non-clustered unique indexes on sql server partitions
Short answer: 

If the clustering key would not be part of the key set then the index uniqueness across all partitions could not be guaranteed. 

The long story:  

The only way to guarantee uniqueness is by enforcing uniqueness via a B-Tree structure. With a B-Tree is really easy to enforce uniqueness, since any key has a deterministic logical position in the B-Tree (between its left neighbor, that is smaller key, and the right neighbor, which is a larger key). Since each key value can be inserted in a single position in the tree then uniqueness is easy to guarantee: simply check during insert if the position is already occupied by an existing row.  

With partitioning uniqueness across all partitions is trickier, since there are many B-Tree structures involved (one for each partition). Each can guarantee uniqueness, within a partition, but together they cannot guarantee global uniqueness across all partitions: a key in partition X may also appear in partition Y. Therefore, in order to preserve the semantics of 'unique index', the clustering column must be part of the key. This way each key value deterministically bound to a specific partition: a key cannot appear in partition X and Y because the partitioning column is part of the key, therefore the key can belong to exactly partition X or Y.

@MarkStoreySmith asked: So why doesn't SQL Server check for uniqueness by navigating each of the B-tree structures, acquiring range locks where the value being tested is not present, and holding until each B-tree has been traversed? Seeking each B-Tree would be a performance hog (think 15,000 partitions to be checked at each insert). And also the locking complications (hopping from one B-Tree to the next) are significant, with huge potential for deadlocks.