IsEntanglingGate

IsEntanglingGate
Determines if a unitary is an entangling gate
Other toolboxes required	none
Related functions	IsProductOperator OperatorSchmidtDecomposition OperatorSchmidtRank
Function category	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 m_i's) and its second row should contain the column dimensions (i.e., the n_i'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.