Vue not updating on property change - vue.js

Question

I've got an index component that contains this, in part:
<template>
<ul class="nav navbar-nav navbar-right">
<li v-if="state.user"><a #click.stop="do_logout" v-link="{ path: '/'}">Logout {{ state.user.display_name }}</a></li>
<li v-else v-link-active><a v-link="{ path: '/login' }">Login</a></li>
</ul>
<router-view></router-view>
<template>
<script>
import state from '../state';
import { logout } from '../actions';
export default {
data() {
return {
state
};
},
methods: {
do_logout() {
logout().then(() => this.$router.go('/'));
}
}
}
</script>
And I've got a login component that looks like this:
<template>
<form class="form-signin" #submit.prevent="do_login">
<input type="email" v-model="email" id="inputEmail" class="form-control" placeholder="Email address" required autofocus>
<input type="password" v-model="password" id="inputPassword" class="form-control" placeholder="Password" required>
<button class="btn btn-lg btn-primary btn-block" type="submit">Sign In</button>
</form>
</template>
<script>
import { login } from '../../actions';
export default {
data() {
return {
email: '',
password: ''
};
},
methods: {
do_login() {
login(this.email, this.password).then(() => this.$router.go('/'));
}
}
}
</script>
My login and logout methods look like this:
import state from './state';
export const login = (email, pwd) => {
return fetch_post_json('/api/login', {
email,
pwd
}).then(j => state.user = j.user);
};
export const logout = () => _fetch('/api/logout', 'GET', null, false).then(() => state.user = null);
Logging out works fine; clicking the link changes the text and presents the "Login" link. Logging in succeeds in logging in, in the Javascript, but doesn't update the link. I'm forced to reload the page to get the page to render correctly. Why?