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AIP conference proceedings

The analysis of quantum qutrit entanglements in a qutrit based hyper-sphere in terms of gluing and combining products

Volkan Duran, Azmi Gençten

Published Jan 1, 2016• 1 citation
Publisher

Abstract

In this research the aim is to analyze quantum qutrit entanglements in a new perspective in terms of the reflection of n-dimensional sphere which can be depicted as the set of points equidistant from a fixed central point in three dimensional Euclidian Space which has also real and imaginary dimensions, that can also be depicted similarly as a two unit spheres having same centre in a dome-shaped projection. In order to analyze quantum qutrit entanglements: i- a new type of n dimensional hyper-sphere which is the extend version of Bloch Sphere to hyper-space, is defined ii- new operators and products such as rotation operator, combining and gluing products in this space are defined, iii-the entangled states are analyzed in terms of those products in order to reach a general formula to depict qutrit entanglements and some new patterns between spheres for the analysis of entanglement for different routes in a more simple way in a four dimensional time independent hypersphere.

DOI
10.1063/1.4944179

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In this research the aim is to analyze quantum qutrit entanglements in a new perspective in terms of the reflection of n-dimensional sphere which can be depicted as the set of points equidistant from a fixed central point in three dimensional Euclidian Space which has also real and imaginary dimensions, that can also be depicted similarly as a two unit spheres having same centre in a dome-shaped projection. In order to analyze quantum qutrit entanglements: i- a new type of n dimensional hyper-sphere which is the extend version of Bloch Sphere to hyper-space, is defined ii- new operators and products such as rotation operator, combining and gluing products in this space are defined, iii-the entangled states are analyzed in terms of those products in order to reach a general formula to depict qutrit entanglements and some new patterns between spheres for the analysis of entanglement for different routes in a more simple way in a four dimensional time independent hypersphere.
In the Bloch sphere based representation of qudits with dimensions greater than two, the Heisenberg-Weyl operator basis is not preferred because of presence of complex Bloch vector components. We try to address this issue and parametrize a qutrit using the Heisenberg-Weyl operators by identifying eight real parameters and separate them as four weight and four angular parameters each. The four weight parameters correspond to the weights in front of the four mutually unbiased bases sets formed by the eigenbases of Heisenberg-Weyl observables and they form a four-dimensional unit radius Bloch hypersphere. Inside the four-dimensional hypersphere all points do not correspond to a physical qutrit state but still it has several other features which indicate that it is a natural extension of the qubit Bloch sphere. We study the purity, rank of three level systems, orthogonality and mutual unbiasedness conditions and the distance between two qutrit states inside the hypersphere. We also analyze the two and three-dimensional sections centered at the origin which gives a close structure for physical qutrit states inside the hypersphere. Significantly, we have applied our representation to find mutually unbiased bases(MUBs) and to characterize the unital maps in three dimensions. It should also be possible to extend this idea in higher dimensions.
We present a novel method to study the Bloch space of the qutrit system by examining the Bloch trajectories in it. Since such system is inherently a three-level quantum system, therefore we use the SU(3) group as the basis group to obtain the Bloch vectors of different configurations of it. The norm of the Bloch space is evaluated from the geometric consideration and also from the dynamics of the Bloch vectors and both results are found to be identical. The analysis of the dynamical evolution of the Bloch vectors reveals an additional feature that, under resonant conditions, the Bloch sphere S7\mathbb{S}^{7}S7 splits into two parts, a four-sphere S4\mathbb{S}^{4}S4 and a two-sphere S2\mathbb{S}^{2}S2. The Bloch trajectories of the two sectors across different configurations exhibit a range of simple to complex curves, highlighting the non-trivial structure of the Bloch space of the qutrit system.
Geometric intuition is a crucial tool to obtain deeper insight into many concepts of physics. A paradigmatic example of its power is the Bloch ball, the geometrical representation for the state space of the simplest possible quantum system, a two-level system (or qubit). However, already for a three-level system (qutrit) the state space has eight dimensions, so that its complexity exceeds the grasp of our three-dimensional space of experience. This is unfortunate, given that the geometric object describing the state space of a qutrit has a much richer structure and is in many ways more representative for a general quantum system than a qubit. In this work we demonstrate that, based on the Bloch representation of quantum states, it is possible to construct a three dimensional model for the qutrit state space that captures most of the essential geometric features of the latter. Besides being of indisputable theoretical value, this opens the door to a new type of representation, thus extending our geometric intuition beyond the simplest quantum systems.
Geometric intuition is a crucial tool to obtain deeper insight into many concepts of physics. A paradigmatic example of its power is the Bloch ball, the geometrical representation for the state space of the simplest possible quantum system, a two-level system (or qubit). However, already for a three-level system (qutrit) the state space has eight dimensions, so that its complexity exceeds the grasp of our three-dimensional space of experience. This is unfortunate, given that the geometric object describing the state space of a qutrit has a much richer structure and is in many ways more representative for a general quantum system than a qubit. In this work we demonstrate that, based on the Bloch representation of quantum states, it is possible to construct a three dimensional model for the qutrit state space that captures most of the essential geometric features of the latter. Besides being of indisputable theoretical value, this opens the door to a new type of representation, thus extending our geometric intuition beyond the simplest quantum systems.
We represent a two-qubit density matrix in the basis of Pauli matrix tensor products, with the coefficients constituting a Bloch matrix, analogous to the single qubit Bloch vector. We find the quantum state positivity requirements on the Bloch matrix components, leading to three important inequalities, allowing us to parametrize and visualize the two-qubit state space. Applying the singular value decomposition naturally separates the degrees of freedom to local and nonlocal, and simplifies the positivity inequalities. It also allows us to geometrically represent a state as two entangled Bloch spheres with superimposed correlation axes. It is shown that unitary transformations, local or nonlocal, have simple interpretations as axis rotations or mixing of certain degrees of freedom. The nonlocal unitary invariants of the state are then derived in terms of local unitary invariants. The positive partial transpose criterion for entanglement is generalized, and interpreted as a reflection, or a change of a single sign. The formalism is used to characterize maximally entangled states, and generalize two qubit isotropic and Werner states.