<p>M. G. Parker and V. Rijmen have studied the quantum entanglement of binary and bipolar sequences in <a href="http://arxiv.org/abs/quant-ph/0107106">arXiv:quant-ph/0107106</a>. 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.  </p>

<p>Notice also that <a href="http://www.quantiki.org/wiki/Graph_states">graph states</a> 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).</p>

CC BY-SA 3.0
2012-06-08T15:11:41.920Z
The class of states corresponds exactly to hypergraph states, where hyper-edges correspond to controlled-controlled-...-controlled-Z gates.
Joe Fitzsimons
CC BY-SA 3.0
2012-06-08T15:24:00.607Z
@JoeFitzsimons: true. I am curious to know if anyone has ever used the name _hypergraph states_.
Alessandro Cosentino
CC BY-SA 3.0
2012-06-08T15:31:02.280Z
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.
Joe Fitzsimons
CC BY-SA 3.0
2012-06-11T15:59:26.053Z
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.
Juan Miguel Arrazola
CC BY-SA 3.0
2012-06-08T15:02:57.117Z
2012-06-08T15:02:57.117Z
Alessandro Cosentino
<p>Recently, I have been studying a quantum protocol for the "Hidden Matching" problem that makes use of states that can be expressed as</p>

<p>$|\psi\rangle=\frac{1}{\sqrt{n}}\sum_{i=1}^n (-1)^{x_i}|i\rangle$,</p>

<p>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?</p>

<p>I am particularly interested in how the reduced density matrix of each qubit and the entanglement of the state depend on the bit values $x_i$.</p>

<p>Thank you!</p>

CC BY-SA 3.0
2012-06-08T03:10:31.940Z
Are you interested in multi-partite ent. or bipartite ent.? In the second case, where would the cut be? Also, if [this](http://cstheory.stackexchange.com/users/7101/juan-miguel-arrazola) is also you, you can ask the moderators to merge the two accounts.
Alessandro Cosentino
CC BY-SA 3.0
2012-06-08T10:16:19.687Z
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
CC BY-SA 3.0
2012-06-15T16:28:04.433Z
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
<reference-request><quantum-computing><quantum-information>
Known properties of a specific class of quantum states
Juan Miguel Arrazola
# Known properties of a specific class of quantum states

(Asked by: username_0 on 2012-06-08)

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!

**Tags:** reference-request, quantum-computing, quantum-information
> **Comments:**
> **username_1**: Are you interested in multi-partite ent. or bipartite ent.? In the second case, where would the cut be? Also, if [this](http://cstheory.stackexchange.com/users/7101/juan-miguel-arrazola) is also you, you can ask the moderators to merge the two accounts.  
> **username_2**: 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.  
> **username_0**: 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  

---

## 1 answers

### Answer by username_1 on 2012-06-08. Score: 6

M. G. Parker and V. Rijmen have studied the quantum entanglement of binary and bipolar sequences in [arXiv:quant-ph/0107106](http://arxiv.org/abs/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](http://www.quantiki.org/wiki/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).

> **Comments:**
> **username_2**: The class of states corresponds exactly to hypergraph states, where hyper-edges correspond to controlled-controlled-...-controlled-Z gates.  
> **username_1**: @JoeFitzsimons: true. I am curious to know if anyone has ever used the name _hypergraph states_.  
> **username_2**: 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.  
> **username_0**: 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.  

---

