SIC-POVMs exist in all dimensions

The most general type of measurement in a quantum system is a Positive Operator Valued Measure (POVM). In order for the POVM to completely determine the quantum state being measured it must be Informationally Complete. Maximal independence of outcomes is critical in Cryptographic applications [7], thus the pairwise inner products of the POVM elements need to be uniform. Thus we are interested in a Symmetric Informationally Complete POVM or SIC-POVM. A related idea is that of Mutually Unbiased Bases (MUBs). A POVM is a set of operators acting on C, and MUBs are a specific set of bases for the space. Maximal sets of MUBs are also an important feature of proposed cryptographic schemes [2]. Wooters [12] discusses some geometric links between MUBs, SICPOVMs and Affine planes. For the physical significance of SIC-POVMs see [1]. Maximal sets of MUBs have been constructed in prime-powered dimensions. It has been conjectured that maximal sets of MUBs exist only in the dimensions for which affine planes also exist[11]. It has been conjectured that SIC-POVMs exist in all finite dimensions [10, 6]. We show that this conjecture is true for all but the trivial case of d = 1. In Section 2 we reproduce with more detail the proof [10] that SIC-POVMs are spherical 2-designs. In Section 3 we show that SIC-POVMs exist in all dimensions ≥ 2. In Section 4 we give explicit constructions for odd and even dimensions [9]. Section 5 is the conclusion.