results for au:McConnell_G in:quant-ph

- Jan 20 2017 quant-ph arXiv:1701.05200v1We give an overview of some remarkable connections between symmetric informationally complete measurements (SIC-POVMs, or SICs) and algebraic number theory, in particular, a connection with Hilbert's 12th problem. The paper is meant to be intelligible to a physicist who has no prior knowledge of either Galois theory or algebraic number theory.
- Let K be a real quadratic field. For certain K with sufficiently small discriminant we produce explicit unit generators for specific ray class fields of K using a numerical method that arose in the study of complete sets of equiangular lines in $\mathbb{C}^d$ (known in quantum information as symmetric informationally complete measurements or SICs). The construction in low dimensions suggests a general recipe for producing unit generators in infinite towers of ray class fields above arbitrary K and we summarise this in a conjecture. Such explicit generators are notoriously difficult to find, so this recipe may be of some interest. In a forthcoming paper we shall publish promising results of numerical comparisons between the logarithms of these canonical units and the values of L-functions associated to the extensions, following the programme laid out in the Stark Conjectures.
- The notion of Symmetric Informationally Complete Positive Operator-Valued Measures (SIC-POVMs) arose in physics as a kind of optimal measurement basis for quantum systems. However the question of their existence is equivalent to that of the existence of a maximal set of \emphcomplex equiangular lines. That is to say, given a complex Hilbert space of dimension $d$, what is the maximal number of (complex) lines one can find which all make a common (real) angle with one another, in the sense that the inner products between unit vectors spanning those lines all have a common absolute value? A maximal set would consist of $d^2$ lines all with a common angle of $\arccos{\frac{1}{\sqrt{d+1}}}$. The same question has been posed in the real case and some partial answers are known. But at the time of writing no unifying theoretical result has been found in the real or the complex case: some sporadic low-dimensional numerical constructions have been converted into algebraic solutions but beyond this very little is known. It is conjectured that such structures always arise as orbits of certain fiducial vectors under the action of the Weyl (or generalised Pauli) group. In this paper we point out some new construction methods in the lowest dimensions ($d=2$ and $d=3$). We should mention that the SIC-POVMs so constructed are all unitarily equivalent to previously known SIC-POVMs.
- Let m and n be any integers with n>m>=2. Using just the entropy function it is possible to define a partial order on S_mn (the symmetric group on mn letters) modulo a subgroup isomorphic to S_m x S_n. We explore this partial order in the case m=2, n=3, where thanks to the outer automorphism the quotient space is actually isomorphic to a parabolic quotient of S_6. Furthermore we show that in this case it has a fairly simple algebraic description in terms of elements of the group ring.
- May 17 2012 quant-ph arXiv:1205.3517v2We study the correlation structure of separable and classical states in 2x2- and 2x3-dimensional quantum systems with fixed spectra. Even for such simple systems the maximal correlation - as measured by mutual information - over the set of unitarily accessible separable states is highly non-trivial to compute; however for the 2x2 case a particular class of spectra admits full analysis and allows us to contrast classical states with more general separable states. We analyse a particular entropic partial order on the set of spectra and prove for the qubit-qutrit case that this partial order alone picks out a unique classical maximum state for mutual information. Moreover the 2x3 case is the largest system with such a property.
- Oct 09 2007 quant-ph arXiv:0710.1502v1We show that in a complex d-dimensional vector space, one can find O(d) bases whose elements form a 2-design. Such vector sets generalize the notion of a maximal collection of mutually unbiased bases (MUBs). MUBs have manifold applications in quantum information theory (e.g. in state tomography, cloning, or cryptography) -- however it is suspected that maximal sets exist only in prime-power dimensions. Our construction offers an efficient alternative for general dimensions. The findings are based on a framework recently established in [A. Roy and A. Scott, J. Math. Phys. 48, 072110 (2007)], which reduces the construction of such bases to the combinatorial problem of finding certain highly nonlinear functions between abelian groups.