IMaPh : Quantum Information Problems 21 22 23 24 25
SIC POVMs and Zauner's Conjecture

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Problem   Background   Partial Results and History   Literature

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Problem

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 [1]. For problem 3 see [2] and [3].

Problem 1: SIC-POVMs

A set of d² normed vectors {|\phi_i>}_i in a Hilbert space of dimension d constitutes a set of equiangular lines if their mutual inner products

|<\phi_i|\phi_j>|²

• the associated projection operators sum to a multiple of unity and thus induce a POVM (up to normalization) and that
• these operators are linearly independent and hence any quantum state can be reconstructed from the measurement statistics p_i := \tr(|\phi_i> <\phi_i| \rho ) of the POVM.

The most general form of the problem is: decide if SIC-POVMs exists in any dimension d.

Problem 2: Covariant SIC-POVMs

For a given basis {|q>}_{q=0... d-1} of the Hilbert space, define the shift operator X and clock operator Z respectively by the relations

X |q>	:=	|q+1>
Z |q>	:=	ei \frac 2 \pi d q |q>,

where arithmetic is modulo d. Further, define the Weyl operators

w(p,q) = Z(p) X(q)      

A vector |\phi> is called a fiducial vector with respect to the Heisenberg group if the set

{w(p,q) |\phi> <\phi| w(p,q)^* }_{p,q=0... d-1}      

The problem: decide if group covariant SIC-POVMs exist in any dimension d.

Problem 3: Zauner's Conjecture

The normalizer of the Heisenberg group within the unitaries U(d) is called the 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^* = w(q-p,-p).

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Background

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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Partial Results and History

• In the context of quantum information, the problem seems to have been tackled first by Gerhard Zauner in his doctorial thesis [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[Note: Refer e.g. to the first vs. the second version of [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. [1]. Building on concepts from 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 spherical 2-design[Note: A finite set X of unit vectors is a 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 [4] and was apparently known to Zauner (see Remark 3 in [4]).

• In [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 [1].

• Appleby in [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 [1] were compatible with Zauner's conjecture and analyzed their stability groups inside the Clifford group[Note: A similar analysis can be performed using discrete Wigner functions, as will be reported in [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 [7]. The same line of thought was pursued by Bengtsson and Ericsson in [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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Literature

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

Questions and comments

Last modified: 21 Apr 2005

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