# Topological Qubits as Carriers of Quantum Information in Optics
## 4 Multiports and Winding Number
\[\widehat{H}(k)=d_{0}(k)I+d(\vec{k})\cdot\vec{\sigma}\, \tag{18}\]  
describing the dynamics in a two dimensional \"internal\" subspace labeled by the two substates present at each lattice site. The two energy levels are then separated by a momentum-dependent gap. Without loss of generality, an appropriate redefinition of states allows the quantum state amplitudes to be made real:  
\[\langle m,a|m,b\rangle = |t_{a}| \tag{19}\] \[\langle m+1,a|m,b\rangle = \langle m,b|m+1,a\rangle = |t_{b}|. \tag{20}\]  
Transitions in which the photon \"reflects off\" the diamond graph, returning to the same subcell add a constant background term to the Hamiltonian and can be ignored for current purposes. The remaining terms are those that take a photon from one subcell to an adjacent subcell in a single time step of the corresponding quantum walk. The nonvanishing coefficients of the Pauli matrices in Equation (18) are now  
Figure 7:0 Figure 7:1  
As \(k\) goes from \(0\) to \(2\pi\), \(\vec{d}\) then traces out paths labeled by their winding numbers \(\
u\) about the origin. These winding numbers are functions of photon hopping amplitudes: \(\
u(v,w)=\
u(|t_{a}|,|t_{b}|)\). Because \(t_{a}\) and \(t_{b}\) vary only weakly with \(k\), the path traced out by \(\vec{d}(\vec{k})\) encloses the origin and, so, has _nonzero winding number_ if \(|t_{b}|>|t_{a}|\). Different values of the phase shifts allow solutions with both zero and nonzero winding numbers to occur: Evaluation of Equations (17), (21) and (22) for a range of \(\phi_{a}\) and \(\phi_{b}\) values readily shows that varying these phases causes the path traced out by \(\vec{d}\) to shift horizontally and change radius, leading to transitions between winding numbers \(0\) and \(1\) due to the lack of enclosure to enclosure of the origin in this space as shown in Figure 7:2 This system is topologically equivalent to the well-known Su-Schreiffer-Heeger (SSH) system [54] and reduces exactly to the pure SSH model when the transmission amplitudes are independent of \(k\). Figure 7:3 In the pure SSH model (transmission coefficients independent of \(k\)), the paths the Hamiltonian takes in momentum space are circular. Varying phases \(\phi_{a}\) and \(\phi_{b}\) causes the path traced out by \(\vec{d}\) to shift horizontally and change radius, leading to transitions between winding numbers \(0\) and \(1\). The right-most path here has winding number \(0\), while the others shown have winding number \(1\). If the transmission amplitudes are \(k\)-dependent, the curves are no longer circular, but similar transitions between \(\
u=0\) and \(\
u=1\) still occur as the phases are varied. Different phase values in the two diamond graphs therefore lead to different topological phases, distinguished by their distinct winding numbers. By attaching two chains of these graphs with different winding numbers on each chain there arise localized, topologically-protected states at their mutual boundaries [7; 8; 9; 18]. With appropriate choices of phase shifts \(\phi_{a}\) and \(\phi_{b}\), a persistent probability of finding the photon at the boundary between the two path topologies appearing in the complex system described in Section 4 can be achieved; a localized quantum optical state protected from noise influences by winding-number [8; 9; 18] arises when two chains of directionally unbiased three-ports, the two chains having different winding numbers [18; 20]. In addition, it has been shown [55; 56] that at the boundaries between regions of different winding number there is strong reflection of the wave function. By appropriate choice of parameters transmission between regions can be made arbitrarily small; thus winding-number bit flip errors can be made negligible. Furthermore, by entangling the winding number with a more easily measured variable, such as polarization, the value of the winding number bit can be read in a straightforward manner by measurement of the variable entangled to it [20].