Seeking SICs: A Workshop on Quantum Frames and Designs - 2008

By definition, SIC-POVMs are symmetric in the sense that the magnitude of the inner product between any pair of vectors is constant. All known constructions are based on additional symmetries, mainly with respect to the Weyl-Heisenberg group. Analyzing solutions for small dimensions, Zauner has identified an additional symmetry of order three and conjectured that these symmetries can be used to construct SIC-POVMs for all dimensions. Appleby has confirmed that all numerical solutions of Renes et al. indeed have that additional symmetry. This leads to the main questions addressed in the talk: Do all SIC-POVMs necessarily possess these symmetries, or can we construct SIC-POVMs without or with other symmetries?

Abstract: Adapting the concept of Wigner functions to finite dimensional systems is no simple matter. Basic issues with existing proposals are that they are either over-complete in the sense that the degrees of freedom in the Wigner function does not match the degrees of freedom in the original density matrix, or that they work only for restricted dimensions, namely for prime powers. We propose a new way to define a Wigner function and associated quantum phase space for some non-prime power dimensions. This allows to perform quantum state tomography from the measurement statistics of a minimal number of yes/no measurements.

As a means of exactly derandomizing certain quantum information processing tasks, unitary designs have become an important concept in quantum information theory. A unitary design is a collection of unitary matrices that approximates the entire unitary group, much like a spherical design approximates the entire unit sphere. We use irreducible representations of the unitary group to find a general lower bound on the size of a unitary t-design in U(d), for any d and t. The tightness of these bounds is then considered, where specific unitary 2-designs are introduced that are analogous to SIC-POVMs and complete sets of MUBs in the complex projective case. Additionally, we catalogue the known constructions of unitary t-designs and give an upper bound on the size of the smallest weighted unitary t-design in U(d). This is joint work with Aidan Roy (Calgary): \'Unitary designs and codes,\' arXiv:0809.3813.

I present three realizations about the SIC problem which excited me several years ago but which did not - unsurprisingly - lead anywhere. 1. In odd dimensions d, the metaplectic representation of SL(2,Z_d) decomposes into two irreducible components, acting on the odd and even parity subspaces respectively. It follows that if a fiducial vector | Psi> possesses some Clifford-symmetry, the same is already true for both its even and its odd parity components |Psi_e>, |Psi_o>. What is more, these components have potentially a larger symmetry group than their sum. Indeed, this effect can be verified when looking at the known numerical solutions in d=5 and d=7. A finding of remarkably little consequence! 2. In composite dimensions d=p_1^r_1 ... p_k^r_k, all elements of the Clifford group factor with respect to some tensor decomposition C^d=C^(p_1^r_1) x ... x C^(p_k^r_k) of the underlying Hilbert space. This structure may potentially be used to simplify the constraints on fiducial vectors. My optimism is vindicated by the following, ground-breaking result: In even dimensions 2d not divisible by four, the Hilbert space is of the form C^2 x C^d. So it makes sense to ask for the Schmidt-coefficients of a fiducial vector with respect to that tensor product structure. They can be computed to be 1/2(1 +/- sqrt{3/(d+1)}), removing one (!) parameter from the problem and establishing that, asymptotically, fiducial vectors are maximally entangled. 3. Becoming slightly more esoteric, I could move on to talk about discrete Wigner functions and show in what sense finding elements of a set of MUBs corresponds to imposing that a certain matrix be positive, while a similar argument for fiducial vectors requires a related matrix to be unitary. Now, positivity has \'local\' consequences: it implies constraints on small sub-matrices. Unitarity, on the other hand, seems to be more \'global\' in that all algebraic consequences of unitarity involve \'many\' matrix elements at the same time. This point of view suggests that SICs are harder to find than MUBs (in case anybody wondered). If we solve the problem by Wednesday, I\'ll talk about quantum expanders.

Abstract: Complete sets of mutually unbiased bases are clearly \'cousins\' of SICs. One difference is that there is a \'theory\' for MUBs, in the sense that they are straightforward to construct in some cases, and probably impossible to construct in others. Moreover complete sets of MUBs do appear naturally in the algebraic geometry of projective space (in particular they come from elliptic curves with certain symmetries). I will describe some unsuccessful attempts I have made to go from MUBs to SICs.

The question whether SICs exist can be viewed as a question about the structure of the convex set of quantum measurements, or turned into one about quantum states, asserting that they must have a high degree of symmetry. I\'ll address Chris Fuchs\' contrast of a \'probability first\' view of the issue with a \'generalized probabilistic theories\' view of it. I\'ll review some of what\'s known about the structure of convex state and measurement spaces with symmetries of a similar flavor, including the quantum one, and speculate on connections to recent SIC triple product results. And I\'ll present some old calculations, which will look familiar to old hands but may be worth contemplating yet again, reducing the Heisenberg-symmetric-SIC existence problem to the existence of solutions to a set of simultaneous polynomials in unit-modulus complex variables.