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An MPS tensor $A^{[s]}$ is called injective if the map $$ X\mapsto\sum_s\mathrm{tr}[XA^{[s]}]\,\lvert s\rangle $$ is injective.¹

In particular, the example you give is not an injective tensor -- e.g., if you set $X=\begin{pmatrix}1 &0\\0&1\end{pmatrix}$, it is mapped to $0$. (In fact, you can already see this from the fact that $s$ can take only two values -- for the map to be injective it would have to take at least $4$ values.)

¹ This map is sometimes denoted as $P(A)$ -- note that unlike what you say in your question, $P(A)$ is in general not a projector, even though in some important examples it is, and the "Projected" in "Projected Entangled Pair States", i.e., PEPS, refers to that.

An MPS tensor $A^{[s]}$ is called injective if the map $$ X\mapsto\sum_s\mathrm{tr}[XA^{[s]}]\,\lvert s\rangle $$ is injective.¹

In particular, the example you give is not an injective tensor -- e.g., if you set $X=\begin{pmatrix}0 &1\\1&0\end{pmatrix}$, it is mapped to $0$. (In fact, you can already see this from the fact that $s$ can take only two values -- for the map to be injective it would have to take at least $4$ values.)

¹ This map is sometimes denoted as $P(A)$ -- note that unlike what you say in your question, $P(A)$ is in general not a projector, even though in some important examples it is, and the "Projected" in "Projected Entangled Pair States", i.e., PEPS, refers to that.

An MPS tensor $A^{[s]}$ is called injective if the map $$ X\to\sum_s\mathrm{tr}[XA^{[s]}]\,\lvert s\rangle $$ is injective.¹

In particular, the example you give is not an injective tensor -- e.g., if you set $X=\begin{pmatrix}1 &0\\0&1\end{pmatrix}$, it is mapped to $0$. (In fact, you can already see this from the fact that $s$ can take only two values -- for the map to be injective it would have to take at least $4$ values.)

¹ This map is sometimes denoted as $P(A)$ -- note that unlike what you say in your question, $P(A)$ is in general not a projector, even though in some important examples it is, and the "Projected" in "Projected Entangled Pair States", i.e., PEPS, refers to that.

An MPS tensor $A^{[s]}$ is called injective if the map $$ X\mapsto\sum_s\mathrm{tr}[XA^{[s]}]\,\lvert s\rangle $$ is injective.¹

In particular, the example you give is not an injective tensor -- e.g., if you set $X=\begin{pmatrix}1 &0\\0&1\end{pmatrix}$, it is mapped to $0$. (In fact, you can already see this from the fact that $s$ can take only two values -- for the map to be injective it would have to take at least $4$ values.)

¹ This map is sometimes denoted as $P(A)$ -- note that unlike what you say in your question, $P(A)$ is in general not a projector, even though in some important examples it is, and the "Projected" in "Projected Entangled Pair States", i.e., PEPS, refers to that.

An MPS tensor $A^{[s]}$ is called injective if the map $$ X\to\sum_s\mathrm{tr}[XA^{[s]}]\,\lvert s\rangle $$ is injective.¹

In particular, the example you give is not an injective tensor -- e.g., if you set $X=\begin{pmatrix}1 &0\\0&1\end{pmatrix}$, it is mapped to $0$. (In fact, you can already see this from the fact that $s$ can take only two values -- for the map to be injective it would have to take at least $4$ values.)

¹ This map is sometimes denoted as $P(A)$ -- note that unlike what you say in your question, $P(A)$ is in general not a projector, even though in some important examples it is, and the "Projected" in "Projected Entangled Pair States", i.e., PEPS, refers to that.)

An MPS tensor $A^{[s]}$ is called injective if the map $$ X\to\sum_s\mathrm{tr}[XA^{[s]}]\,\lvert s\rangle $$ is injective.¹

In particular, the example you give is not an injective tensor -- e.g., if you set $X=\begin{pmatrix}1 &0\\0&1\end{pmatrix}$, it is mapped to $0$. (In fact, you can already see this from the fact that $s$ can take only two values -- for the map to be injective it would have to take at least $4$ values.)

¹ This map is sometimes denoted as $P(A)$ -- note that unlike what you say in your question, $P(A)$ is in general not a projector, even though in some important examples it is, and the "Projected" in "Projected Entangled Pair States", i.e., PEPS, refers to that.