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arXiv:quant-ph/0209098 · Physical Review A

Entanglement gauge and the non-Abelian geometric phase with two photonic qubits

Karl-Peter Marzlin, Stephen D. Bartlett, Barry C. Sanders

Published Feb 28, 2003• 8 citations
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Abstract

We introduce the entanglement gauge describing the combined effects of local operations and nonlocal unitary transformations on bipartite quantum systems. The entanglement gauge exploits the invariance of nonlocal properties for bipartite systems under local (gauge) transformations. This new formalism yields observable effects arising from the gauge geometry of the bipartite system. In particular, we propose a non-Abelian gauge theory realized via two separated spatial modes of the quantized electromagnetic field manipulated by linear optics. In this linear optical realization, a bipartite state of two separated spatial modes can acquire a non-Abelian geometric phase.

DOI
10.1103/physreva.67.022316

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We introduce the entanglement gauge describing the combined effects of local operations and nonlocal unitary transformations on bipartite quantum systems. The entanglement gauge exploits the invariance of nonlocal properties for bipartite systems under local (gauge) transformations. This new formalism yields observable effects arising from the gauge geometry of the bipartite system. In particular, we propose a non-Abelian gauge theory realized via two separated spatial modes of the quantized electromagnetic field manipulated by linear optics. In this linear optical realization, a bipartite state of two separated spatial modes can acquire a non-Abelian geometric phase.
We theoretically introduce synthetic non-Abelian gauge fields for topological quantum walks. The photonic mesh lattice configuration is generalized with polarization multiplexing to achieve a four-dimensional Hilbert space, based on which we provide photonic building blocks for realizing various quantum walks in non-Abelian gauge fields. It is found that SU(2) gauge fields can lead to Peierls substitution in both momenta and quasienergy. In one and two dimensions, we describe detailed photonic setups to realize topological quantum walk protocols whose Floquet winding numbers and Rudner-Lindner-Berg-Levin invariants can be effectively controlled by the gauge fields. Finally, we show how non-Abelian gauge fields facilitate convenient simulation of entanglement in conjunction with polarization-dependent and spatial-mode-dependent coin operations. Our results shed light on the study of synthetic non-Abelian gauge fields in photonic Floquet systems.
Floquet systems like quantum walks can exhibit anomalous topological boundary modes that have no stationary counterpart. Although the topological properties of synthetic gauge fields have been explored in quantum walks, these studies have so far focused on Abelian gauge fields. We theoretically introduce synthetic non-Abelian gauge fields for topological quantum walks and study their topological consequences. The photonic mesh lattice configuration is generalized with polarization multiplexing to achieve a four-dimensional Hilbert space, based on which we provide photonic building blocks for realizing various quantum walks in non-Abelian gauge fields. It is found that SU(2) gauge fields can lead to Peierls substitution in both momenta and quasienergy. In one and two dimensions, we describe detailed photonic setups to realize topological quantum walk protocols whose Floquet winding numbers and Rudner–Lindner–Berg–Levin invariants can be effectively controlled by the gauge fields. Finally, we show how non-Abelian gauge fields facilitate convenient simulation of entanglement in conjunction with polarization-dependent and spatial-mode-dependent coin operations. Our results shed light on the study of synthetic non-Abelian gauge fields in photonic Floquet systems and hold implications for time-multiplexed optical networks.
Geometrical and topological phases play a fundamental role in quantum theory. Geometric phases have been proposed as a tool for implementing unitary gates for quantum computation. A fractional topological phase has been recently discovered for bipartite systems. The dimension of the Hilbert space determines the topological phase of entangled qudits under local unitary operations. Here we investigate fractional topological phases acquired by photonic entangled qudits. Photon pairs prepared as spatial qudits are operated inside a Sagnac interferometer and the two-photon interference pattern reveals the topological phase as fringes shifts when local operations are performed. Dimensions d=2d=2d=2, 3, and 4 were tested, showing the expected theoretical values.
When a multi-qubit state evolves under local unitaries it may obtain a geometric phase, a feature dependent on the geometry of the state projective Hilbert space. A correction term to this geometric phase, in addition to the local subsystem phases, may appear from correlations between the subsystems. We find that this correction term can be characterized completely either by the entanglement or by the classical correlations for several classes of entangled state. States belonging to the former set are W states and their mixtures, while members of the latter set are cluster states, GHZ states and two classes of bound entangled state. We probe the structures of these states more finely using local invariants and suggest that the cause of the entanglement correction is a recently introduced gauge field-like SL(2,ℂ)-invariant called twist.
When a multi-qubit state evolves under local unitaries it may obtain a geometric phase, a feature dependent on the geometry of the state's projective Hilbert space. A correction term to this geometric phase in addition to the local subsystem phases may appear from correlations between the subsystems. We find this correction term can be characterized completely either by the entanglement or completely by the classical correlations for several classes of entangled state. States belonging to the former set are W states and their mixtures, while members of the latter set are cluster states, GHZ states and two classes of bound entangled state. We probe the structures of these states more finely using local invariants and suggest the cause of the entanglement correction is a gauge field like SL(2,C)SL(2,\mathbb{C})SL(2,C) invariant recently introduced named twist.