Ancilla dimensions needed to carry out positive-operator-valued measurement

To implement a positive-operator-valued measurement (POVM), which is defined on the $d_S$-dimensional Hilbert space of a physical system, one has to extend the Hilbert space to include $d_A$ additional dimensions (called the ancilla). This is done via either the tensor product extension (TPE) or the direct sum extension (DSE). The implementation of a POVM utilizes the available resources more efficiently if it requires fewer additional dimensions. To determine how to implement a POVM with the least additional dimensions is, therefore, an important task in quantum information. We have determined the necessary and sufficient (hence minimal) number of the additional dimensions needed to implement the same POVM by the TPE and the DSE, respectively. If the POVM has $n$ elements and $r_i$ is the rank of the $i$th element, then the dimension of the minimal ancilla is $d_A={\sum }_{i=1}^n{r}_i-d_S$ for the DSE implementation, and this represents a lower bound for the added dimensions in the TPE implementation. In the proof, we explicitly construct the DSE implementation of a general POVM with elements of arbitrary rank. As an example, we determine $d_A$ for the unambiguous discrimination of $N$ linearly independent states and provide the full DSE implementation of a state-discriminating POVM for $N=2$.