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Physical Review A

Complex projection of unitary dynamics of quaternionic pure states

M. Asorey, G. Scolarici, Luigi Solombrino

Published Jul 17, 2007• 8 citations
Publisher

Abstract

Quaternionic quantum mechanics has been revealed to be a very useful framework to describe quantum phenomena. In the case of two qubit compound systems we show that the complex projection of quaternionic pure states and quaternionic unitary maps permits the description of interesting phenomena such as decoherence and optimal entanglement generation. The approach, however, presents severe limitations for the case of multipartite or higher dimensional bipartite quantum systems as we point out.

DOI
10.1103/physreva.76.012111

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Quaternionic quantum mechanics has been revealed to be a very useful framework to describe quantum phenomena. In the case of two qubit compound systems we show that the complex projection of quaternionic pure states and quaternionic unitary maps permits the description of interesting phenomena such as decoherence and optimal entanglement generation. The approach, however, presents severe limitations for the case of multipartite or higher dimensional bipartite quantum systems as we point out.
The quaterbit is the smallest unit of information of Quaternion quantum mechanics. Their study brings to light fundamental problems of Quantum Mechanics that also have implications in Information Theory and Quantum Computation. In this work, it was investigated whether a pair of quaterbits may be entangled in Quantum Mechanics of complex numbers. To verify this hypothesis, two arbitrary quaterbits were used as singlets and we did the tensorial product. From this tensorial product, we establish the necessary conditions for the state to be only in Complex Quantum Mechanics. Within this procedure, two particular cases were analyzed: pure state and a simple case of mixed state. For pure states, we nd that the resulting separable bipartite qubit is not entangled, and therefore we conclude that the tensor product of two pure quaterbits does not result in a bipartite pure qubit entangled. Finally, we consider a mixed state of disentangled quaterbits (with some restrictions) and establish the conditions so that it lies only in Complex Quantum Mechanics, and therefore we conclude by using the Wootters concurrence that the state is disentangled.
The two-qubit pure state is explicitly parameterized by three unit 2-spheres and a phase factor. For separable states, two of the three unit spheres are the Bloch spheres of each qubit. The third sphere parameterizes the degree and phase of concurrence, an entanglement measure. This sphere may be considered a variable complex imaginary unit where the stereographic projection maps the qubit-A Bloch sphere to a complex plane with this variable imaginary unit. This Bloch sphere model gives a consistent description of the two-qubit pure states for both separable and entangled states. We argue that the third sphere (entanglement sphere) parameterizes the nonlocal properties, entanglement and a nonlocal relative phase, while the local relative phases are parameterized by the azimuth angles of the two quasi-Bloch spheres. On these three unit spheres and a phase factor, the two-qubit pure states and unitary gates can be geometrically represented. Accomplished by means of Hopf fibration, the complex amplitudes of a two-qubit pure state and the Bloch sphere parameters are related by a single quaternionic relation.
Complex Hopf fibration and quaternionic Hopf fibration provide distinct ways of describing the state space for two-qubit quantum states. In this research, we have studied the geometry of quantum states for a two-level quantum system, along with the correspondence between complex Hopf bundle and quaternionic Hopf bundle. In the first part of our study, we investigate the behaviour of local coordinates for both Hopf bundles under different degree of entanglement such as entangled states and non-entangled states. Fubini-Study metric for complex projective space is also obtained. Its form suggests that for the intermediate entangled states, complex projective space CP3 can be described as a set of flat three-tori parametrized by a three-sphere. The local inhomogeneous coordinate of CP3 (base space of complex Hopf fibration) is found to carry the description of both subsystems A and B, whereas in the case of maximally entangled state, basis elements of local coordinates CP3 is not linearly independent of each other. Next, we construct a base space map between CP3 and S4, which is denoted as η map. After the mapping, we obtained phases in the base space manifold, different sections and coordinate charts are related by transition functions. We found that there is an inherent symmetry of coordinate transformation corresponds to different sections of CP3, which is expressed in terms of transition functions having the U(1) group structure. Also under η map, phases and transition function in S4 is doubled over that of CP3, indicating subtle symmetric changes after the mapping. The base space coordinates of quaternionic Hopf bundle are consist of two parts, whereby the first part is invariant to the coordinate transformation in CP3 but sensitive to the coordinate transformation in S4.
It has been recently pointed out by Caves, Fuchs, and Rungta [1] that real quantum mechanics (that is, quantum mechanics defined over real vector spaces [2–5]) provides an interesting foil theory whose study may shed some light on just which particular aspects of quantum entanglement are unique to standard quantum theory and which are more generic over other physical theories endowed with this phenomenon. Following this work, some entanglement properties of two-rebit systems are discussed and a comparison with the basic properties of two-qubit systems, i.e., the systems described by standard complex quantum mechanics, is made. The use of quaternionic quantum mechanics as applied to the phenomenon of entanglement is also discussed.