1 (5 centimeter) piece of fresh ginger, thinly sliced
Make the pomegranate and ginger syrup. Combine the pomegranate juice, sugar, lime juice, and ginger in a small pot and bring to a boil. Reduce the heat and simmer for five to eight minutes, until the liquid begins to thicken slightly, periodically pressing the ginger with the back of a spoon to help bring out its flavour.
Remove the pot from the heat and let cool for 30 minutes. Strain the syrup through a fine-mesh sieve into an airtight container. The syrup will keep in the refrigerator for up to two weeks.
Make the cocktail. Add the syrup, vodka and bitters to a cocktail shaker. Add the ice and shake for 10 to 15 seconds.
Strain the cocktail into a highball glass filled with fresh ice cubes. Top with soda water and garnish with pomegranate arils.