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

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Let D and E be subspaces of the tensor product of the finite-dimensional Hilbert spaces Cᵐ ⊗ Cⁿ. We show that the number of product vectors in D with their partial conjugates in E is uniformly bounded depending only on m and n whenever it is finite. We also give an upper bound in qubit-qunit case which we expect to be sharp.