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 → ρi1 ⊗ ρi2 ⊗ ... ⊗ ρin. The output from the block encoding is thus also a product state Λ(ρi1) ⊗ Λ(ρi2) ⊗ ... ⊗ Λ(ρin). 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 {pi(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.

Category:Quantum Communication

Last modified: 
Monday, October 26, 2015 - 17:56