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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 analyze 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 analyze 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. 

We investigate the connection between the concept of a-balancedness introduced in [Phys. Rev A. 85, 032112 (2012)] and polynomial local SU invariants and the appearance of topological phases respectively. It is found that different types of a-balancedness correspond to different types of local SU invariants analogously to how different types of balancedness as defined in [New J. Phys. 12, 075025 (2010)] correspond to different types of local SL invariants. These different types of SU invariants distinguish between states exhibiting different topological phases. In the case of three qubits the different kinds of topological phases are fully distinguished by the three-tangle together with one more invariant. Using this we present a qualitative classification scheme based on balancedness of a state. While balancedness and local SL invariants of bidegree $(2n,0)$ classify the SL-semistable states [New J. Phys. 12, 075025 (2010), Phys. Rev. A 83, 052330 (2011)], a-balancedness and local SU invariants of bidegree (2n-m,m) gives a more fine grained classification. In this scheme the a-balanced states form a bridge from the genuine entanglement of balanced states, invariant under the SL-group, towards the entanglement of unbalanced states characterized by U invariants of bidegree (n,n). As a by-product we obtain generalizations to the W-state, states that are entangled, but contain only globally distributed entanglement of parts of the system.

In their recent paper, Yan-Xiong Du {\it et al.} [Phys. Rev. A {\bf 84}, 034103 (2011)] claim to have found a non-Abelian adiabatic geometric phase associated with the energy eigenstates of a large-detuned $\Lambda$ three-level system. They further propose a test to detect the non-commutative feature of this geometric phase. On the contrary, we show that the non-Abelian geometric phase picked up by the energy eigenstates of a $\Lambda$ system is trivial in the adiabatic approximation, while, in the exact treatment of the time evolution, this phase is very small and cannot be separated from the non-Abelian dynamical phase acquired along the path in parameter space.

The robustness to different sources of error of the scheme for non-adiabatic holonomic gates proposed in [arXiv:1107.5127v2] is investigated. Open system effects as well as errors in the driving fields are considered. It is found that the gates can be made more error resilient by using sufficiently short pulses. The principal limit of how short the pulses can be made is given by the breakdown of the quasi-monochromatic approximation. A comparison with the resilience of adiabatic gates is carried out.

Global phase factors of topological origin, resulting from cyclic local $\rm{SU}$ evolution, called topological phases, were first described in [Phys. Rev. Lett. {\bf 90}, 230403 (2003)], in the case of entangled qubit pairs. In this paper we investigate topological phases in multi-qubit systems as the result of cyclic local $\rm{SU(2)}$ evolution. These phases originate from the topological structure of the local $\rm{SU(2)}$-orbits and are an attribute of most entangled multi-qubit systems. We discuss the relation between topological phases and SLOCC-invariant polynomials and give examples where topological phases appear. A general method to find the values of the topological phases in an $n$-qubit system is described and a complete list of these phases for up to seven qubits is given.

In the context of two-particle interferometry, we construct a parallel transport condition that is based on the maximization of coincidence intensity with respect to local unitary operations on one of the subsystems. The dependence on correlation is investigated and it is found that the holonomy group is generally non-Abelian, but Abelian for uncorrelated systems. It is found that our framework contains the Lévay geometric phase (2004 J. Phys. A: Math. Gen. 37 1821) in the case of two-qubit systems undergoing local SU(2) evolutions.

We explore a geometric approach to generating local SU(2) and SL(2,C) invariants for a collection of qubits inspired by lattice gauge theory. Each local invariant or 'gauge' invariant is associated to a distinct closed path (or plaquette) joining some or all of the qubits. In lattice gauge theory, the lattice points are the discrete space-time points, the transformations between the points of the lattice are defined by parallel transporters and the gauge invariant observable associated to a particular closed path is given by the Wilson loop. In our approach the points of the lattice are qubits, the link-transformations between the qubits are defined by the correlations between them and the gauge invariant observable, the local invariants associated to a particular closed path are also given by a Wilson loop-like construction. The link transformations share many of the properties of parallel transporters although they are not undone when one retraces one's steps through the lattice. This feature is used to generate many of the invariants. We consider a pure three qubit state as a test case and find we can generate a complete set of algebraically independent local invariants in this way, however the framework given here is applicable to mixed states composed of any number of d level quantum systems. We give an operational interpretation of these invariants in terms of observables.

We introduce a novel bridge between the familiar gauge field theory approaches used in many areas of modern physics such as quantum field theory and the SLOCC protocols familiar in quantum information. Although the mathematical methods are the same the meaning of the gauge group will be different. The measure we introduce, `twist', is constructed as a Wilson loop from a correlation induced holonomy. The measure can be understood as the global asymmetry of the bipartite correlations in a loop of three or more qubits; if the holonomy is trivial (the identity matrix), the bipartite correlations can be globally untwisted using general local qubit operations, the gauge group of our theory, which turns out to be the group of Lorentz transformations familiar from special relativity. If it is not possible to globally untwist the bipartite correlations in a state globally using local operations, the twistedness is given by a non-trivial element of the Lorentz group, the correlation induced holonomy. We provide several analytical examples of twisted and untwisted states for three qubits, the most elementary non-trivial loop one can imagine.

The coherence properties of a three-level $\Lambda$-system influenced by a Markovian environment are analyzed. A coherence vector formalism is used and a vector form of the Lindblad equation is derived. Together with decay channels from the upper state, open system channels acting on the subspace of the two lower states are investigated, i.e., depolarization, dephasing, and amplitude damping channels. We derive an analytic expression for the coherence vector and the concomitant optical susceptibility, and analyze how the different channels influence the optical response. This response depends non-trivially on the type of open system interaction present, and even gain can be obtained. We also present a geometrical visualization of the coherence vector as an aid to understand the system response.

Nodal free geometric phases are the eigenvalues of the final member of a parallel transporting family of unitary operators. These phases are gauge invariant, always well defined, and can be measured interferometrically. Nodal free geometric phases can be used to construct various types of quantum phase gates.