Abstract
We argue that the usual Bloch sphere is insufficient in various aspects for the representation of qubits in quantum information theory. For example, spin flip operations with the quaternions I J K = e 2 π i 2 = − 1 and J I K = + 1 cannot be distinguished on the Bloch sphere. We show that a simple knot theoretic extension of the Bloch sphere representation is sufficient to track all unitary operations for single qubits. Next, we extend the Bloch sphere representation to entangled states using knot theory. As applications, we first discuss contextuality in quantum physics—in particular the Kochen-Specker theorem. Finally, we discuss some arguments against many-worlds theories within our knot theoretic model of entanglement. The key ingredients of our approach are symmetries and geometric properties of the unitary group.
Highlights
With the rapidly increasing importance of quantum communication and quantum computation, and in particular the increasing ability to manipulate systems of entangled qubits, the representation of qubits and operations with qubits is an important issue, in particular for writing codes for quantum computers [1]
We introduce the Heegard-splitting of the Lie algebra SU (2), and show that the Dirac belt construction is just a topological property of rotations in R3, not necessarily related to quantum physics
Symmetry 2020, 12, 1135 In Section 4, we review the topological model of entanglement recently introduced in Reference [3], and apply this model in Section 5 to give a geometric interpretation of the Kochen-Specker theorem, which states that even for commuting observables, the eigenvalues of a given quantum state depend on the context
Summary
With the rapidly increasing importance of quantum communication and quantum computation, and in particular the increasing ability to manipulate systems of entangled qubits, the representation of qubits and operations with qubits is an important issue, in particular for writing codes for quantum computers [1]. We want to show that the encoding of operations using the Bloch-sphere representation is insufficient for several reasons. We propose several generalizations of the Bloch sphere representation from a group theoretical and a knot theoretical perspective. We propose a minimal extension of the Bloch sphere representation to encode spin flip operations using a simple paper strip model in the (4π )-realm [2]. It turns out that the difference between 2π and 4π-rotations, which cannot be resolved in the usual Bloch-sphere representation, lies at the heart of contextuality in quantum physics. We show how the transition of an entangled state to a mixed state can naturally be modeled using the paper strip model of entanglement, leading naturally to an ensemble interpretation of amplitudes in quantum physics. In the summary and outlook, we discuss further possible applications of this topological approach to quantum physics, and their merits for physics education
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