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arXiv:quant-ph/0506247 · Europhysics Letters (EPL)

Effects of entanglement on off-diagonal geometric phases

H. T. Cui, L. C. Wang, X. X. Yi

Published May 4, 2006• 2 citations
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Abstract

The effect of entanglement on off-diagonal geometric phases is investigated in the paper. Two spin-(1/2) particles in magnetic fields along the y direction are taken as an example. Three parameters (the purity of state r , the mixing angle θ and the relative phase β) are chosen to characterize the initial states. The nodal points at which the usual geometric phases disappear are calculated and illustrated as a function of the three parameters.

DOI
10.1209/epl/i2005-10587-x

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The effect of entanglement on off-diagonal geometric phases is investigated in the paper. Two spin-(1/2) particles in magnetic fields along the y direction are taken as an example. Three parameters (the purity of state r , the mixing angle θ and the relative phase β) are chosen to characterize the initial states. The nodal points at which the usual geometric phases disappear are calculated and illustrated as a function of the three parameters.
The entanglement dependence of the noncyclic geometric phase is analyzed. A pair of noninteracting spin-12\frac{1}{2}21​ particles prepared in an arbitrarily entangled state and precessing in an external time-independent uniform magnetic field is considered. It is shown that the geometric phase reduces to a sum of one-particle geometric phases for product states and takes on the two values corresponding to the phase factors $\ifmmode\pm\else\textpm\fi{}1$ for maximally entangled states. If only one of the particles is affected by the magnetic field it is demonstrated that the influence of entanglement on the geometric phase may be interpreted as an effective reduction of the degree of polarization of the affected particle. The generalization to more than two precessing spin-12\frac{1}{2}21​ particles, in the particular case where Schmidt decompositions exists, is briefly outlined. The geometric phase for a pair of spin-12\frac{1}{2}21​ particles with a spin-spin interaction is calculated. In this model we show that the noncyclic geometric phase for a certain class of states may be interpreted solely in terms of the solid angle enclosed by the geodesically closed curve on a two-sphere parametrized by the evolving Schmidt coefficients. This suggests a geometric interpretation of Schmidt decompositions for spin-12\frac{1}{2}21​ pairs analogous to that of the Poincar\'e sphere for a single spin 12.\frac{1}{2}.21​.
The geometric phase for states of two spin-1/2 particles in rotating magnetic field is calculated, in particular, the noncyclic and cyclic non-adiabatic phases for the general case are explicitly derived and discussed. We find that the cyclic geometric phase for the entangled state can always be written as a sum of the phases of the two particles respectively; the same cannot be said for the noncyclic phase. We also investigate the geometric phase of mixed state of one particle in a biparticle system, and we find that the geometric phase for one subsystem of an entangled system is always affected by another subsystem of the entangled system.
The entanglement dependence of Berry's geometric phase is studied in the case of two spin-1/2 particles with a spin-spin interaction.Berry's phases for two spin-1/2 entanglement particles in the presence of spin-spin interaction and noninteraction are calculated,respectively.The expression of calculating geometric phase,when entanglement vanishes,is obtained.Moreover,the geometric phase is discussed when only one of the two particles is affected by the external magnetic field.It is shown that its geometric phase is not equal to that of single particle which is acted by the external magnetic field,and the result is related toαthat affect the strength of entanglement. A general entanglement-dependence geometric phase is formulated.
We study the geometric Uhlmann phase of entangled mixed states in a composite system made of two coupled spin-12\frac{1}{2}21​ particles with a magnetic field acting on one of them. Within a uniform noisy quantum channel setup, an exact analytical expression for this geometric phase in each subsystem is derived. We find an explicit connection to the concurrence of the density matrix of such a channel, which allows us to characterize the features of the Uhlmann phase in terms of the degree of entanglement in the system. In the space of field direction and coupling parameter, it exhibits a phase singularity revealing a topological transition between orders characterized by different winding numbers. The transition occurs for fields lying in the equator of the sphere of directions and at critical values of the coupling which can be controlled by tuning the depolarization strength. Notably, under these conditions the concurrence of the composite system is bounded to the range [0,1/2][0,1/2][0,1/2]. We also compare the calculated Uhlmann phase to an interferometric phase, which has been formulated as an alternative for density matrices. The latter does not present a phase vortex, although they coincide in the weak entanglement regime for vanishing depolarization (pure states). Otherwise, they behave clearly different in the strong-entanglement regime.
The concept of off-diagonal geometric phases for mixed quantal states in unitary evolution is developed. We show that these phases arise from three basic ideas: (1) fulfillment of quantum parallel transport of a complete basis, (2) a concept of mixed-state orthogonality adapted to unitary evolution, and (3) a normalization condition. We provide a method for computing the off-diagonal mixed-state phases to any order for unitarities that divide the parallel transported basis of Hilbert space into two parts: one part where each basis vector undergoes cyclic evolution and one part where all basis vectors are permuted among each other. We also demonstrate a purification based experimental procedure for the two lowest-order mixed-state phases and consider a physical scenario for a full characterization of the qubit mixed-state geometric phases in terms of polarization-entangled photon pairs. An alternative second order off-diagonal mixed-state geometric phase, which can be tested in single-particle experiments, is proposed.