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 $\backslash Lambda$ utilizing product state encoding is given by the formula, :$\backslash chi\; =\; \backslash max\_\{\; \backslash \{\; p\_i,\; \backslash rho^i\; \backslash \}\; \}\; S\backslash Big(\; \backslash sum\_i\; p\_i\; \backslash Lambda(\backslash rho^i)\; \backslash Big)\; -\; \backslash sum\_i\; p\_i\; S\backslash big(\backslash Lambda(\backslash rho^i)\backslash big)$ for $S(\backslash omega)$ the von Neumann entropy of the state $\backslash omega$. Product state encoding is a form of block coding that takes $n$ copies of a channel $\backslash Lambda^\{\backslash otimes\; n\}$ and encodes the input message into a product state codeword $m\; \backslash rightarrow\; \backslash rho^\{i\_1\}\; \backslash otimes\; \backslash rho^\{i\_2\}\; \backslash otimes\; ...\; \backslash otimes\; \backslash rho^\{i\_n\}$. The output from the block encoding is thus also a product state $\backslash Lambda(\backslash rho^\{i\_1\})\; \backslash otimes\; \backslash Lambda(\backslash rho^\{i\_2\})\; \backslash otimes\; ...\; \backslash otimes\; \backslash Lambda(\backslash rho^\{i\_n\})$. However, there is nothing stopping the sender encoding the message into states that are ''entangled across channels'' such that the input $\backslash rho^i\_\{(n)\}$ and possibly the output $\backslash big(\; \backslash Lambda\; \backslash otimes\; \backslash Lambda\; \backslash otimes\; ...\; \backslash otimes\; \backslash Lambda)\; (\backslash rho^i\_\{(n)\})$ 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, :$\backslash chi\_n\; =\; \backslash max\_\{\; \backslash \{\; p^\{(n)\}\_i,\; \backslash rho\_\{(n)\}^i\; \backslash \}\; \}\; \backslash frac\{1\}\{n\}\; \backslash Big[\; S\backslash Big(\; \backslash sum\_i\; p^\{(n)\}\_i\; \backslash Lambda^\{\backslash otimes\; n\}(\backslash rho\_\{(n)\}^i)\; \backslash Big)\; -\; \backslash sum\_i\; p^\{(n)\}\_i\; S\backslash big(\backslash Lambda^\{\backslash otimes\; n\}(\backslash rho\_\{(n)\}^i)\backslash big)\; \backslash Big]$ where the ensemble of states $\backslash \{\; p^\{(n)\}\_i,\; \backslash rho\_\{(n)\}^i\; \backslash \}$ may be entangled across blocks of $n$ channels. Asymptotically, this leads to the expression for the classical capacity in regularized form, :$C(\backslash Lambda)\; =\; \backslash lim\_\{n\; \backslash rightarrow\; \backslash infty\}\; \backslash max\_\{\; \backslash \{\; p^\{(n)\}\_i,\; \backslash rho\_\{(n)\}^i\; \backslash \}\; \}\; \backslash frac\{1\}\{n\}\; \backslash Big[\; S\backslash Big(\; \backslash sum\_i\; p^\{(n)\}\_i\; \backslash Lambda^\{\backslash otimes\; n\}(\backslash rho\_\{(n)\}^i)\; \backslash Big)\; -\; \backslash sum\_i\; p^\{(n)\}\_i\; S\backslash big(\backslash Lambda^\{\backslash otimes\; n\}(\backslash rho\_\{(n)\}^i)\backslash big)\; \backslash Big]$ It is straightforward to see that $C(\backslash Lambda)\; \backslash geq\; \backslash chi$. The ''additivity'' problem is then to prove (or disprove) the conjecture that $C\backslash big(\backslash Lambda\backslash big)\; =\; \backslash chi$, giving a "single-letter" formula for the classical capacity of a memoryless quantum channel. [[Category:Quantum Communication]]