# IsEntanglingGate

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 Other toolboxes required IsEntanglingGate Determines if a unitary is an entangling gate none IsProductOperatorOperatorSchmidtDecompositionOperatorSchmidtRank Entanglement and separability

IsEntanglingGate is a function that determines if a bipartite or multipartite unitary is an entangling gate or not (i.e., whether or not there is a product vector that is no longer a product vector after the unitary is applied). If it is an entangling gate, a product vector that is entangled by the gate can be provided, and if it is not an entangling gate then a decomposition that proves that it is not entangling can be provided.

## Syntax

• EG = IsEntanglingGate(U)
• EG = IsEntanglingGate(U,DIM)
• [EG,WIT] = IsEntanglingGate(U,DIM)

## Argument descriptions

### Input arguments

• U: An operator (typically a unitary operator) that acts on a bipartite or multipartite Hilbert space.
• DIM (optional, by default has two subsystems of equal dimension): A specification of the dimensions of the subsystems that U lives on. DIM can be provided in one of three ways:
• If DIM is a scalar, it is assumed that U lives on the tensor product of two spaces, the first of which has dimension DIM and the second of which has dimension length(U)/DIM.
• If $U \in M_{n_1} \otimes \cdots \otimes M_{n_p}$ then DIM should be a row vector containing the dimensions (i.e., DIM = [n_1, ..., n_p]).
• If the subsystems aren't square (i.e., $U \in M_{m_1, n_1} \otimes \cdots \otimes M_{m_p, n_p}$) then DIM should be a matrix with two rows. The first row of DIM should contain the row dimensions of the subsystems (i.e., the mi's) and its second row should contain the column dimensions (i.e., the ni's). In other words, you should set DIM = [m_1, ..., m_p; n_1, ..., n_p].

### Output arguments

• EG: Either 1 or 0, indicating that U is or is not an entangling gate.
• WIT (optional): If EG = 0 (i.e., U is not an entangling gate), then WIT is a struct with fields WIT.perm and WIT.dec such that PermutationOperator(DIM,WIT.perm)*Tensor(WIT.dec) equals U and thus verifies that U is indeed not an entangling gate. If EG = 1 (i.e., U is an entangling gate) then WIT is a (sparse) product pure state that is entangled by U. Furthermore, WIT will always have 2 or fewer non-zero entries in this case.

## Examples

### A bipartite example: the CNOT gate

The following code verifies that the CNOT gate is an entangling gate. The following code also requests a product pure state wit that is entangled by the CNOT gate. The product pure state returned in this case is $(|0\rangle + |1\rangle) \otimes |0\rangle/\sqrt{2}$, which maps to the Bell state $(|00\rangle + |11\rangle)/\sqrt{2}$:

>> U = [1 0 0 0;0 1 0 0;0 0 0 1;0 0 1 0];
>> [eg,wit] = IsEntanglingGate(U)

eg =

1

wit =

(1,1)       0.7071
(3,1)       0.7071

>> U*wit

ans =

0.7071
0
0
0.7071

## Notes

Despite the name and description of the function, it can be used on non-unitary (and even non-square) operators just fine. If U is not unitary, the function has the exact same interpretation: it determines whether or not there is a product vector that is mapped to a non-product vector.

## Source code

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