# Known properties of a specific class of quantum states

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!

• Are you interested in multi-partite ent. or bipartite ent.? In the second case, where would the cut be? Also, if this is also you, you can ask the moderators to merge the two accounts. – Alessandro Cosentino Jun 8 '12 at 3:10
• Such states appear fairly frequently in quantum information. For example, they occur in quantum fingerprinting (quant-ph/0102001). I'm sure there is a name for this particular class of states, but I can't remember what it is. – Joe Fitzsimons Jun 8 '12 at 10:16
• I am interested in multipartite entanglement. Also, you are right, that account is also me and I have to thank you for reminding me of this! How exactly can I contact moderators to have them merged? Thanks – Juan Miguel Arrazola Jun 15 '12 at 16:28

Notice also that graph states are a special case of your states. In particular, $\psi$ is a graph state if and only if the sequence $x$ is generated by a quadratic polynomial in GF(2).