# Multilinear maps that preserve unitarity

Let $M_1, M_2, M_3$ be spaces of square complex matrices, respectively acting on finite-dimensional Hilbert spaces $V_1, V_2$, and $V_3 = V_1 \otimes V_2$. Consider bilinear maps

$$\phi: M_1 \times M_2 \to M_3$$

with the property that if $U_1 \in M_1, U_2 \in M_2$ are unitary, then $\phi(U_1, U_2)$ is also unitary. Then Marcus' Theorem from the theory of linear preserver problems implies that the following two decompositions (up to transposes) hold simultaneously

$$\phi(U_1, U_2) = f(U_2) (U_1\otimes I_2) g(U_2)$$ $$\phi(U_1, U_2) = h(U_1) (I_1 \otimes U_2) q(U_1),$$

for some (not necessarily linear) functions $f, g : M_2 \to M_3$ and $h, q: M_1 \to M_3$ that send unitaries to unitaries.

My interest in this question is inspired by my research in quantum information theory. Has such a problem been studied before in mathematics? Is anything known about the characterization of the functions $f, g, h, q$ that allow $\phi$ to preserve unitarity? Or are there perhaps methods that are well-suited to the study of this problem?

There exists "non-trivial" solutions to this problem: for example the so-called quantum switch (https://doi.org/10.1103/PhysRevA.88.022318). Let $V_1$ and $V_2$ be two-dimensional, with $|0\rangle$, $|1\rangle$ as orthonormal bases for these Hilbert spaces. Then

$$f(U_2) = I \otimes |1\rangle \langle 1| + U_2 \otimes |0\rangle \langle 0|$$ $$g(U_2) = I \otimes |0\rangle \langle 0| + U_2 \otimes |1\rangle \langle 1|$$ defines a $\phi$ that satisfies the desired property.

I am also interested in the more general situation of mulitilinear maps $\phi: M_1\times M_2\times \cdots \times M_n \to M_1 \otimes \cdots \otimes M_n$ that have a similar unitarity-preserving property.

PS: I had originally posted this question on the Mathematics Stackexchange (https://math.stackexchange.com/questions/2241844/multilinear-maps-that-preserve-unitarity), but after searching the web some more, I believe that this is a research level question.