2015/08/24 by Vishakh Hegde, Hegde, Vishakh, Prabha Mandayam +1
Computer Science · Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Matrix Theory and Algorithms #math-ph #math.MP #quant-ph
paper · pdf · doi:10.48550/arxiv.1508.05892
12 pages, 2 figures, v1
arxiv created 2015/08/24 · arxiv updated 2015/08/25
A set of mutually unbiased bases (MUBs) is said to be unextendible if there does not exist another basis that is unbiased with respect to the given set. Here, we prove the existence of smaller sets of MUBs in prime-squared dimensions (d=p2) that cannot be extended to a complete set using the generalized Pauli operators. We further observe an interesting connection between the existence of unextendible sets and the tightness of entropic uncertainty relations (EURs) in these dimensions. In particular, we show that our construction of unextendible sets of MUBs naturally leads to sets of p+1 MUBs that saturate both a Shannon (H1) and a collision (H2) entropic lower bound. Such an identification of smaller sets of MUBs satisfying tight EURs is crucial for cryptographic applications as well as constructing optimal entanglement witnesses for higher dimensional systems.