SIC POVMs and Zauner's Conjecture

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Problem

SIC POVMs and Zauner's Conjecture

We will give three variants of the problem, each being stronger than its predecessor. The terminology of problems 1 and 2 is taken mainly from \cite{1}. For problem 3 see \cite{2} and \cite{3}.

\subsection*{Problem 1: SIC-POVMs}

A set of $d^2$ normed vectors $\{\Ket{\phi_i}\}_i$ in a Hilbert space of dimension $d$ constitutes a set of \emph{equiangular lines} if their mutual inner products $\left Background Partial Results Solution Literature _d. We will refer to the group generated by (\ref{weyl}) as the \emph{Heisenberg group}. It is also known as the \emph{Weyl-Heisenberg group} or \emph{Generalized Pauli group}. A vector \Ket{\phi} is called a \emph{fiducial vector} with respect to the Heisenberg group if the set $$\left\{w(p,q) \, \Ket{\phi} \Bra{\phi} \, w(p,q)^\ast \right\}_{p,q=0\ldots d-1} \label{cov}$$ induces a SIC-POVM. Such a SIC-POVM is said to be \emph{group covariant}. The definition makes sense for any group of order at least d^2. However, we will focus on the Heisenberg group in what follows. The problem: decide if group covariant SIC-POVMs exist in any dimension d. \subsection*{Problem 3: Zauner's Conjecture} The normalizer of the Heisenberg group within the unitaries U(d) is called the \emph{Clifford group}. There exists an element z of the Clifford group which is defined via its action on the Weyl operators as \[ z \,w(p,q) z^\ast = w(q-p,-p).$ Zauner's conjecture, as formulated in \cite{3}, runs: in any dimension $d$, a fiducial vector can be found among the eigenvectors of $Z$.

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Besides their mathematical appeal, SIC-POVMs have obvious applications to quantum state tomography. The symmetry condition assures that the possible measurement outcomes are in some sense maximally complementary.

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• In the context of quantum information, the problem seems to have been tackled first by Gerhard Zauner in his doctorial thesis \cite{2} in 1999. To our knowledge, the results were neither published nor translated into English, which caused some confusion in the English literature, as to what Zauner had actually conjectured\footnote{Refer e.g. to the first vs. the second version of \cite{3} on the arXiv server.}. Zauner analyzed the spectrum of $z$. He listed analytical expressions for fiducial vectors in dimension 2, 3, 4, 5 and numerical expressions for $d=6, 7$. He noted that for dimension 8 an analytic SIC-POVM is known, which is covariant under the action of the threefold tensor product of the two dimensional Heisenberg group.
• Wide interest in the problem arose with the 2003 paper by Renes et. al. \cite{1}. Building on concepts from \emph{frame theory}, the authors reduced the task of numerically finding fiducial vectors to a non-convex global optimization problem. Using this method, they presented numerical fiducial vectors for all dimensions up to 45 and counted the number of distinct covariant SIC-POVMs up to dimension 7. The question of whether those vectors were eigenstates of a Clifford operation was left open (but see below). Further, four groups other than the Heisenberg group were numerically found to induce SIC-POVMs in the sense of (\ref{cov}). The authors showed that a SIC-POVM corresponds to a \emph{spherical 2-design}\footnote{A finite set $X$ of unit vectors is a \emph{t-design} if the average of any t-th order polynomial over $X$ is the same as the average of that polynomial over the entire unit sphere.}. The same assertion was proven by Klappenecker and Rötteler in \cite{4} and was apparently known to Zauner (see Remark 3 in \cite{4}).
• In \cite{5} Grassl used a computer algebra system capable of symbolic calculations to prove Zauner's conjecture for $d=6$. He remarked that elements of the Clifford group map fiducial vectors onto fiducial vectors. Building on that observation, he could account for all 96 covariant SIC-POVMs that were reported to exist for $d=6$ in \cite{1}.
• Appleby in \cite{3} gave a detailed description of the Clifford group and extended it by allowing for anti-unitary operators. He verified that the numeric solutions of \cite{1} were compatible with Zauner's conjecture and analyzed their stability groups inside the Clifford group\footnote{A similar analysis can be performed using discrete Wigner functions, as will be reported in \cite{6}.}. Appleby goes on to present analytical expressions for fiducial vectors in dimension 7 and 19 and specifies an infinite sequence of dimensions for which he conjectures that solutions can be found more easily.
• Inspired by a construction that links finite geometries to MUBs, there have been some speculations by Wootters about whether SIC-POVMs can be linked to finite affine planes \cite{7}. The same line of thought was pursued by Bengtsson and Ericsson in \cite{8}. However, the existence of such a construction remains an open problem. The results by Grassl are of some relevance here, as it is known that affine planes of order 6 do not exist.

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 [HHHo02] J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, \bibtitle{Symmetric Informationally Complete Quantum Measurements}, J. Math. Phys. \textbf{45}, 2171 (2004) and \href{http://xxx.lanl.gov/abs/quant-ph/0310075}{quant-ph/0310075} (2003). [DW] G. Zauner, \bibtitle{Quantendesigns -- Grundzüge einer nichtkommutativen Designtheorie}, Doctorial thesis, University of Vienna, 1999 (available online at \href{http://www.mat.univie.ac.at/~neum/papers/physpapers.html}{http://www.mat.univie.ac.at/\~{}neum/papers/physpapers.html}). [HHHO05] D. M. Appleby, \bibtitle{SIC-POVMs and the Extended Clifford Group}, \href{http://xxx.lanl.gov/abs/quant-ph/0412001}{quant-ph/0412001} (2004). [HPHH05] A. Klappenecker, and M. Rötteler, \bibtitle{Mutually Unbiased Bases are Complex Projective 2-Designs}, \href{http://xxx.lanl.gov/abs/quant-ph/0502031}{quant-ph/0502031} (2005). [HLLO05] M. Grassl, \bibtitle{On SIC-POVMs and MUBs in dimension 6}, \href{http://xxx.lanl.gov/abs/quant-ph/0406175}{quant-ph/0406175} (2004). [HLLO05] D. Gross, Diploma thesis, University of Potsdam, 2005. [HLLO05] W. K. Wootters, \bibtitle{Quantum measurements and finite geometry}, \href{http://xxx.lanl.gov/abs/quant-ph/0406032}{quant-ph/0406032} (2004). [HLLO05] I. Bengtsson and \AA. Ericsson, \bibtitle{Mutually Unbiased Bases and The Complementarity Polytope}, \href{http://xxx.lanl.gov/abs/quant-ph/0410120}{quant-ph/0410120} (2004).

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