# Can we restore a state of a whole system from its subsystem?

I am thinking about the deletion error correcting codes for quantum information.

In classical information theory, there exist some deletion error correcting codes. An easy example is the following situation:

1. Alice prepares a set $$\{ 00,11 \}$$ and sends an element of the set to Bob. Here Bob knows the set.
2. One deletion error occurs at the message so $$00$$ becomes $$0$$ and $$11$$ becomes $$1$$.
3. Then Bob receives the message which is $$0$$ or $$1$$.
4. Bob can correctly get the message Alice sent because he can restore $$0$$ to $$00$$ and $$1$$ to $$11$$.

If Alice prepared a set $$\{ 01,11 \}$$, Bob could not correctly get the messsage when he received $$1$$. That is why the set $$\{ 00,11 \}$$ is called a deletion error correcting code.

I would like to construct the quantum version of deletion error correcting codes by replacing the deletion error in the classical theory to the partial trace in the quantum information theory. The situation I expect is as follows:

1. Alice prepares a quantum state $$\left | \psi \right >\in \mathbb{C}^{2 \otimes n}$$. This state is represented as a density matrix $$\rho=\left | \psi \right >\left < \psi \right |$$.
2. One deletion error occurs at $$\rho$$ so that the state becomes $$\rho'=Tr_i(\rho)$$.
3. Then Bob can perform some unitary operations and some measurements to obtain $$\rho$$. I wonder this is possible or not.

If Alice prepares a separable state such as $$\left | \psi \right >=\left | 0\right >\otimes \left | 0\right >\in \mathbb{C}^{2 \otimes 2}$$, it can obviously be restored by preparing $$\left | 0\right >$$. This is not interesting. So I would like to construct some non-trivial solutions, which means Alice prepares an entangled state. I tried to find a specific example, but I could not find it at all.

Would someone give me an example to satisfy the situation above? Perhaps, such states do not exist. If so, I would like to know the reason.

• It is a well known result that a quantum error correction code (QECC) which can correct for $n$ errors can also correct for $2n$ erasure errors. -- But a key point is that you have to be able to encode any 1-qubit state, so your "separable state" example above does not make sense. – Norbert Schuch Jun 23 at 7:54
• Is this of any help: arxiv.org/abs/quant-ph/9610042 – Norbert Schuch Jun 23 at 7:56
• I appreciate your answer, but the deletion error is completely different from the erasure error. In the classical coding theory, an erasure error is like $012$ to $0 \epsilon 2$ where $\epsilon$ is the erasure symbol. However, a deletion error is like $012$ to $02$. – Jack Jun 23 at 8:14
• Well, basically this still can be expressed in terms of the standard Quantum Error Correction Conditions $\langle i|E_a^\dagger E_b|j\rangle=\delta_{ij}c_{ab}$, and one can look for solution. Would be surprised if no one has looked into that. – Norbert Schuch Jun 24 at 10:23

An example for a quantum erasure code is given in Grassl, Beth, Pellizzari: Codes for the Quantum Erasure Channel. They show that $$4$$ qubits are necessary for this task, and devise a $$4$$-qubit code which can correct for erasure of one qubit. This should be contrasted with quantum error correction codes for general errors, for which at least 5 qubits are needed. Remarkably, they show that $$4$$ qubits are also sufficient to protect $$2$$ logical qubits against erasure of $$1$$ qubit.