17 research outputs found
Non-Abelian generalization of off-diagonal geometric phases
If a quantum system evolves in a noncyclic fashion the corresponding
geometric phase or holonomy may not be fully defined. Off-diagonal geometric
phases have been developed to deal with such cases. Here, we generalize these
phases to the non-Abelian case, by introducing off-diagonal holonomies that
involve evolution of more than one subspace of the underlying Hilbert space.
Physical realizations of the off-diagonal holonomies in adiabatic evolution and
interferometry are put forward.Comment: Additional material, journal reference adde
Nodal free geometric phases: Concept and application to geometric quantum computation
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.Comment: Small typo corrected, journal reference adde
Noncyclic geometric changes of quantum states
Non-Abelian quantum holonomies, i.e., unitary state changes solely induced by
geometric properties of a quantum system, have been much under focus in the
physics community as generalizations of the Abelian Berry phase. Apart from
being a general phenomenon displayed in various subfields of quantum physics,
the use of holonomies has lately been suggested as a robust technique to obtain
quantum gates; the building blocks of quantum computers. Non-Abelian holonomies
are usually associated with cyclic changes of quantum systems, but here we
consider a generalization to noncyclic evolutions. We argue that this open-path
holonomy can be used to construct quantum gates. We also show that a structure
of partially defined holonomies emerges from the open-path holonomy. This
structure has no counterpart in the Abelian setting. We illustrate the general
ideas using an example that may be accessible to tests in various physical
systems.Comment: Extended version, new title, journal reference adde
Holonomy for Quantum Channels
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 rotation symmetry of the
neutron spin, can be viewed as early experimental realizations of the channel
holonomy.Comment: Minor changes, journal reference adde
Operational approach to the Uhlmann holonomy
We suggest a physical interpretation of the Uhlmann amplitude of a density
operator. Given this interpretation we propose an operational approach to
obtain the Uhlmann condition for parallelity. This allows us to realize
parallel transport along a sequence of density operators by an iterative
preparation procedure. At the final step the resulting Uhlmann holonomy can be
determined via interferometric measurements.Comment: Added material, references, and journal reference
Manifestations of quantum holonomy in interferometry
Abelian and non-Abelian geometric phases, known as quantum holonomies, have
attracted considerable attention in the past. Here, we show that it is possible
to associate nonequivalent holonomies to discrete sequences of subspaces in a
Hilbert space. We consider two such holonomies that arise naturally in
interferometer settings. For sequences approximating smooth paths in the base
(Grassmann) manifold, these holonomies both approach the standard holonomy. In
the one-dimensional case the two types of holonomies are Abelian and coincide
with Pancharatnam's geometric phase factor. The theory is illustrated with a
model example of projective measurements involving angular momentum coherent
states.Comment: Some changes, journal reference adde
The quantum adiabatic search with decoherence in the instantaneous energy eigenbasis
In Phys. Rev. A {\bf 71}, 060312(R) (2005) the robustness of the local
adiabatic quantum search to decoherence in the instantaneous eigenbasis of the
search Hamiltonian was examined. We expand this analysis to include the case of
the global adiabatic quantum search. As in the case of the local search the
asymptotic time complexity for the global search is the same as for the ideal
closed case, as long as the Hamiltonian dynamics is present. In the case of
pure decoherence, where the environment monitors the search Hamiltonian, we
find that the time complexity of the global quantum adiabatic search scales
like , where is the list length. We moreover extend the analysis
to include success probabilities and prove bounds on the run time with
the same scaling as in the conditions for the limit. We supplement the
analytical results by numerical simulations of the global and local search.Comment: Material added, journal reference adde
Quantum Holonomies : Concepts and Applications to Quantum Computing and Interferometry
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