Quantum Holonomy

Vera Neef, Matthias Heinrich, Tom A. W. Wolterink +1 more

Holonomies are of great interest to quantum computation and simulation. The geometrical nature of holonomies offers increased stability to quantum gates. Furthermore, symmetries of particle physics are naturally reflected in holonomies, making them ideally suited for quantum simulation of quantum chromodynamics and grand unified theories. Yet, practically designing quantum holonomies with the required properties and scale is challenging. Here, we construct a new class of holonomies by increasing the particle number. We show that multiparticle holonomies can even exist in systems devoid of any single-particle holonomies. We present a comprehensive framework for multiparticle quantum holonomies and experimentally realize various two-particle holonomies in integrated photonics. Our results enable the number of particles to be harnessed as a design parameter, offering increased freedom in constructing holonomic systems.

Deep Bhattacharjee

Paolo Zanardi A, Mario Rasetti

David Kult, Johan Åberg, Erik Sjöqvist

A quantum holonomy reflects the curvature of some underlying structure of quantum-mechanical systems, such as that associated with quantum states. Here, we extend the notion of holonomy to families of quantum channels, i.e., trace-preserving completely positive maps. By the use of the Jamio\l{}kowski isomorphism, we show that the proposed channel holonomy is related to the Uhlmann holonomy. The general theory is illustrated for specific examples. We put forward a physical realization of the channel holonomy in terms of interferometry. This enables us to identify a gauge-invariant physical object that directly relates to the channel holonomy. Parallel transport condition and concomitant gauge structure are delineated in the case of smoothly parametrized families of channels. Finally, we point out that interferometer tests that have been carried out in the past to confirm the $4\ensuremath{\pi}$ rotation symmetry of the neutron spin can be viewed as early experimental realizations of the channel holonomy.

David Kult

Quantum holonomies are investigated in different contexts. A geometric phase is proposed for decomposition dependent evolution, where each component of a given decomposition of a mixed state evolves independently. It is shown that this geometric phase only depends on the path traversed in the space of decompositions. A holonomy is associated to general paths of subspaces of a Hilbert space, both discrete and continuous. This opens up the possibility of constructing quantum holonomic gates in the open path setting. In the discrete case it is shown that it is possible to associate two distinct holonomies to a given path. Interferometric setups for measuring both holonomies are provided. It is further shown that there are cases when the holonomy is only partially defined. This has no counterpart in the Abelian setting. An operational interpretation of amplitudes of density operators is provided. This allows for a direct interferometric realization of Uhlmann's parallelity condition, and the possibility of measuring the Uhlmann holonomy for sequences of density operators. Off-diagonal geometric phases are generalized to the non-Abelian case. These off-diagonal holonomies are undefined for cyclic evolution, but must contain members of non-zero rank if all standard holonomies are undefined. Experimental setups for measuring the off-diagonal holonomies are proposed. The concept of nodal free geometric phases is introduced. These are constructed from gauge invariant quantities, but do not share the nodal point structure of geometric phases and off-diagonal geometric phases. An interferometric setup for measuring nodal free geometric phases is provided, and it is shown that these phases could be useful in geometric quantum computation. A holonomy associated to a sequence of quantum maps is introduced. It is shown that this holonomy is related to the Uhlmann holonomy. Explicit examples are provided to illustrate the general idea.

Miguel Orszag