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arXiv:quant-ph/0702080 · Physical Review A

Composite geometric phase for multipartite entangled states

M. S. Williamson, Vlatko Vedral

Published Sep 24, 2007• 17 citations
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Abstract

When an entangled state evolves under local unitaries, the entanglement in the state remains fixed. Here we show that the dynamical phase acquired by an entangled state in such a scenario can always be understood as the sum of the dynamical phases of its subsystems. In contrast, the equivalent statement for the geometric phase is not generally true unless the state is separable. For an entangled state an additional term is present, the mutual geometric phase, that measures the change the additional correlations present in the entangled state make to the geometry of the state space. For $N$ qubit states we find that this change can be explained solely by classical correlations for states with a Schmidt decomposition and solely by quantum correlations for $W$ states.

DOI
10.1103/physreva.76.032115

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When an entangled state evolves under local unitaries, the entanglement in the state remains fixed. Here we show that the dynamical phase acquired by an entangled state in such a scenario can always be understood as the sum of the dynamical phases of its subsystems. In contrast, the equivalent statement for the geometric phase is not generally true unless the state is separable. For an entangled state an additional term is present, the mutual geometric phase, that measures the change the additional correlations present in the entangled state make to the geometry of the state space. For NNN qubit states we find that this change can be explained solely by classical correlations for states with a Schmidt decomposition and solely by quantum correlations for WWW states.
When a multi-qubit state evolves under local unitaries it may obtain a geometric phase, a feature dependent on the geometry of the state projective Hilbert space. A correction term to this geometric phase, in addition to the local subsystem phases, may appear from correlations between the subsystems. We find that this correction term can be characterized completely either by the entanglement or by the classical correlations for several classes of entangled state. States belonging to the former set are W states and their mixtures, while members of the latter set are cluster states, GHZ states and two classes of bound entangled state. We probe the structures of these states more finely using local invariants and suggest that the cause of the entanglement correction is a recently introduced gauge field-like SL(2,ℂ)-invariant called twist.
When a multi-qubit state evolves under local unitaries it may obtain a geometric phase, a feature dependent on the geometry of the state's projective Hilbert space. A correction term to this geometric phase in addition to the local subsystem phases may appear from correlations between the subsystems. We find this correction term can be characterized completely either by the entanglement or completely by the classical correlations for several classes of entangled state. States belonging to the former set are W states and their mixtures, while members of the latter set are cluster states, GHZ states and two classes of bound entangled state. We probe the structures of these states more finely using local invariants and suggest the cause of the entanglement correction is a gauge field like SL(2,C)SL(2,\mathbb{C})SL(2,C) invariant recently introduced named twist.
This paper focuses on the geometric phase of entangled states of bipartite systems under bilocal unitary evolution. We investigate the relation between the geometric phase of the system and those of the subsystems. It is shown that (1) the geometric phase of cyclic entangled states with nondegenerate eigenvalues can always be decomposed into a sum of weighted nonmodular pure state phases pertaining to the separable components of the Schmidt decomposition, although the same cannot be said in the noncyclic case, and (2) the geometric phase of the mixed state of one subsystem is generally different from that of the entangled state even if the other subsystem is kept fixed, but the two phases are the same when the evolution operator satisfies conditions where each component in the Schmidt decomposition is parallel transported.
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​.
It is known that an interacting bipartite system evolves as an entangled state in general, even if it is initially in a separable state. Due to the entanglement of the state, the geometric phase of the system is not equal to the sum of the geometric phases of its two subsystems. However, there may exist a set of states in which the nonlocal interaction does not affect the separability of the states, and the geometric phase of the bipartite system is then always equal to the sum of the geometric phases of its subsystems. In this article, we illustrate this point by investigating a well-known physical model. We give a necessary and sufficient condition in which a separable state remains separable so that the geometric phase of the system is always equal to the sum of the geometric phases of its subsystems.
We introduce a connection between entanglement induced by interaction and geometric phases acquired by a composite quantum spin system. We begin by analyzing the evaluation of cyclic (Aharonov-Anandan) and noncyclic (Mukunda-Simon) geometric phases for general spin chains evolving in the presence of time-independent magnetic fields. Then, by considering Heisenberg chains, we show that the interaction geometric phase, namely, the total geometric phase with subtraction of free spin contributions, is directly related to the global (Meyer-Wallach) entanglement exhibited by an initially separable state during its evolution in Hilbert space. This is analytically shown for N=2N=2N=2 spins and numerically illustrated for larger chains. This relationship promotes the interaction geometric phase to an indicator of global entanglement in the system, which may constitute a useful tool for quantum tasks based on entanglement as a resource to their performance.