# Additivity

The *additivity* of the classical capacity of any memoryless quantum channel remains an open problem in quantum information theory. The Holevo-Schumacher-Westmoreland (HSW) theorem states that the classical capacity of a memoryless quantum channel Λ utilizing product state encoding is given by the formula,

$$\chi = \max_{ \{ p_i, \rho^i \} } S\Big( \sum_i p_i \Lambda(\rho^i) \Big) - \sum_i p_i S\big(\Lambda(\rho^i)\big)$$

for *S*(*ω*) the von Neumann entropy of the state *ω*.

Product state encoding is a form of block coding that takes *n* copies of a channel Λ ⊗ *n* and encodes the input message into a product state codeword *m* → *ρ**i*1 ⊗ *ρ**i*2 ⊗ ... ⊗ *ρ**i**n*. The output from the block encoding is thus also a product state Λ(*ρ**i*1) ⊗ Λ(*ρ**i*2) ⊗ ... ⊗ Λ(*ρ**i**n*). However, there is nothing stopping the sender encoding the message into states that are *entangled across channels* such that the input *ρ*(*n*)*i* and possibly the output $\big( \Lambda \otimes \Lambda \otimes ... \otimes \Lambda) (\rho^i_{(n)})$ are entangled. The HSW theorem states that the capacity for input states that are product states for blocks of channels of size *n*, is thus,

$$\chi_n = \max_{ \{ p^{(n)}_i, \rho_{(n)}^i \} } \frac{1}{n} \Big[ S\Big( \sum_i p^{(n)}_i \Lambda^{\otimes n}(\rho_{(n)}^i) \Big) - \sum_i p^{(n)}_i S\big(\Lambda^{\otimes n}(\rho_{(n)}^i)\big) \Big]$$

where the ensemble of states {*p**i*(*n*), *ρ*(*n*)*i*} may be entangled across blocks of *n* channels.

Asymptotically, this leads to the expression for the classical capacity in regularized form,

$$C(\Lambda) = \lim_{n \rightarrow \infty} \max_{ \{ p^{(n)}_i, \rho_{(n)}^i \} } \frac{1}{n} \Big[ S\Big( \sum_i p^{(n)}_i \Lambda^{\otimes n}(\rho_{(n)}^i) \Big) - \sum_i p^{(n)}_i S\big(\Lambda^{\otimes n}(\rho_{(n)}^i)\big) \Big]$$

It is straightforward to see that *C*(Λ) ≥ *χ*. The *additivity* problem is then to prove (or disprove) the conjecture that $C\big(\Lambda\big) = \chi$, giving a "single-letter" formula for the classical capacity of a memoryless quantum channel.