# Coherent information

The **coherent information** is a quantity used in quantum communication theory. It is a property of a quantum state and a quantum channel Λ; intuitively, it attempts to describe how much of the quantum information in the state will remain after the state goes through the channel. In this sense, it is analogous to the classical mutual information of classical information theory. As the classical mutual information gives the capacity of a classical channel, the coherent information gives the capacity of a quantum channel. The coherent information is written *I*(*A*⟩*B*).

### Definition

The coherent information is defined as *I*(*A*⟩*B*) ≡ *S*(*B*) − *S*(*A**B*) where *S*(*B*) is the Von Neumann entropy of the output state *ρ**B* and *S*(*A**B*) is the entropy of the channel environment (equivalently, the joint entropy of the output state, and the purification *ρ**A*) of the input state. One may think of inputting part of a pure state through the channel. The part which remains at the input is *ρ**A*, and the channel produces *ρ**B* at the output.

### Properties

### Uses

The coherent information, when maximised over input states *ρ**A*, gives the quantum capacity of the quantum channel. This is known as the LSD Theorem after Lloyd, Shor and Devetak. The quantity is not additive, and one must go to large block size (i.e. many quantum uses of the channel).

The coherent information, gives also the one way distillable entanglement when maximised over local operations performed by Alice, and if maximised over LOCC operations, it gives the two-way distillable entanglement.

### History

The coherent information was introduced by Benjamin Schumacher and Michael A. Nielsen in a 1996 paper *Quantum data processing and error correction*, which appeared in Physical Review A.

### See Also

### References

- Nielsen, Michael A. and Isaac L. Chuang (2000).
*Quantum Computation and Quantum Information*, Cambridge University Press, ISBN 0-521-63503-9

- Nielsen, Michael A. and Benjamin Schumacher (1996). Quantum data processing and error correction.
*Physical Review A.*,**54**(4), 2629-2635.

Category:Quantum Information Theory Category:Quantum Communication Category:Handbook of Quantum Information