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arXiv:quant-ph/0608185 · Physical Review A

Operational approach to the Uhlmann holonomy

Johan Åberg, David Kult, Erik Sjöqvist, Daniel K. L. Oi

Published Mar 8, 2007• 21 citations
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Abstract

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.

DOI
10.1103/physreva.75.032106

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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.
A unifying framework for identifying distance and holonomy for decompositions of density operators is introduced. Parallelity between quantum ensembles is defined by minimizing this distance over allowed decompositions. The minimum is a property of a pair of states and coincides with the Bures distance. The parallelity condition imposes a connection (rule for parallel transport) that results in the Uhlmann holonomy for sequences of density operators. A distance and holonomy for spectral decompositions of density operators is identified as a sub-group restriction of the full decomposition freedom. These spectral concepts are gauge invariant (decomposition independent) properties of mixed quantum ensembles, as long as the corresponding density operators are non-degenerate. A gauge invariant spectral geometric phase for discrete sequences of mixed quantum states is obtained as the phase of the trace of the spectral holonomy. This geometric phase differs from the interferometric mixed state geometric phase in the continuous limit.
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.
We accept the implicit challenge of A. Uhlmann in his 1994 paper, Parallel Lifts and Holonomy along Density Operators: Computable Examples Using O(3)-Orbits, by, in fact, computing the holonomy invariants for rotations of certain n-level Gibbsian density matrices (n = 2,...,11). From these, we derive, by the tracing operation, the associated geometric phases and visibilities, which we analyze and display. We then proceed analogously, implementing the alternative methodology presented by E. Sjoqvist et al in their letter, Geometric Phases for Mixed States in Interferometry (Phys. Rev. Lett. 85, 2845 [2000]). For the Uhlmann case, we are able to also compute several higher-order invariants. We compare the various geometric phases and visibilities for different values of n, and also directly compare the two forms of analysis. By setting one parameter (a) in the Uhlmann analysis to zero, we find that the so-reduced form of the first-order holonomy invariant is simply equal to (-1)^{n+1} times the holonomy invariant in the Sjoqvist et al method. Additional phenomena of interest are reported too.