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1.

Andersson, Ole

et al.

Stockholm University, Faculty of Science, Department of Physics.

Bengtsson, Ingemar

Stockholm University, Faculty of Science, Department of Physics.

CLIFFORD TORI AND UNBIASED VECTORS2017In: Reports on mathematical physics, ISSN 0034-4877, E-ISSN 1879-0674, Vol. 79, no 1, p. 33-51Article in journal (Refereed)

Abstract [en]

The existence problem for mutually unbiased bases is an unsolved problem in quantum information theory. A related question is whether every pair of bases admits vectors that are unbiased to both. Mathematically this translates to the question whether two Lagrangian Clifford tori intersect, and a body of results exists concerning it. These results are however rather weak from the point of view of the first problem. We make a detailed study of how the intersections behave in the simplest nontrivial case, that of complex projective 2-space (the qutrit), for which the set of pairs of Clifford tori can be usefully parametrized by the unistochastic subset of Birkhoff's polytope. Pairs that do not intersect transversally are located. Some calculations in higher dimensions are included to see which results are special to the qutrit.

Stockholm University, Faculty of Science, Department of Physics.

Bengtsson, Ingemar

Stockholm University, Faculty of Science, Department of Physics.

Ericsson, Marie

Sjöqvist, Erik

Geometric phases for mixed states of the Kitaev chain2016In: Philosophical Transactions. Series A: Mathematical, physical, and engineering science, ISSN 1364-503X, E-ISSN 1471-2962, Vol. 374, no 2068, article id 20150231Article in journal (Refereed)

Abstract [en]

The Berry phase has found applications in building topological order parameters for certain condensed matter systems. The question whether some geometric phase for mixed states can serve the same purpose has been raised, and proposals are on the table. We analyse the intricate behaviour of Uhlmann's geometric phase in the Kitaev chain at finite temperature, and then argue that it captures quite different physics from that intended. We also analyse the behaviour of a geometric phase introduced in the context of interferometry. For the Kitaev chain, this phase closely mirrors that of the Berry phase, and we argue that it merits further investigation.

Alignment is a geometric relation between pairs of Weyl-Heisenberg SICs, one in dimension d and another in dimension d(d - 2), manifesting a well-founded conjecture about a number-theoretical connection between the SICs. In this paper, we prove that if d is even, the SIC in dimension d(d - 2) of an aligned pair can be partitioned into (d - 2)(2) tight d(2)-frames of rank d(d - 1)/2 and, alternatively, into d(2) tight (d - 2)(2) -frames of rank (d - 1) (d - 2)/2. The corresponding result for odd d is already known, but the proof for odd d relies on results which are not available for even d. We develop methods that allow us to overcome this issue. In addition, we provide a relatively detailed study of parity operators in the Clifford group, emphasizing differences in the theory of parity operators in even and odd dimensions and discussing consequences due to such differences. In a final section, we study implications of alignment for the symmetry of the SIC.

We use tools from the theory of dynamical systems with symmetries to stratify Uhlmann's standard purification bundle and derive a new connection for mixed quantum states. For unitarily evolving systems, this connection gives rise to the 'interferometric' geometric phase of Sjqvist et al (2000 Phys. Rev. Lett. 85 2845-9), and for more generally evolving open systems it gives rise to the generalization of the interferometric geometric phase due to Tong et al (2004 Phys. Rev. Lett. 93 080405).

Geometric phase has found a broad spectrum of applications in both classical and quantum physics. In this work we discuss a geometric phase for mixed quantum states based on traces of spectral weighted holonomies. Our approach applies to general unitary evolutions of both nondegenerate and degenerate mixed states, and it generalizes the standard definition of geometric phase for mixed states, which is based on quantum interferometry. We provide an explicit formula for the geometric phase that can be easily implemented for computations in quantum physics, and we discuss higher order analogs of the geometric phase that might be defined at points where the ordinary geometric phase is undefined.

Geometrical structures of quantum mechanics provide us with new insightful results about the nature of quantum theory. In this work we consider mixed quantum states represented by finite rank density operators. We review our geometrical framework that provide the space of density operators with Riemannian and symplectic structures, and we derive a geometric uncertainty relation for observables acting on mixed quantum states. We also give an example that visualizes the geometric uncertainty relation for spin-1/2 particles.