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Let $D$ and $E$ be subspaces of the tensor product of the finite-dimensionalHilbert spaces $\mathbb{C}^m \otimes \mathbb{C}^n$. We show that the number ofproduct vectors in $D$ with their partial conjugates in $E$ is uniformlybounded depending only on $m$ and $n$ whenever it is finite. We also give anupper bound in qubit-qunit case which we expect to be sharp.

The number of product vectors and their partial conjugates in a pair of spaces