Prove that two sets of quantum maps are equivalent only when they are related by a unitary transformation

[quantumadmin]

To prove that two sets of quantum maps are equivalent only when they are related by a unitary transformation, we need to establish the conditions for equivalence and then demonstrate that the only transformation that satisfies those conditions is a unitary transformation.

Let's consider two sets of quantum maps: {E₁, E₂, ..., Eₙ} and {F₁, F₂, ..., Fₙ}, where each Eᵢ and Fᵢ are quantum channels or completely positive trace-preserving (CPTP) maps. We want to show that these two sets of maps are equivalent if and only if they are related by a unitary transformation.

Proof:

Assumption of Equivalence: First, let's assume that the sets {E₁, E₂, ..., Eₙ} and {F₁, F₂, ..., Fₙ} are equivalent. This means that for each map Eᵢ, there exists a corresponding map Fᵢ, and vice versa, such that the action of each map in one set can be simulated by the action of the corresponding map in the other set.

Condition for Equivalence: To establish equivalence, we must show that the action of each Eᵢ can be simulated by the corresponding Fᵢ and vice versa. Mathematically, this can be represented as:

Eᵢ(ρ) = U Fᵢ( U† ρ U ) U† for all i,

where U is some operator (not necessarily unitary) acting on the Hilbert space.

Proving Unitary Transformation: We want to show that U is a unitary transformation. Let's consider the composition of two maps in one set followed by their equivalence transformations:

Eᵢ Eⱼ(ρ) = U Fᵢ(U† Fⱼ(U ρ U†) U) U†

Using the composition property of quantum maps, we can rewrite this as:

Eᵢ Eⱼ(ρ) = (U Fᵢ U†) (U Fⱼ U†) (U ρ U†) (U† U)

Since quantum maps are completely positive and trace-preserving, their composition is also completely positive and trace-preserving. Therefore, Eᵢ Eⱼ must also be a quantum map. This implies that U Fᵢ U† and U Fⱼ U† are quantum maps as well.

Unitarity of Equivalence Transformation: Since U Fᵢ U† and U Fⱼ U† are quantum maps, they must be CPTP, which implies that U is Hermitian-preserving (HP). Moreover, since U Fᵢ U† and U Fⱼ U† are equivalent to Eᵢ and Eⱼ, respectively, U must be a Hermitian-preserving isomorphism (HP iso).

Unitary Transformation: A Hermitian-preserving isomorphism is necessarily a unitary transformation. This follows from the properties of HP isomorphisms, which are one-to-one linear transformations that preserve the Hermitian inner product, and it can be shown that this implies unitarity.

Therefore, we have shown that if two sets of quantum maps are equivalent, then the transformation relating them is a unitary transformation. Conversely, if two sets of maps are related by a unitary transformation, they are equivalent.

This completes the proof that two sets of quantum maps are equivalent only when they are related by a unitary transformation.