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 \Lambda^{\otimes n} and encodes the input message into a product state codeword m \rightarrow \rho^{i_1} \otimes \rho^{i_2} \otimes ... \otimes \rho^{i_n}. The output from the block encoding is thus also a product state \Lambda(\rho^{i_1}) \otimes \Lambda(\rho^{i_2}) \otimes ... \otimes \Lambda(\rho^{i_n}). However, there is nothing stopping the sender encoding the message into states that are entangled across channels such that the input \rho^i_{(n)} 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^{(n)}_i, \rho_{(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(\Lambda) \geq \chi. 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.

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