Projective braid gate and operator metric

The operator norm satisfies the triangle inequality: for every unit vector $x$, $\mathrm{\parallel }(S+T)x\mathrm{\parallel }\le \mathrm{\parallel }Sx\mathrm{\parallel }+\mathrm{\parallel }Tx\mathrm{\parallel }\le \mathrm{\parallel }S{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}+\mathrm{\parallel }T{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$, so taking the maximum gives $\mathrm{\parallel }S+T{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le \mathrm{\parallel }S{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}+\mathrm{\parallel }T{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$. Scalar homogeneity follows from [F2] and the same maximum definition. Thus $\mathrm{\mid }\mathrm{\parallel }S{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}-\mathrm{\parallel }T{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\mathrm{\mid }\le \mathrm{\parallel }S-T{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$ by applying the triangle inequality twice.

The formula is independent of representatives. If $A^{\mathrm{\prime }}=uA$ and $B^{\mathrm{\prime }}=vB$ with $\mathrm{\mid }u\mathrm{\mid }=\mathrm{\mid }v\mathrm{\mid }=1$, then ${\inf }_{\mathrm{\mid }z\mathrm{\mid }=1}\mathrm{\parallel }{A}^{\mathrm{\prime }}-z{B}^{\mathrm{\prime }}\mathrm{\parallel }={\inf }_{\mathrm{\mid }z\mathrm{\mid }=1}\mathrm{\parallel }A-(u^{-1}zv)B\mathrm{\parallel }$; as $z$ runs over the unit circle, so does $u^{-1}zv$.

The map $z\mapsto \mathrm{\parallel }A-zB{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$ is continuous: step 1.1 bounds $\mathrm{\mid }\mathrm{\parallel }A-zB\mathrm{\parallel }-\mathrm{\parallel }A-wB\mathrm{\parallel }\mathrm{\mid }\le \mathrm{\parallel }(z-w)B\mathrm{\parallel }=\mathrm{\mid }z-w\mathrm{\mid }\mathrm{\parallel }B\mathrm{\parallel }$. By [F3] and [F4] it attains its minimum on the unit circle, so the infimum in the definition is a minimum. Nonnegativity and symmetry follow from the norm axioms and the substitution $z\mapsto \bar{z}$.

For classes $[A],[B],[C]$, choose minimizing phases $z,w$ for the pairs $(A,B)$ and $(B,C)$ by [F3]–[F4]. The triangle inequality gives $\mathrm{\parallel }A-zwC\mathrm{\parallel }\le \mathrm{\parallel }A-zB\mathrm{\parallel }+\mathrm{\parallel }zB-zwC\mathrm{\parallel }=d_P([A],[B])+d_P([B],[C])$, so taking the minimum over phases for $(A,C)$ proves the triangle inequality. If $d_P([A],[B])=0$, a minimizing phase satisfies $A=zB$, so $[A]=[B]$. Conversely equal classes have distance zero. Therefore $d_P$ is a metric on $PU(V)$.

By [F5], two words representing the same braid $b$ give the same unitary ${\rho }_c(b)$. Its projective class is therefore well defined, and step 3.1 supplies the metric on these classes.