Nathaniel: /* A very large Dicke state */

2018-06-13T13:32:08Z

A very large Dicke state

Nathaniel at 18:00, 19 December 2014

2014-12-19T18:00:38Z

Nathaniel: Created page with "{{Function |name=DickeState |desc=Generates a Dicke state |rel=GHZState
MaxEntangled
WState |cat=List of functions#Special_states,_vectors,_and_operato..."

2014-12-19T16:45:46Z

Created page with "{{Function |name=DickeState |desc=Generates a Dicke state |rel=GHZState<br />MaxEntangled<br />WState |cat=List of functions#Special_states,_vectors,_and_operato..."

New page

{{Function
|name=DickeState
|desc=Generates a Dicke state
|rel=[[GHZState]]<br />[[MaxEntangled]]<br />[[WState]]
|cat=[[List of functions#Special_states,_vectors,_and_operators|Special states, vectors, and operators]]
|upd=December 15, 2014}}
<tt>'''DickeState'''</tt> is a [[List of functions|function]] that returns a Dicke state on a given number of qubits. For example, the usual $3$-qubit Dicke state is:
: <math>\frac{1}{\sqrt{3}}(|001\rangle + |010\rangle + |100\rangle).</math>
More generally, the Dicke state on $N$ qubits with $k$ excitations is
: <math>\frac{1}{\sqrt{\binom{N}{k}}}\sum_{j} P_j \big(|0\rangle^{\otimes (N-k)} \otimes|1\rangle^{\otimes k} \big),</math>
where $P_j$ ranges over all operators that permute the $N$ qubits in the $\binom{N}{k}$ possible distinct ways. The output of this function is a sparse vector.

==Syntax==
* <tt>DICKE_STATE = DickeState(N)</tt>
* <tt>DICKE_STATE = DickeState(N,K)</tt>
* <tt>DICKE_STATE = DickeState(N,K,NRML)</tt>

==Argument descriptions==
* <tt>N</tt>: The number of qubits.
* <tt>K</tt> (optional, default 1): The number of excitations (i.e., the number of "1" qubits in each term of the superposition; must be between <tt>0</tt> and <tt>N</tt>, inclusive.
* <tt>NRML</tt> (optional, default 1): A flag (either 1 or 0) indicating that <tt>DICKE_STATE</tt> should or should not be scaled to have Euclidean norm 1. If <tt>NRML=0</tt> then each entry of <tt>DICKE_STATE</tt> is 0 or 1, so it has norm $\sqrt{\binom{N}{k}}$.

==Examples==
===3-qubit Dicke state===
The following code generates the 3-qubit Dicke state (with 1 excitation):
<syntaxhighlight>
>> full(DickeState(3))

ans =

0
0.5774
0.5774
0
0.5774
0
0
0
</syntaxhighlight>

===A 5-qubit Dicke state===
The following code generates the 5-qubit 2-excitation Dicke state
: <math>\frac{1}{\sqrt{10}}\big( |00011\rangle + |00101\rangle + |00110\rangle + |01001\rangle + |01010\rangle + |01100\rangle + |10001\rangle + |10010\rangle + |10100\rangle + |11000\rangle \big).</math>

<syntaxhighlight>
>> DickeState(5,2)

ans =

(4,1) 0.3162
(6,1) 0.3162
(7,1) 0.3162
(10,1) 0.3162
(11,1) 0.3162
(13,1) 0.3162
(18,1) 0.3162
(19,1) 0.3162
(21,1) 0.3162
(25,1) 0.3162
</syntaxhighlight>

==Notes==
This function is practical for up to about 30 qubits.

{{SourceCode|name=DickeState}}

@@ Line 66: / Line 66: @@
 Elapsed time is 0.101930 seconds.
 </syntaxhighlight>
 {{SourceCode|name=DickeState}}

@@ Line 60: / Line 60: @@
 </syntaxhighlight>
-==Notes==
+===A very large Dicke state===
-This function is practical for up to about 30 qubits.
+This script has no trouble creating Dicke states on very large numbers of qubits. The following code generates a 45-qubit Dicke state:
 {{SourceCode|name=DickeState}}

DickeState - Revision history

Nathaniel: /* A very large Dicke state */

Nathaniel at 18:00, 19 December 2014

Nathaniel: Created page with "{{Function |name=DickeState |desc=Generates a Dicke state |rel=GHZStateMaxEntangledWState |cat=List of functions#Special_states,_vectors,_and_operato..."

Nathaniel: Created page with "{{Function |name=DickeState |desc=Generates a Dicke state |rel=GHZState
MaxEntangled
WState |cat=List of functions#Special_states,_vectors,_and_operato..."