To prove that a quantum channel NA→B is entanglement-breaking if (idR ⊗NA→B ) (ΦRA) is a separable state, where ΦRA is a maximally entangled state, we can follow these steps:

1. Definition: A quantum channel NA→B is called entanglement-breaking if for any system R and any state ρRA, the state (idR ⊗NA→B ) (ρRA) is separable.

2. Given: We are given that (idR ⊗NA→B ) (ΦRA) is a separable state, where ΦRA is a maximally entangled state.

3. Maximally Entangled State: A maximally entangled state is a state that is maximally entangled if it is as entangled as possible. This means that it cannot be transformed into another state by local operations and classical communication (LOCC).

4. Separable State: A state is separable if it can be written as a convex combination of product states.

5. Proof: To prove that NA→B is entanglement-breaking, we need to show that for any system R and any state ρRA, the state (idR ⊗NA→B ) (ρRA) is separable.

6. Since we are given that (idR ⊗NA→B ) (ΦRA) is a separable state, where ΦRA is a maximally entangled state, this means that the quantum channel NA→B has the property that it transforms a maximally entangled state into a separable state.

7. Therefore, by definition, the quantum channel NA→B is entanglement-breaking.

This is a high-level proof and the actual mathematical proof would involve more detailed calculations and manipulations of the quantum states and channels.

Question

To prove that a quantum channel NA→B is entanglement-breaking if (idR ⊗NA→B ) (ΦRA) is a separable state, where ΦRA is a maximally entangled state, we can follow these steps:

1. Definition: A quantum channel NA→B is called entanglement-breaking if for any system R and any state ρRA, the state (idR ⊗NA→B ) (ρRA) is separable.

2. Given: We are given that (idR ⊗NA→B ) (ΦRA) is a separable state, where ΦRA is a maximally entangled state.

3. Maximally Entangled State: A maximally entangled state is a state that is maximally entangled if it is as entangled as possible. This means that it cannot be transformed into another state by local operations and classical communication (LOCC).

4. Separable State: A state is separable if it can be written as a convex combination of product states.

5. Proof: To prove that NA→B is entanglement-breaking, we need to show that for any system R and any state ρRA, the state (idR ⊗NA→B ) (ρRA) is separable.

6. Since we are given that (idR ⊗NA→B ) (ΦRA) is a separable state, where ΦRA is a maximally entangled state, this means that the quantum channel NA→B has the property that it transforms a maximally entangled state into a separable state.

7. Therefore, by definition, the quantum channel NA→B is entanglement-breaking.

This is a high-level proof and the actual mathematical proof would involve more detailed calculations and manipulations of the quantum states and channels.

Knowee AI · Accepted Answer