# What is the best way to extend a state $\rho_S$ to a tensor product of spaces ${\cal H}_S\otimes{\cal H}_A?$

Let $$\Phi_S$$ be an operator acting on a space $$\mathcal H_S$$. If we introduce an ancilla $$A$$, the total space becomes $$\mathcal H_S\otimes \mathcal H_A$$ and I can naturally extend the operator $$\Phi_S$$ to act on the whole space by defining $$\Phi_{SA}=\Phi_S\otimes \mathbb{I}_A$$, where $$\mathbb{I}_A$$ is the identity on $$A$$.

What is the correct way of extending a state or more generally a density operator $$\rho_S$$ to the space $$\mathcal H_S\otimes \mathcal H_A$$? Since $$\rho_S$$ is also an operator, I would write again $$\rho_{SA}=\rho_S\otimes \mathbb I_A$$. Is this correct?

The way that you propose does not preserve the trace of the state -- if you compute the trace you'll find it equal to the dimension of $$\mathcal{A}$$.

There are many ways to extend your state $$\rho_S$$ to the joint system such that the marginal state on $$S$$ stays the same. For example, if the systems $$S$$ and $$A$$ are assumed to be independent then you can pick any state $$\rho_A$$ on system $$A$$ and define $$\rho_{SA} = \rho_S \otimes \rho_A.$$

Another possibility would be to take $$A$$ to be a purifying system of $$S$$ and define $$\rho_{SA}$$ to be the purification of $$\rho_S$$ (assuming the dimension of the system $$A$$ is large enough.

• Notably, in the spirit of the question being posed, one can choose $\rho_A=\mathbb{I}/\mathrm{Tr}(\mathbb{I})$. Commented Jul 14, 2021 at 19:36
• @QuantumMechanic This is not at all in the spirit of the question. Commented Jul 23, 2021 at 17:04
• @NorbertSchuch you don't think? The question asked about using $\rho_{SA}=\rho_S\otimes\mathbb{I}$... ahh, your answer is much more in the spirit of the question, this is explicitly just the letter Commented Jul 23, 2021 at 19:20

This is not a meaningful question -- in the sense that there is no "well-defined" way to extend a state to a larger system, without further assumptions.

In analogy, it is like asking what is written on the second page of a book after having read the first page. (Maybe more accurately, what's written in the second chapter of a collection of essays after having read the first.)

On the contrary, extending operations is well-defined: Just as it is clear how to extend the operation "strike out the first page" to a whole book: You still just strike out the first page, and leave the rest of the book untouched.

• This analogy is quite useful Commented Jul 23, 2021 at 19:23

There is no more natural way of doing the extension. Yet, two are most often used:

1. You do not want to make hypotheses about the state of A besides its dimension so that the state of SA will be $$\rho_S \otimes I / N$$ where $$I$$ is the identity matrix, and $$N$$ is the dimension of A.

2. You want to purify your system $$S$$ so that the global state of SA is now pure. This is useful when you want to argue that the state of the system --that you are about to purify-- could be viewed as a resulting from a unitary transformation acting on SA. In such case, a purification can be obtained by diagonalizing $$\rho_S = \sum_i \lambda_i | e_i\rangle\langle e_i|$$ and writing $$\rho_{SA} = \sum_i \sqrt{\lambda_i} |e_i\rangle\otimes|f_i\rangle$$ where {$$|f_i\rangle$$} is any orthonormal set of state vectors on $$A$$.