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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!

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    $\begingroup$ 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. $\endgroup$
    – Alessandro Cosentino
    Commented Jun 8, 2012 at 3:10
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    $\begingroup$ 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. $\endgroup$
    – Joe Fitzsimons
    Commented Jun 8, 2012 at 10:16
  • $\begingroup$ 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 $\endgroup$
    – Juan Miguel Arrazola
    Commented Jun 15, 2012 at 16:28

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M. G. Parker and V. Rijmen have studied the quantum entanglement of binary and bipolar sequences in arXiv:quant-ph/0107106. I remember reading their paper long ago. They use a lot of terminology from coding theory, so now by just skimming through, I can't understand exactly what their result is, but you can have a deeper look.

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).

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    $\begingroup$ The class of states corresponds exactly to hypergraph states, where hyper-edges correspond to controlled-controlled-...-controlled-Z gates. $\endgroup$
    – Joe Fitzsimons
    Commented Jun 8, 2012 at 15:11
  • $\begingroup$ @JoeFitzsimons: true. I am curious to know if anyone has ever used the name hypergraph states. $\endgroup$
    – Alessandro Cosentino
    Commented Jun 8, 2012 at 15:24
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    $\begingroup$ I'm not sure if it appears in the literature or not, but certainly I have used it and heard it used in private correspondence. $\endgroup$
    – Joe Fitzsimons
    Commented Jun 8, 2012 at 15:31
  • $\begingroup$ Alesssandro, thank you for your comment, this is precisely the type of connection I suspected would exist.I will definitely look at this more carefully. Also, a scholar google search of "hypergraph states" yields only 3 results, so I do not think it is a standard term in the literature. $\endgroup$
    – Juan Miguel Arrazola
    Commented Jun 11, 2012 at 15:59

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