# DickeState

DickeState | |

Generates a Dicke state | |

Other toolboxes required | none |
---|---|

Related functions | GHZState MaxEntangled WState |

Function category | Special states, vectors, and operators |

` DickeState` is a function that returns a Dicke state on a given number of qubits. For example, the usual $3$-qubit Dicke state is:
\[\frac{1}{\sqrt{3}}(|001\rangle + |010\rangle + |100\rangle).\]
More generally, the Dicke state on $N$ qubits with $k$ excitations is
\[\frac{1}{\sqrt{\binom{N}{k}}}\sum_{j} P_j \big(|0\rangle^{\otimes (N-k)} \otimes|1\rangle^{\otimes k} \big),\]
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

`DICKE_STATE = DickeState(N)``DICKE_STATE = DickeState(N,K)``DICKE_STATE = DickeState(N,K,NRML)`

## Argument descriptions

`N`: The number of qubits.`K`(optional, default 1): The number of excitations (i.e., the number of "1" qubits in each term of the superposition; must be between`0`and`N`, inclusive.`NRML`(optional, default 1): A flag (either 1 or 0) indicating that`DICKE_STATE`should or should not be scaled to have Euclidean norm 1. If`NRML=0`then each entry of`DICKE_STATE`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):

```
>> full(DickeState(3))
ans =
0
0.5774
0.5774
0
0.5774
0
0
0
```

### A 5-qubit Dicke state

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

```
>> 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
```

### A very large Dicke state

This script has no trouble creating Dicke states on very large numbers of qubits. The following code generates a 45-qubit Dicke state:

```
>> tic; DickeState(45); toc
Elapsed time is 0.101930 seconds.
```

Note that Octave has a size limit of 2^31-1, so `DickeState(30)` is the limit of this function there. `DickeState(31)` and larger will produce an error in Octave (but not MATLAB).

## Source code

Click here to view this function's source code on github.