Session W29: Symmetric Discrete Structures for Finite Dimensional Quantum Systems

Pairwise complementary observables and their mutually unbiased bases

Pairs of complementary observables (PCO) characterize all quantum degrees of freedom and are central to a technical formulation of Bohr's principle of complementarity. A defining property of such pairs are their mutually unbiased bases (MUB) of eigenstates. MUB have found many applications for tasks in quantum information processing. Maximal sets of PCO and MUB are known, by explicit construction, for degrees of freedom that live in finite-dimensional Hilbert space whose dimension is a power of a prime; continuous sets of MUB are also known for most continuous degrees of freedom. I will review the situation and mention a couple of open problems.   

Quantum States as Probabilities from Symmetric Informationally Complete Measurements

If you pick $d^2$ symmetrically spread vectors in a $d$-dimensional Hilbert space, you get a symmetric informationally complete set of quantum states (or SIC for short). SICs have applications within quantum information science, such as to quantum state tomography and quantum cryptography, and are also of interest for foundational studies of quantum mechanics. In this talk I will review the representation of quantum states as probability distributions over the outcomes of a SIC measurement. Not all probability distributions correspond to quantum states, thus quantum state space is a restricted subset of all potentially available probabilities. We will explore how this restriction can be characterized. A recent publication (Fuchs and Schack, arXiv:0906.2187) advocates the SIC-representation and suggests that the Born rule rewritten in this language can be taken as a postulate for quantum mechanics. This motivates the introduction of so-called maximally consistent sets (Appleby, Ericsson, and Fuchs, arXiv:0910.2750); one such set is quantum state space.   

The Lie Algebraic Significance of Symmetric Informationally Complete Measurements

Examples of symmetric informationally complete positive operator valued measures (SIC-POVMs) have been constructed in every dimension less than 68. However, it remains an open question whether they exist in all finite dimensions. A SIC-POVM is usually thought of as a highly symmetric structure in quantum state space. However, its elements can equally well be regarded as a basis for the Lie algebra gl(d,C). We examine the resulting structure constants, which are calculated from the traces of the triple products of the SIC-POVM elements and which, it turns out, characterize the SIC-POVM up to unitary equivalence. We show that the structure constants have numerous remarkable properties. In particular we show that the existence of a SIC-POVM in dimension d is equivalent to the existence of a certain structure in the adjoint representation of gl(d,C). We hope that transforming the problem in this way, from a question about quantum state space to a question about Lie algebras, may help to make the existence problem tractable. This is joint work with M. Appleby and C. Fuchs.   

Experimental access to higher-dimensional discrete quantum systems, towards realizing SIC-POVM and MUB measurements, using integrated optics

The aim of our work is to access and explore higher-dimensional photonic quantum systems. In terms of stability and complexity, normal bulk-optic setups greatly limit the capabilities of reaching higher-dimensional systems. However, the recent development in integrated photonic circuits has opened new possibilities [1]. Our approach is to use integrated photonic circuits on-chip, as well as in fiber, to reach photonic states of higher dimension. We are working toward a fully integrated realization of a multiport [2], a device which can apply any unitary transformation based on tunable internal parameters. Our first step is to realize a multiport in dimension four, implementing any unitary transformation on Qubits, Qutrits and Ququarts. Furthermore, we have built an integrated source using purely in-fiber components for creating higher-dimensional entangled photons. The combination of this source with the multiport yields a very general system applicable to a variety of experiments in higher dimensional Hilbert spaces. It is possible to realize different experimental setups by setting the device for different incoming entangled states, and subsequently applying unitary transformations. For example, this opens the possibility to observe new types of higher-order perfect correlations [3], or to realize full SIC-POVM measurements in higher dimensions.\\[4pt] [1] J.L.O'Brien, G.J.Pryde, A.G.White, T.C.Ralph and D.Branning, Nature Vol.426, pp264-267 (2003) \newline [2] M.Reck and A.Zeilinger, PRL Vol.73, No.1 (1994) \newline [3] M.Zukowski, A.Zeilinger and M.A.Horne, PRA Vol.55, No.1 (1997)   

Isotropic States in Discrete Phase Space

An energy eigenstate of a harmonic oscillator is isotropic in phase space, in the sense that the state looks the same along any ray emanating from the origin. It is possible to extend this notion of ``isotropic'' to quantum systems with finite-dimensional state spaces---the rays are then rays in discrete phase space. In this talk I present examples of discrete isotropic states and discuss their properties. One can show that every isotropic state minimizes a specific information-theoretic measure of uncertainty with respect to a complete set of mutually unbiased bases. Numerical results on a certain class of isotropic state vectors suggest that their components, in any of those same mutually unbiased bases, exhibit a semicircular distribution when the dimension of the state space is large.