Recently, I have been studying a quantum protocol for the Hidden Matching problem that makes use of states that can be expressed as$|\psi\rangle=\frac{1}{\sqrt{n}}\sum_{i=1}^n (-1)^{x_i}|i\rangle$,where $x_i\in \{0,1\}$ and $n$ is a power of 2. Note that instead of an $n$-level system we can consider these as states of $\log_2 n$ qubits. I suspect these states to have appeared in other contexts as they are a straightforward and useful way to map an $n$ bit string into an exponentially smaller amount of qubits. Hence I am wondering: have the properties of these types of states been already extensively studied?I am particularly interested in how the reduced density matrix of each qubit and the entanglement of the state depend on the bit values $x_i$.Thank you!
Known properties of a specific class of quantum states
reference request;quantum computing;quantum information