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arXiv:1202.0716 · Physical Review A

Topological phases and multiqubit entanglement

Markus Johansson, Marie Ericsson, Kuldip Singh, Erik Sjöqvist, Mark S. Williamson

Published Mar 12, 2012• 29 citations
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Abstract

Global phase factors of topological origin, resulting from cyclic local $\mathit{SU}$ evolution, called topological phases, were first described in [Phys. Rev. Lett. 90, 230403 (2003)], in the case of entangled qubit pairs. In this paper we investigate topological phases in multiqubit systems as the result of cyclic local $\mathit{SU}(2)$ evolution. These phases originate from the topological structure of the local $\mathit{SU}(2)$ orbits and are an attribute of most entangled multiqubit systems. We discuss the relation between topological phases and stochastic local operations and classical communication (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.

DOI
10.1103/physreva.85.032112

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Global phase factors of topological origin, resulting from cyclic local SU\mathit{SU}SU evolution, called topological phases, were first described in [Phys. Rev. Lett. 90, 230403 (2003)], in the case of entangled qubit pairs. In this paper we investigate topological phases in multiqubit systems as the result of cyclic local SU(2)\mathit{SU}(2)SU(2) evolution. These phases originate from the topological structure of the local SU(2)\mathit{SU}(2)SU(2) orbits and are an attribute of most entangled multiqubit systems. We discuss the relation between topological phases and stochastic local operations and classical communication (SLOCC)-invariant polynomials and give examples where topological phases appear. A general method to find the values of the topological phases in an nnn-qubit system is described and a complete list of these phases for up to seven qubits is given.
We propose an experiment to observe the topological phases associated with cyclic evolutions, generated by local SU(2) operations, on three-qubit entangled states prepared on different degrees of freedom of entangled photon pairs. The topological phases reveal the nontrivial topological structure of the local SU(2) orbits. We describe how to prepare states showing different topological phases and discuss their relation to entanglement. In particular, the presence of a $\ensuremath{\pi}/2$ phase shift is a signature of genuine tripartite entanglement in the sense that it does not exist for two-qubit systems.
We discuss the appearance of fractional topological phases on cyclic evolutions of entangled qudits. The original result reported by Oxman and Khoury [Phys. Rev. Lett. 106, 240503 (2011)] is detailed and extended to qudits of different dimensions. The topological nature of the phase evolution and its restriction to fractional values are related to both the structure of the projective space of states and entanglement. For maximally entangled states of qudits with the same Hilbert-space dimension, the fractional geometric phases are the only ones attainable under local SU(ddd) operations, an effect that can be experimentally observed through conditional interference.
In this thesis we investigate geometric and topological structures in the context of entanglement and quantum computation. A parallel transport condition is introduced in the context of Franson interferometry based on the maximization of two-particle coincidence intensity. The dependence on correlations is investigated and it is found that the holonomy group is in general non-Abelian, but Abelian for uncorrelated systems. It is found that this framework contains a parallel transport condition developed by Levay in the case of two-qubit systems undergoing local SU(2) evolutions. Global phase factors of topological origin, resulting from cyclic local SU(2) evolution, called topological phases, are investigated in the context of multi-qubit systems. These phases originate from the topological structure of the local SU(2)-orbits and are an attribute of most entangled multi-qubit systems. The relation between topological phases and SLOCC-invariant polynomials is discussed. A general method to find the values of the topological phases in an n-qubit system is described. A non-adiabatic generalization of holonomic quantum computation is developed in which high-speed universal quantum gates can be realized by using non-Abelian geometric phases. It is shown how a set of non-adiabatic holonomic one- and two-qubit gates can be implemented by utilizing transitions in a generic three-level Λ configuration. The robustness of the proposed scheme to different sources of error is investigated through numerical simulation. It is found that the gates can be made robust to a variety of errors if the operation time of the gate can be made sufficiently short. This scheme opens up for universal holonomic quantum computation on qubits characterized by short coherence times.
We investigate the connection between the concept of affine balancedness (a-balancedness) introduced by M. Johansson et al. [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 by A. Osterloh and J. Siewert, [New J. Phys. 12, 075025 (2010)], correspond to different types of local special linear (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)(2n,0)(2n,0) classify the SL-semistable states [A. Osterloh and J. Siewert, New J. Phys. 12, 075025 (2010); O. Viehmann et al., Phys. Rev. A 83, 052330 (2011)], a-balancedness and local SU invariants of bidegree $(2n\ensuremath{-}m,m)$ give 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)(n,n)(n,n). As a byproduct, we obtain generalizations to the W state, i.e., states that are entangled, but contain only globally distributed entanglement of parts of the system.
Kauffman and Lomonaco (New J Phys 4:73.1–73.18, 2002. arXiv:quant-ph/0401090 , New J Phys 6:134.1–134.40, 2004) explored the idea of understanding quantum entanglement (the non-local correlation of certain properties of particles) topologically by viewing unitary entangling operators as braiding operators. In Alagic et al. (Yang–Baxter operators need quantum entanglement to distinguish knots, 2015. arXiv:1507.05979v1 ), it is shown that entanglement is a necessary condition for forming non-trivial invariants of knots from braid closures via solutions to the Yang–Baxter equation. We show that the arguments used by Alagic et al. (2015) generalize to essentially the same results for quantum invariant state summation models of knots. In one case (the unoriented swap case) we give an example of a Yang–Baxter operator, and associated quantum invariant, that can detect the Hopf link. Again this is analogous to the results of Alagic et al. (2015). We also give a class of R matrices that are entangling and are weak invariants of classical knots and links yet strong invariants of virtual knots and links. We also give an example of an SU(2) representation of the three-strand braid group that models the Jones polynomial for closures of three-strand braids. This invariant is a quantum model for the Jones polynomial restricted to three-strand braids, and it does not involve quantum entanglement. These relationships between topological braiding and quantum entanglement can be used as a framework for future work in understanding the properties of entangling gates in topological quantum computing. The paper ends with a discussion of the Aravind hypothesis about the direct relationship of knots and quantum entanglement and the ER=EPRER = EPRER=EPR E R = E P R hypothesis about the relationship of quantum entanglement with the connectivity of space. We describe how, given a background space and a quantum tensor network, to construct a new topological space that welds the network and the background space together. This construction embodies the principle that quantum entanglement and topological connectivity are intimately related.
Classification of entanglement is an important problem in quantum resource theory. In this paper we discuss an embedding of this problem in the context of topological quantum field theories (TQFTs). This approach allows classifying entanglement patterns in terms of topological equivalence classes. In the bipartite case a classification equivalent to the one by stochastic local operations and classical communication (SLOCC) is constructed by restricting to a simple class of connectivity diagrams. Such diagrams characterize quantum states of TQFT up to braiding and tangling of the ``connectome.'' In the multipartite case the same restricted topological classification only captures a part of the SLOCC classes, in particular, it does not see the WWW entanglement of three qubits. Nonlocal braiding of connections may solve the problem, but no finite classification is attempted in this case. Despite incompleteness, the connectome classification has a straightforward generalization to any number and dimension of parties and has a very intuitive interpretation, which might be useful for understanding specific properties of entanglement and for design of new quantum resources.