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Geometric Phases of the Uhlmann Type for n-Level Gibbsian Density Matrices

Paul B. Slater

Published Dec 21, 2001• 0 citations
OpenAlex

Abstract

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.

Related papers

Geometric Phases of the Uhlmann and Sjoqvist et al Types for O(3)-Orbits of n-Level Gibbsian Density Matrices

Paul B. Slater

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 Sjöqvist et al method. Additional phenomena of interest are reported too.

arXiv:math-ph/0112054•Dec 21, 2001•2 citations

Comment on "Geometric Phases for Mixed States in Interferometry"

Paul B. Slater

We find for the unitary evolution of spin-1/2 systems that the "purely mathematical mixed state holonomy of Uhlmann limitedly agrees, in the case of evolution over geodesic spherical triangles, with the holonomy "in the experimental context of quantum interferometry" recently proposed by Sjoqvist, Pati, Ekert, Anandan, Ericsson, Oi and Vedral (Phys. Rev. Lett. 85 [2000], 2845-2848).

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For full-rank density matrices the state space is contractible, the Uhlmann bundle is topologically trivial, and the holonomy admits no quantized invariants; although the Uhlmann phase is genuinely geometric, this topological poverty has kept it from serving as a robust physical diagnostic of mixed-state matter. Mixed states of fixed rank below the Hilbert-space dimension, however, are ubiquitous: reduced states of constrained subsystems, states confined to invariant sectors, and states supported on decoherence-free subspaces can have a support smaller than the Hilbert space, a support that can vary with parameters. Their geometry is far richer: the support carries genuine curvature, non-Abelian holonomy, and Chern topology. We formulate Uhlmann's theory directly on the manifold of rank-$k$ density matrices: minimal purifications make the fixed-rank stratum the base of a principal $U(k)$-bundle whose Uhlmann connection is uniquely determined by a Sylvester equation, and in an eigenframe this connection takes a closed form reducing to the Berry, Wilczek--Zee, and faithful Uhlmann connections at $k=1$, at equal weights, and at $k=N$. The fixed-rank bundle inherits the topology of the Grassmannian, and non-factorizable higher Chern topology requires failure of the global eigenline splitting, which in the present spectral setting requires degeneracy, minimally realized by a rank-2 Yang monopole whose quantized second Chern number links the geometry to the four-dimensional quantum Hall response. Solvable models, from a genuinely non-Abelian holonomy to a dissipatively driven orbit whose Uhlmann holonomy is steered by the reservoir through an elliptic-integral spectral dressing, illustrate the content: the supporting subspace carries the topology, the spectral weights shape the transport, and Uhlmann holonomy provides a direct geometric probe of the resulting mixed-state structures.

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Quantum Holonomies Concepts and Applications to Quantum Computing and Interferometry

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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.

•Jan 1, 2007•0 citations
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.
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 Sjöqvist et al method. Additional phenomena of interest are reported too.
For full-rank density matrices the state space is contractible, the Uhlmann bundle is topologically trivial, and the holonomy admits no quantized invariants; although the Uhlmann phase is genuinely geometric, this topological poverty has kept it from serving as a robust physical diagnostic of mixed-state matter. Mixed states of fixed rank below the Hilbert-space dimension, however, are ubiquitous: reduced states of constrained subsystems, states confined to invariant sectors, and states supported on decoherence-free subspaces can have a support smaller than the Hilbert space, a support that can vary with parameters. Their geometry is far richer: the support carries genuine curvature, non-Abelian holonomy, and Chern topology. We formulate Uhlmann's theory directly on the manifold of rank-kkk density matrices: minimal purifications make the fixed-rank stratum the base of a principal U(k)U(k)U(k)-bundle whose Uhlmann connection is uniquely determined by a Sylvester equation, and in an eigenframe this connection takes a closed form reducing to the Berry, Wilczek--Zee, and faithful Uhlmann connections at k=1k=1k=1, at equal weights, and at k=Nk=Nk=N. The fixed-rank bundle inherits the topology of the Grassmannian, and non-factorizable higher Chern topology requires failure of the global eigenline splitting, which in the present spectral setting requires degeneracy, minimally realized by a rank-2 Yang monopole whose quantized second Chern number links the geometry to the four-dimensional quantum Hall response. Solvable models, from a genuinely non-Abelian holonomy to a dissipatively driven orbit whose Uhlmann holonomy is steered by the reservoir through an elliptic-integral spectral dressing, illustrate the content: the supporting subspace carries the topology, the spectral weights shape the transport, and Uhlmann holonomy provides a direct geometric probe of the resulting mixed-state structures.
In this thesis we provide a uniform treatment of two non-adiabatic geometric phases for dynamical systems of mixed quantum states, namely those of Uhlmann and of Sjöqvist et al. We develop a holonomy theory for the latter which we also relate to the already existing theory for the former. This makes it clear what the similarities and differences between the two geometric phases are. We discuss and motivate constraints on the two phases. Furthermore, we discuss some topological properties of the holonomy of `real' quantum systems, and we introduce higher-order geometric phases for not necessarily cyclic dynamical systems of mixed states. In a final chapter, we apply the theory developed for the geometric phase of Sjöqvist et al. to geometric uncertainty relations, including some new "quantum speed limits".
The Uhlmann phase extends geometric phases to mixed quantum states via a parallel-transport condition on purification amplitudes, yet its direct implementation under standard Hamiltonian dynamics is obstructed by the non-Hermitian nature of the purification. We establish that for any smooth one-dimensional closed loop of full-rank qubit density matrices, there exists a four-level Hermitian parent Hamiltonian whose doubly degenerate ground-state subspace carries a Wilczek--Zee connection exactly equal to the Uhlmann connection. Consequently, the Uhlmann holonomy is faithfully reproduced by adiabatic evolution in the enlarged system. We further prove that this auxiliary-field construction is obstructed in generic two-dimensional parameter spaces by a Frobenius integrability condition, which we derive explicitly. The one-dimensional Uhlmann phase is thus placed on the same footing as the non-Abelian Berry phase, offering a purely Hermitian, Hamiltonian-based route to simulating mixed-state geometric phases. Numerical integration of the adiabatic dynamics confirms the exact correspondence and validates the convergence to the Uhlmann holonomy in the large-gap 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.