Coherent information ( $I_{c} (Λ, ρ)$ )

For a channel output

Q

and complementary environment output

E

, coherent information is

I_{c} (Λ, ρ) = S (Q) - S (E) = - H (R ∣ Q)

for a purification reference

R

.

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Mutual information and coherent information identity ( $I (R : B) = S (ρ) + I_{c} (Λ, ρ)$ )

For a purified input with reference

R

and channel output

B

,

I (R : B) = S (R) + S (B) - S (RB) = S (ρ) + I_{c} (Λ, ρ)

. The reference entropy is fixed through subsequent channels. Thus data-processing inequality for coherent information immediately implies data processing for quantum mutual information between that reference and successive outputs.

Coherent information upper bound by input entropy ( $I_{c} (Λ, ρ) = S (ρ) - I (R : E) \leq S (ρ)$ )

In a purified Stinespring dilation, coherent information satisfies

I_{c} = S (RE) - S (E) = S (ρ) - I (R : E)

. Nonnegativity of quantum mutual information implies

I_{c} \leq S (ρ)

. Equality means that the reference and environment are uncorrelated, so the environment learns nothing about the purified input reference.

Coherent information as an environment conditional entropy ( $I_{c} (Λ, ρ) = S (R ∣ E) = - S (R ∣ B)$ )

Purify the input on

R A

and dilate a channel by

A \to BE

. The joint state

RBE

is pure, so complementary entropies obey

S (B) = S (RE)

and

S (RB) = S (E)

. Therefore coherent information is

I_{c} = S (B) - S (RB) = S (R ∣ E) = - S (R ∣ B)

. The positive environment conditional entropy and negative receiving-system conditional entropy are two equivalent forms.

Data-processing inequality for coherent information ( $I (A ⟩ B) - I (A ⟩ B') = I (A : E ∣ B') \geq 0$ )

A quantum channel on the receiving system cannot increase coherent information:

I (A ⟩ B)_{ρ} \geq I (A ⟩ B^{'})_{σ}

. Dilate the channel to a linear isometry of Hilbert spaces

B \to B^{'} E

. The difference is

I (A : E ∣ B^{'})

of the dilated state, nonnegative by Strong subadditivity of Von Neumann entropy.

Anti-degradable quantum channel

A quantum channel is anti-degradable when its receiver output can be simulated from the complementary environment output. Data processing then forces every coherent information to be nonpositive, and no-cloning prevents perfect quantum transmission.

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Coherent information is a concept derived from quantum information theory, particularly in the context of quantum communication and quantum error correction. It describes a specific type of information that can be transmitted or processed coherently through a quantum channel, taking advantage of the unique properties of quantum mechanics, such as superposition and entanglement. In classical information theory, information is typically concerned with bits—units that can exist in one of two states (0 or 1).

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