I am struggling to understand what is an injective matrix product state (MPS). From the definition, it is said that an injective MPS $|M(A)\rangle$is one where the tensor $A$ has a projector $P(A)$ with a left inverse. I try to think of the example with the GHZ state with

 $$A^{[0]} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, A^{[1]} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$$

but it's difficult to visualize the left inverse since it's a 4-rank tensor. Is there a procedure to break it down and visualize it, as to understand why it is called "injective" and verify if one is? Thank you!

I am struggling to understand what is an injective matrix product state (MPS). From the definition, it is said that an injective MPS $|M(A)\rangle$is one where the tensor $A$ has a projector $P(A)$ with a left inverse. I try to think of the example with the GHZ state with  $$A^{[0]} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, A^{[1]} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$$ but it's difficult to visualize the left inverse since it's a 4-rank tensor. Is there a procedure to break it down and visualize it, as to understand why it is called "injective" and verify if one is?

I am struggling to understand what is an injective Matrix product state (MPS). From the definition, it is said that an injective MPS $|M(A)\rangle$is one where the tensor $A$ has a projector $P(A)$ with a left inverse. I try to think of the example with the GHZ state with

$$A^{[0]} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, A^{[1]} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$$

but it's difficult to visualize the left inverse since it's a 4-rank tensor. Is there a procedure to break it down and visualize it, as to understand why it is called "injective" and verify if one is? Thank you!

I am struggling to understand what is an injective matrix product state (MPS). From the definition, it is said that an injective MPS $|M(A)\rangle$is one where the tensor $A$ has a projector $P(A)$ with a left inverse. I try to think of the example with the GHZ state with

$$A^{[0]} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, A^{[1]} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$$

but it's difficult to visualize the left inverse since it's a 4-rank tensor. Is there a procedure to break it down and visualize it, as to understand why it is called "injective" and verify if one is? Thank you!

I am struggling to understand what is an injective MPS. From the definition, it is said that an injective MPS $|M(A)\rangle$is one where the tensor $A$ has a projector $P(A)$ with a left inverse. I try to think of the example with the GHZ state with

$$A^{[0]} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, A^{[1]} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$$

but it's difficult to visualize the left inverse since it's a 4-rank tensor. Is there a procedure to break it down and visualize it, as to understand why it is called "injective" and verify if one is? Thank you!

I am struggling to understand what is an injective Matrix product state (MPS). From the definition, it is said that an injective MPS $|M(A)\rangle$is one where the tensor $A$ has a projector $P(A)$ with a left inverse. I try to think of the example with the GHZ state with

$$A^{[0]} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, A^{[1]} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$$

but it's difficult to visualize the left inverse since it's a 4-rank tensor. Is there a procedure to break it down and visualize it, as to understand why it is called "injective" and verify if one is? Thank you!