# DickeState

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 Other toolboxes required DickeState Generates a Dicke state none GHZStateMaxEntangledWState 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).