/2$ negative eignevalues, where $_{AB}$ is a maximally entangled state. For a map $\mathcal E$ to be physical, any extension of the form $\mathcal E\mathcal I$ has therefore to be positive. This is the definition of complete positivity. The Stinespring dilation theorem then shows that this necessary condition is sufficient, since it gives a recipe to implement any CPTP map with an ancillary subsystem and and a unitary.