# A Knot Theoretic Extension of the Bloch Sphere Representation for Qubits in Hilbert Space and Its Application to Contextuality and Many-Worlds Theories

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## Abstract

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## 1. Introduction

## 2. Geometry of Rotations in Real Space ${\mathit{R}}^{\mathbf{3}}$

## 3. Group Theoretic View on a Qubit in a Magnetic Field

#### 3.1. On the Relation between SU(2) and SO(3)

#### 3.2. On the Relation between Operators and Amplitudes in the Group SU(2)

#### 3.3. Spin Flip Operations in a Generalized Bloch Sphere Representation

#### 3.4. Modeling Amplitudes with Closed Paper Strips

## 4. Paper Strip Model Model for a Pair of Entangled Qubits

## 5. Kochen-Specker Theorem

## 6. Many-Worlds Theories of Quantum Physics

## 7. Summary And Outlook

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. The Dirac Belt in S 3 for Spin j

## Appendix B. Stratification of S_{7}

**Figure A1.**The 15 non-trivial generators of the Lie algebra $su\left(4\right)$ can be decomposed in 6 local transformations $su\left(2\right)\times su\left(2\right)$ and 9 non-local transformations $su\left(4\right)/\left(su\right(2)\times su(2\left)\right)$. Note that the generators defined in Figure 11 used for the proof of the Kochen-Specker theorem are a subset of these generators.

## Appendix C. Paper Strip Model for W- and GHZ States

**Figure A2.**Homotopic loops emerging upon rotation around the z-axis in case of three entangled qubits. Taking the partial trace leads to a mixed state, which can directly be read off from the topology without calculation by summing over all possibilities to trace out one twist. Compare also with Figure 8 for the case of entangled Bell states.

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**Figure 1.**Left: The Lie algebra $so\left(3\right)$ of the real group of rotations in three dimensions $SO\left(3\right)$ can be parametrized by the $\pi $-ball $\vartheta \overrightarrow{e}$, where $\overrightarrow{e}$ is the rotation axis, and $\vartheta $ is the rotation angle around this axis. Right: Action of the group element $g(\overrightarrow{e},\vartheta )=exp\left[\vartheta \phantom{\rule{4pt}{0ex}}{e}_{k}{X}_{k}\right]$ on a given vector ${\overrightarrow{n}}_{0}$ in ${R}^{3}$. The angle between ${\overrightarrow{n}}_{0}$ and $\overrightarrow{e}$ is denoted as $\theta $.

**Figure 2.**Since a rotation with $\pm \pi $ in ${R}^{3}$ leads to the same final state, the points $\pm \pi \overrightarrow{e}$ in the Lie algebra of $SO\left(3\right)$ are identified. Only after a second traversal, the closed path is isomorphic to the Null homotopy in $SO\left(3\right)$. The Lie algebra of the covering group $SU\left(2\right)$ can be described by $\frac{\vartheta}{2}\overrightarrow{e}$, corresponding to $two$$\pi $-balls, where the boundaries are identified. This is the so-called Heegard-splitting on the level of the algebra $SU\left(2\right)$. The action of the quaternions $I,J,K$ correspond to rotations around the axis $x,y,z$ with angle $\pi $, respectively.

**Figure 3.**The equivalence of a $4\pi $-rotation with the Null homotopy in ${R}^{3}$ can be illustrated by a ball which is attached by (elastic) paper strips to an outer sphere with very large radius. Each rotation by $\pi $ induces a $twist$. Two twists cannot be deformed to the identity. In contrast, four twists are homotopically equivalent to the identity.

**Figure 4.**Qubits are isomorphic to the group $SU\left(2\right)$: A given qubit can be represented by the vector $\overrightarrow{n}$ on the Bloch sphere. The corresponding group elements $exp\left[\frac{\pm i\vartheta}{2}\overrightarrow{n}\right]$ generate rotations around the axis $\overrightarrow{n}$.

**Figure 5.**The operations $IJK={e}^{\left(2\pi i\right)/2}$ and $JKI=1$ in the Heegard splitting: The path between 0 and $2\pi $ is $not$ homotopically equivalent to the path between 0 and $4\pi $.

**Figure 6.**Minimal knot theoretic extension of the Bloch sphere representation of a single qubit: Time development, described by any unitary operation ${\mathbf{U}}_{\overrightarrow{e}}\left(\vartheta \left(t\right)\right)$, is tracked by the paper strip attached to the qubit. In such a way, the Null homotopy $JKI$ can be distinguished from the path $IJK=-1$ connecting 0 and $2\pi $ in the time development described by $SU\left(2\right)$.

**Figure 7.**In contrast to the minimal extension of the Bloch sphere representation, where only the unitary time development acting on $|\mathsf{\psi}\rangle $ is encoded in knots, we may also represent the quantum state itself in Hilbert space. The Dirac belts within ${S}_{3}$ are weighted by amplitudes $0\le \lambda \le 1$. The maximal amplitude $\lambda =1$ corresponds in the Bloch sphere to the ’direction of spin’ $\overrightarrow{n}$. For the state $|0\rangle $, we show a representative of the fibre of Dirac belts emerging on the equatorial line with $\lambda =1/\sqrt{2}$. The double traversal of the Dirac belt is equivalent to a Möbius strip [4].

**Figure 8.**Homotopic loops around the z-axis and the x-axis of the Bell state $|{\mathsf{\Psi}}^{+}\rangle $. For any maximally entangled pair of qubit, there exists at least one homotopic loop with constant phase.

**Figure 9.**Exchanging the role of the $\pi $-balls in the Heegard splitting leads to a global minus sign for $|{\mathsf{\Psi}}^{-}\rangle $. As a constant phase is homotopically equivalent to a combination of one R and one L twist, this phase change can also be seen as exchange of this pair of (virtual) particles.

**Figure 10.**The constant phase is homotopically equivalent to one right (R) and one left (L) twist. In the superposition $\frac{1}{\sqrt{2}}\left(\right|01\rangle +|10\rangle $, the particles loose the individual characteristic, as they together merge to a single quantum state with constant phase on the homotopic loop $({e}^{\pm i\frac{\vartheta}{2}{\sigma}_{z}}\times {e}^{\pm i\frac{\vartheta}{2}{\sigma}_{z}})$ with $0\le \vartheta \le 4\pi $.

**Figure 11.**Mermin’s square, expressed with spin-flip operations: The three operators in each row and in column mutually commute and can be measured simultaneously. The product of all nine operators is $-1$, compare also Figure 5 for the corresponding homotopies in the Lie algebra $SU\left(2\right)$.

**Figure 12.**A constant phase is homotopically equivalent to one R and one L twist. As long as the state is entangled, all homotopically equivalent configurations coexist, which can be expressed as $\frac{1}{\sqrt{2}}{\left(\right|0\rangle}_{A}{|1\rangle}_{B}+{|1\rangle}_{A}{|0\rangle}_{B}$. After splitting into two particles with phases R and L (corresponding to the states $|0\rangle ,|1\rangle $), there are two possibilities for detection of these states in detector A or B: either ${|0\rangle}_{A}{|1\rangle}_{B}$ or ${|0\rangle}_{B}{|1\rangle}_{A}$. However, once the labeling is done (with $p=1/2$ for each case), no ’remaining’ part in some parallel universe can emerge, because this would imply that one more pair of particles with phases R and L would have been created, which is not the case.

**Figure 13.**All pure qubit states can be described with the hypersphere ${S}_{7}$. Within this hypersphere, slices with fixed concurrence $c=2|{z}_{00}{z}_{11}-{z}_{01}{z}_{10}|$ define the degree of entanglement between two qubits. For $c=0$, the qubits $A,B$ are separate. For $c=1$, we obtain the maximally entangled Bell-states. For $c=1$, the labels $A,B$ loose their individual meaning.

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**MDPI and ACS Style**

Heusler, S.; Schlummer, P.; Ubben, M.S.
A Knot Theoretic Extension of the Bloch Sphere Representation for Qubits in Hilbert Space and Its Application to Contextuality and Many-Worlds Theories. *Symmetry* **2020**, *12*, 1135.
https://doi.org/10.3390/sym12071135

**AMA Style**

Heusler S, Schlummer P, Ubben MS.
A Knot Theoretic Extension of the Bloch Sphere Representation for Qubits in Hilbert Space and Its Application to Contextuality and Many-Worlds Theories. *Symmetry*. 2020; 12(7):1135.
https://doi.org/10.3390/sym12071135

**Chicago/Turabian Style**

Heusler, Stefan, Paul Schlummer, and Malte S. Ubben.
2020. "A Knot Theoretic Extension of the Bloch Sphere Representation for Qubits in Hilbert Space and Its Application to Contextuality and Many-Worlds Theories" *Symmetry* 12, no. 7: 1135.
https://doi.org/10.3390/sym12071135